We propose to create a center dedicated to the study of Statistical Mechanics and Complex Systems. The center will be located in Rome. The main objective of the center will be the study of the collective emergent properties of systems with a very large number of components which show a complex behavior. This problem is central in the study of many physical systems and its full mastering is instrumental to the possibility of developing many applications of physical methodology to many other fields (a very partial list of examples ranges from traffic to the immune system, from the Internet to memory and cognitive processes, from earthquakes to finance).

The main mission of the Center will be to promote the study of the following fields:

• Statistical mechanics of systems with very slow dynamics (typically structural glasses and spin glasses).
  
• The physics of strongly correlated quantum systems.
  
• Dynamical structures (the ones appearing for example in turbulence and in surface growth).
  

The Center will try to be a self-consistent, strong and visible reality, being at the same time very open to contributions from outside. Its main mission will be to become a reference point for the researchers working in these fields not only in Italy, but in all of Europe and hopefully worldwide, and to form and train young brilliant researchers. We believe this is a realistic, ambitious and appropriate goal, based on the available skills and on the support we plan to acquire and organize.

The three main research lines we have quoted are already present and strong in our Institution, but we believe that the Center we are proposing will give them a very large added value. The common scientific grounds that are behind them will be strengthened by exploiting potential synergies that are clearly very strong.

The Center will organize the activity of the local scientist, and the ones of students, postdoctoral fellows and visitors. We plan to constantly host a large number of postdoctoral fellows, and to systematically invite senior researchers from Italy and abroad for stays of medium and long duration. We especially hope to attract people during their sabbatical year. We see, to summarize, a multiple level structure: permanent researchers and professors, PhD students and postdoctoral fellows, senior visitors for short, average and long stays, dedicated programs including schools. We believe the structure we are proposing will be able to exploit and enhance synergies among these different levels, and to guarantee a large level of visibility to the INFM Center.

Let us detail a bit more on the structure of the programs we plan to organize: the Center will organize programs dedicated to a given subject in which the leading expert in the field will be invited to give a series of seminars. Discussions among experimentalists and theoreticians will be especially welcomed. Each program may will typically last around one month (but longer programs will be possible) and may involve 15-20 people from outside: the precise format will be adjusted after the first experiences. This type of activity has been very successful in institutions like Santa Barbara and The Newton Institute (Cambridge): we plan to build on these experiences, even if with a completely new and original experience (that we hope we are succeeding to describe here at least partially, and that will be determined by the scientific and logistic environment in which we move). The tentative number of programs will be of 2-3 per year. Inside programs (and also independently from them) we will organize workshops. A reasonable estimate is that we will be able to organize 4 workshop per year.

The Center will have a high visibility and its activity will have to be deeply rooted in the national and international research. At this end the director of the Center will be assisted by a Scientific Committee, which will support him in deciding the activity of the Center, selecting the programs and the program directors. This Committee will be tentatively composed by 12 physicists (some theoreticians and some experimentalist), plus the director: 6 of them will be Italian physicists not from La Sapienza and 6 will be foreign senior scientists. The board will meet periodically and it will be in touch with the director via e-mail.

We plan to dedicate many energies to training (see the section about training). Among other activities we plan to organize a yearly doctorate school, with 4 courses of 20 hours each, divided in two (intense) working weeks, one in June and one in September, on subjects that will change every year.

The Center will be located in the Dipartimento di Fisica of the Universita' "La Sapienza" (see the section about logistic for many more details): Rome University La Sapienza is committing to give to the Center strong support (see the statement of the Department Head at http://chimera.roma1.infn.it/CDE/LETTERE/Department_Head.html
and the statement of Dean at http://chimera.roma1.infn.it/CDE/LETTERE/dean.jpg )

Other resources will come from other INFM programs, from the University "La Sapienza", from MURST (Italian Ministry for research), from the EEC, from the European Science Foundation, and hopefully from more organizations.

Our aim is to work in close relation with all the major Centers that investigate problems in Complexity and Statistical Mechanics: we have collected some letters expressing support and precise collaboration commitments.

Slow Dynamics, Disorder and Complexity.

The group in Rome has a longtime experience on slow dynamics, disorder and complexity, both from numerical as well as analytical point of view. The first works in this field date to more than 20 years ago. Here we report only the most recent of our achievements on this field.

One of the most important results in these recent years in statistical mechanics has been the emergence of a new definition of the temperature in off-equilibrium system: the temperature depends on the time scale of the observation. Correspondingly the theory predicts that the usual fluctuation-dissipation relations are no more valid in the "aging" regime and there are substituted by new generalized fluctuation relations as function of this off-equilibrium temperature. These results are very interesting, because these new relations can be observed experimentally. We have been the first to study of these relations in many disordered systems, spin glasses, disordered ferromagnets and structural glasses. The theoretical basis of these new relations and the deep connections with equilibrium thermodynamics has been elucidated in details.

We have studied with care the behavior of three-dimensional spin glasses. We have been asking if the main features of the mean field solution are in agreement with the numerical simulations. We have investigated both the simplified Ising version and a more realistic Heisenberg one (taking care of the anisotropies), We have used large scale numerical methods and we have concentrated our attention on the equilibrium properties at finite temperature, on the study of the approach to equilibrium and on the exact computations of the ground state in samples containing thousands of spins. This last method is quite effective and it allows a very detailed comparison of the theory.

We have also tried to understand more about the glassy state, from the point of view of the inherent structure formalism (which in this case coincide with the replica approach, if one assumes one step Replica Symmetry Breaking). As usual we have used two tools: analytic computations and numerical simulations.

The first problem we studied is the behavior of the systems at low temperatures, in the glassy phase. A first principle computations of the glass transition temperature and of the configurational entropy in simple atomic models as been done using tools like the Hyper-netted Chain approximation. The results of the computations have been successfully compared to simulations.

In order to have a proper thermodynamic description of super-cooled liquid we must control the number of basins in the potential energy surface sampled by the liquid and the intra-basin free energy. The possibility of writing the free-energy of the liquid as a configurational entropy term (number of basins) and of an intra-basin term clarifies the connection between the two contributions to the free energy and the two-step relaxation which characterize the dynamics and allows a very precise comparison of the theory with the response of system in the aging regime.

An other crucial point is the study of the behavior of glasses near the mode coupling temperature (the so called dynamical transition). Our group has a wide experience with the mode coupling theory. In the past we have studied in great care, using molecular dynamics simulations, the dynamical properties of well defined models of liquids (both strong and fragile) in the early super-cooled states and we have carefully compared the correlation functions with the corresponding quantities predicted by the mode coupling theory.

We have shown the relation of the mode-coupling transition with the properties of appropriate equilibrium quantities. Moreover we have used general argument to predict the presence of large scale heterogeneities, and a related divergent correlation length, whose existence has been observed in numerical simulations and quite recently in experiments.

Recently we have started to analyze the connection between the slowing down of the dynamics in the early super-cooled states with topological properties of the potential energy landscape of the corresponding models. We have performed detailed studies based on the instantaneous normal mode approach, showing that the apparent divergence of the characteristic times at the so-called dynamical critical temperature is associated to the vanishing of the number of directions allowing free exploration of the configuration space (as opposed to hopping dynamics).

Granular Matter.

Granular matter is a very interesting fields in which many of the concepts that have been developed in the study of glasses and spin glasses may be used in this field.

On a general ground we have introduced a class of statistical models describing particles undergoing inelastic collisions. These models show two essential aspects: (i) The clustering instability, i.e. the breaking of the spatial homogeneity with the formation of clusters with a fractal structure. (ii) The velocity distribution is not described by the Maxwell-Boltzmann function. Additional research focused on the study of the response properties of lattice models that reproduce many features of granular materials: slow-relaxations during compaction, segregation, dilatancy properties, memory, aging, internal avalanching and coarsening.

A complementary aspect that we have investigated is the role of entropic concepts for the relaxation dynamics in granular systems. We have provided a numerical support for the the Edwards's hypothesis that macroscopic observables of dense granular media can be evaluated from averages over typical blocked configurations. We have shown that this measure is able to reproduce the dynamical sampling of the out-of-equilibrium compaction dynamics for various observables. The connection of Edwards' ensemble with the dynamical FDT temperature immediately suggests experiments to check the validity of these ideas, for example by studying diffusion and mobility of different tracer particles within driven granular media.

Turbulence.

In the past, we have introduced the multi-fractal model for the fully developed turbulence. Our approach had been widely used, and cited, for the description and modeling of turbulence, and more general complex phenomena in chaotic and disordered systems. Another one of our relevant contributions has been the study of the fully developed turbulence in terms of dynamical systems using simplified models (shell models) and direct numerical simulations.

Transport and Diffusion.

We had been among the first to study transport and diffusion in laminar velocity fields. We showed the relevance of the Lagrangian chaos for the spatial structures of magnetic dynamo and the non trivial dependence of the diffusion coefficients on the details of the velocity field. In addition we worked on the anomalous diffusion. Using the FSLE and exit time approach we have treated the transport processes in realistic cases in geophysical context, e.g. closed basins and finite open systems.

Chaos and Complexity.

The multi-fractal description, originally introduced for the fully developed turbulence, had been widely used for the characterization of the attractors of dynamical systems and their finite time properties (with the introduction of the generalized Lyapunov exponents). With the aim of studying the finite resolution properties for predictability problem, we introduced the concept of FSLE. This technique allowed us for a successful characterization of non trivial dynamical systems with many different characteristic temporal and spatial scales. More recently we introduced an efficient method, in terms of exit time statistics, for the computation of the epsilon-entropy.

Fractals and Non-Equilibrium Critical Phenomena.

Our activity in this field results in a cooperative effort on data analysis, numerical simulations and analytical work. We used concepts of modern statistical physics to study the clustering of galaxies in the available catalogues as well as in computer simulations.

We have analyzed a large number of models producing fractal structures and dynamic scaling: Diffusion Limited Aggregation (DLA), the Dielectric Breakdown Model (DBM) Cluster-Cluster Aggregation, Sandpile models, the Kardar-Parisi-Zhang (KPZ) model of surface growth, lattice models of fracture, Invasion Percolation and the Bak-Sneppen model for biological evolution.

We have introduced a theoretical scheme based on a "Fixed Scale Transformation (FST)" that allows to deal with the irreversible dynamics of these processes and to calculate analytically the fractal dimension.

Stripe-Quantum-Critical-Point and Pseudo-gap Phase in Cuprates.

Our group strongly contributed to the debate on the possible realization of a Luttinger liquid in 2D by showing that non-singular interactions lead to a FL already in 1+epsilon dimensions. The major achievement of the last few years is the formulation of the Stripe-Quantum-Critical-Point scenario, which is presently a candidate for the description of the Cuprates. More recently our group elaborated a two-gap model to account for the peculiar pseudo-gap opening in the under-doped Cuprates.

The Non-Adiabatic Superconductivity.

As main results of our activity in this field we have shown that an unbiased analysis of the experimental data of fullerides points out the intrinsic inconsistency of the conventional adiabatic Migdal-Eliashberg theory and the dominant role played by the non-adiabatic processes to determine super-conducting and normal state properties. We have identified characteristic features peculiar of a non-adiabatic el-ph coupling and proposed experimental measurements to test the non-adiabatic theory.

Electron-Phonon Correlations.

The phase diagram of the adiabatic Holstein model, has been completely derived in the high-dimensionality and adiabatic limit. The crossover phenomenon in the disordered and ordered polaronic state has been clarified.

Preliminary results of the competition between super-conductive and charge-order fluctuations have been derived in the attractive Hubbard model.

Numerical Approaches to Strong Correlations.

We carried out the QMC calculation of the static response function of quantum fluids, and, most notably of the electron gas in 2d and in 3d. We also developed the "reptation QMC", a path-integral algorithm for the calculation of unbiased ground state properties and imaginary-time correlations.

Slow Dynamics, Disorder and Complexity

Senior Researchers: Daniel Amit, Enzo Marinari, Giorgio Parisi and Piero Tartaglia.

3 Ph.D. students, 6 Postdoctoral fellows, and 5 coworkers.

In recent years physicists have been deeply interested in studying the behavior of complex systems. Complex systems are often characterized by the existence of an extremely slow approach to equilibrium (metastability) and many observable quantities display a remarkable dependence on the previous history of the sample. The system can therefore be trapped in many different slightly off-equilibrium states and slowly move from one state to another one.

It is now known that quite different complex systems, ranging from ordinary glass to proteins, do exhibit similar behaviors, commonly named glassy. One of the most striking aspect is that disorder in not a necessary ingredient for a glassy behavior, the latter following from the complex topology of the energy or free-energy landscape. This explains why so different systems may have a similar behavior. The understanding of the relationship between the complex energy or free-energy topology and the macroscopic behavior is one of the most challenging topics of equilibrium and non-equilibrium statistical mechanics.

These features indicate that at the microscopic level there are many local equilibrium configurations (which are called in the literature with many different names, for example valleys or inherent structures) and the rich behavior of the system is a macroscopic reflex of this microscopic structure.

In order to understand in general these features (which in the real world are quite common) one has to go through two main steps:

• To construct a statistical mechanics framework in which the existence of many local equilibrium configurations can be studied at the microscopical level.

• To generalize statistical mechanics to systems which are slightly off-equilibrium and to find the appropriate tool to study it.

The general approach to deal with both tasks have been forged in the past. In the first case the replica method (and replica symmetry breaking) has been very successful, while in the second case a crucial ingredient has been the introduction of a time dependent temperature which generalizes the concept of temperature.

The two general formalisms (both introduced for the first time in Rome) are deeply related one with the other: this link allows us to construct a general theory of complex systems.

At the same time than the analytical technique development, also the numerical ones have received an important push. As a matter of fact, in this field numerical simulations play, in some sense, a role of integration and often of replacement for experiments that can be very difficult to be realized. We want to stress the driving role of numerical simulations, where they anticipate and organize the preparation and the realization of real experiments.

Using numerical simulations it is possible to do a systematic analysis of the potential energy landscape of several atomic models as well as of simpler models which are taught to relevant for the understanding of the glass transition. The study of the topological properties of the energy or the free-energy surface is one of the most promising approach to the glassy behavior.

Of course we cannot limit ourself to the study of general principles: we have to find concrete applications of the theory. For each particular case we have to develop the appropriate technical tools, to make concrete predictions which should be tested in numerical simulations and eventually in real experiments. The theory has been developed mainly in the study of spin glasses: some concepts (for example the one of inherent structures or of configurational entropy) have been developed independently in study of structural glasses (a nice example of convergent evolution). It is not strange that the main subjects of our group will be spin glasses and structural glasses. We have to add to these two the study of granular material, a new field in rapid development, and of other complex complex systems.

Although there is an unity in the theoretical approach, for reader convenience the main research which we plan to do in the future on slow dynamics, disorder and complexity can be divided, with some overlaps, into the following main topics.

1. Slow Dynamics and Equilibrium Properties of Spin Glasses.

   Establishing the correct theoretical description of finite dimensional random systems with frustration is an important and difficult goal: we are dedicating many efforts to try to achieve it. The general setting in which we shall move is replica theory.

   Replica theory makes exact predictions in the mean field case and the its extension beyond mean field contains points which are actually investigated. We want to sharpen the theoretical predictions and to compare them with the most advanced numerical simulations and with the large body of experimental results

   There are a few main points which we plan to develop:

   • To obtain a more precise analytic description of the theory. Indeed there are two main properties which are not yet clarified, which are on the reach of present day techniques. The first is the variation of the free energy landscape when we change the temperature (chaos in temperature): this problem is quite relevant for understanding the many puzzling experiments on temperature cycling that have been done in recent times. The second is the detailed quantitative dependence of the correlation and of the response on time in the aging regime, which is well observed numerically and experimentally but still lacks a quantitative understanding.

   • To make progress in the spin wave theory for spin glasses using techniques like the renormalization group. This could allow to obtain crucial critical exponents for different quantities in the three dimensional world.

   • A detail comparison of the theoretical predictions with large scale numerical simulations.

     We plan to use equilibrium methods, and dynamical approaches to investigate the properties of spin glasses. They give complementary information on slightly different region of phase space. We plan to investigate both Ising models and more realistic anisotropic Heisenberg models in order to understand better the comparison of the theoretical predictions with experimental data. We will concentrate our attention both on the behavior in magnetic field and on thermal cycling experiments.

     A new field we are investigating is the one based on computing ground states of disordered systems. We have computed and analyzed couples of ground states of 3D spin glass systems, and we have established that the picture based on RSB correctly describes the behavior of 3D Spin Glasses.

   • Careful study of existing experiments, finalizing the previous described approaches to a better understanding of the data. Moreover it will be crucial to project new types of experiments, to propose them to the experimental groups and to push for their realization (as we already did in the past in a few cases).

2. Complex Systems and the Glassy State.

   A better understanding of the physics of the glassy state is one of the crucial goals we have in mind for the Center. We believe that analytic techniques merging ideas coming from the Replica Symmetry Broken (RSB) mean field theory of spin glasses, from the Inherent Structures approach and from the Mode Coupling Theory, together with large scale numerical simulations, will help in obtaining remarkable progresses.

   The project concerns the study of static and dynamical properties of both realistic atomic models and simplified spin models for structural glasses. The systematic numerical study of equilibrium and out-of-equilibrium behavior of glassy systems will allow us to analyze the part played by activated processes and their relevance for glassy dynamics. Such an approach will make us able to compare the results both, on one side, with the ones on structural glass out-of-equilibrium properties recently obtained by statistical mechanics techniques and, on the other side, with experimental results. Moreover the understanding of the fundamental mechanisms of glassy dynamics will make possible the development of very effective algorithms and numerical codes for the study of structural glasses at low temperatures.

   Another dominant feature of our research has been centered on the applicability of replica theory to glassy systems, which turned out into the development of a theoretical approach to the thermodynamic of structural glasses based both on ideas initially introduced for spin glass models and on classical liquid theory approximations. Recent progresses based on the use of the replica method also allow a definition and the evaluation of the configurational entropy (or complexity), which should be compare with that derived from the inherent structure formalism introduced by Stillinger and Weber and recently applied to disordered spin systems.

   Another consequence of the spin glass approach to the glassy transition is the possibility of deriving generalized fluctuation dissipation relations that can be directly tested in experiments with present day technology (efforts in this direction are under way). This kind of analysis, that started within the spin glass research field, was recently extended to realistic models for structural glasses, and is one of the main research field that will be pursued in the center.

   One of the promising approach to the study of the glass transition focus on the properties of the potential energy landscape (PES), in particular its topological properties.

   The increased computational facilities have significantly improved the early efforts of studying the PES. Nowadays, an exhaustive search for all basins of the PES has been performed for clusters and complete maps of the local minima energies are available for several potential models. For clusters, as well as for small proteins, the connectivity between all basins has also been evaluated, to provide a very informative map both of the thermodynamics as well as of the dynamics in these small systems.

   The extension of these approaches to bulky system is taking place in these days and may offer novel insight on the onset of the slow-dynamics and on the connections between dynamical and thermodynamical quantities. In particular, by pursuing the study of the PES in bulk samples of several atomic models, ranging from LJ and OTP (prototypes of fragile liquids) to water (whose classification in the strong-fragile scheme is still under debate). and Si02 (prototype of strong liquid) we hope to be able to fully understand the connections between Arrhenius dynamics and topology of the PES. We expect that the evaluation of the configurational entropy of the liquid --- defined as the logarithm of the number of different basins visited by the system --- and its relation to diffusivity will allow us to identify the importance of the geometrical properties of the PES in controlling the molecular dynamics.

   We plan to explore the temperature range around the dynamical critical temperature Tc of different models. The relevance of this temperature was first predicted by Mode Coupling Theory and later on by analytic solutions of p-spin models in mean field. At Tc, a change in the T-dependence of the characteristic times is observed, as well as a separation of relaxation processes (as observed for example in dielectric relaxation experiments). The physical processes determining such crossover in the dynamical behavior are not well known yet and very different models have been proposed to account for such changes. Moreover we plan to calculate both the temperature and the density dependence of configurational entropy, i.e. the number of basins in configuration space visited in equilibrium. Such a study will allow us to perform stringent tests on thermodynamic theories of the glass transition and to correlate dynamical properties (like the molecular diffusivity) with thermodynamical ones. Such a comparison, performed for different models, will allow us to clarify the role of the topology of the configurational space on the degree of fragility of the liquid.

   The richness of the previous approach should be complemented by a systematic analytic and first principles studies of the properties of simple glass forming liquids.

   In studying equilibrium thermodynamics of glasses one focuses onto first principle computations in simple fragile glasses. The replica formalism gives a general setting in which the previous described physical concepts can be used to perform quantitative computations. At the technical level there are many replica based approach: in one of them we can translate this problem into that of a gas of interacting molecules: the results, particularly those concerning the Kauzmann temperature and the configurational entropy, can be successfully compared to recent numerical simulations. The extension of these ideas to other systems, displaying the also a strong behavior, would be extremely interesting. Moreover it would be important to develop techniques that could be applied also to systems like hard sphere, which are very hard to be effectively studied in this approach.

   Generally speaking it is clear that an quantitative control of the glass phase cannot be reached if we do not have under our command the properties of the liquid. Already the computation of the static structure functions in the liquid is not a simple task. A fist principle computation of the dynamical structure functions is a more difficult task: the mod coupling theory give us many information, but it would be very interesting to connect it to a more microscopical formulation (a first step in this direction has already been done).

   A crucial properly in the liquid is the spectrum of Instantaneous Normal Modes. In spite of its importance (from the analytic point of view) this problem has been under investigated in the literature. We are developing a new approach for the analytic computation of the spectrum and of the properties of the eigenvalues (localized or extended). These properties play an important role in the understanding of the dynamical behavior near the Mode Coupling Temperature and in controlling the thermodynamical properties below. We hope to make quite relevant progress in this field, using a panoply of analytic tools that are in our inventory.

   An we have already remarked an important property is the behavior of the free energy landscape near the mode coupling transition. The study of the saddle points of the energy in this region is particular interesting and it gives us very useful information. Also in this case we plan to be able to perform analytic computations, which should complement the information obtained using other techniques.

   Last, but not the least, it would be extremely interesting to find out how this emerging picture is relevant at very low temperatures, in the deep quantum regime, and if and how the two level picture is modified. This is a rather difficult and ambitious program, which require a very good command of the properties of glasses in the classical regime. Some preliminary steps in this direction have already been done.

3. Glassy Behavior of Granular Materials.

   The main objective of this search is to explore the possibility of defining a set of statistical ensembles suitable to describe the phenomenology of granular matter and it is connected to the general theory of glassy systems (granular material will also be studied in the activity, but from a different point of view).

   The classical way to go from the microscopic dynamics to statistical mechanics proceeds in two steps: one first identifies a distribution that is left invariant by the dynamics (e.g. the micro-canonical ensemble), and then assumes that this distribution will be reached by the system, under suitable conditions of 'ergodicity'. For granular systems this approach seems doomed from the outset: because energy is lost through internal friction, and gained by a non-thermal source such as tapping or shearing, the dynamical equations do not leave the micro-canonical or any other known ensemble invariant. Moreover, the compaction dynamics is extremely slow and does not approach any stationary state on experimental time scales. This raises the question of characterizing the typical configurations or the region of phase space visited dynamically.

   We have investigated the role of entropic concepts for the relaxation dynamics in granular systems. In the framework of a class of mean-field models introduced for the compaction phenomenon, it is possible to explicitly construct a ``free-energy''-like functional which decreases along the trajectories of the dynamics and which allows to account for the asymptotic behavior: e.g. density profile, segregation phenomena.

   Several years ago S. F. Edwards, for the case of dense granular media, proposed a modified micro-canonical ensemble in which the only relevant configurations are the ``stable'' or blocked ones, i.e. those in which all the particles are unable to move. The strong assumption in this hypothesis is that the blocked configurations are treated as equivalent from the dynamic point of view.

   Numerical simulations support the Edwards approach, and show that macroscopic observables of dense granular media can be evaluated from averages over typical blocked configurations. To this end it has been introduced a method to construct the corresponding measure for different classes of finite-dimensional systems with frustrated interactions, whose static and dynamical properties exhibit interesting features that are common to granular packings, structural glasses and spin glasses. One can thus compare the predictions coming from these measure for various observables with the outcome of the out of equilibrium dynamics at large times.

   Despite this success it is important to to analyze the extension of the validity of the Edwards' approach by analyzing the behavior of models for which the Edwards' construction could be inappropriate, even though they may have a very slow, ``glassy'' dynamics. The general question to answer is therefore whether a long-time configuration is well-reproduced by the typical 'blocked' configuration of the same energy. A natural criterion in this direction, suggested by glass theory, consists in studying how a system explores its phase space, i.e. its chaoticity properties.

   Another important point is the comparison with similar approaches. In particular the connection with the effective temperatures that appear in out of equilibrium glass theories, as well as at the perspective of the so-called `inherent structures' (a partition of the phase-space in terms of the blocked configurations) where assumptions, similar in spirit but not quite equivalent to Edwards', also come into question.

   There is a remarkable synergy among the issues we have discussed under this item and the study of granular matter described in the chaos activity: obviously we bet on the joint efforts of the two research developments to increase our payoff (in terms of research results).

4. Other Complexities.

   Slow and glassy behavior is quite common in nature, just to cite few examples: tiling, RNA and protein structure.

   It is possible to use the techniques described before to study these problems and this has been done in the past. In order to make further progresses one should identify a problem which is enough complex (at the microscopic level) in order to be interesting and simultaneously simple enough (at the microscopic level) in order to be studied in details. The problem of RNA folding seems to be particular interesting in this respect. We have already done some progresses in this field: we have used a simple model for RNA folding and we plan to further investigate it in greater detail.

   An other problem of great interest is the study of complex system from the point of view of optimization theory. In this field a prominent role is played by the statistical mechanics of the random K-satisfiability (K-SAT) problem, that has been the object of many studies in the last years. The K-SAT was the first problem to be shown to be NP complete and the importance of such a model is that it provides a prototype for all the Non-deterministic Polynomial (NP) complete problems in complexity theory of computer science as well as in statistical mechanics of disordered and glassy systems, in computational biology and in other fields. (We recall that he NP complete are those problems whose solution, or the certainty that they have no solutions, can only be found in the worst case by algorithms with a running time of computation that likely grows faster than polynomially with the number of variables N of the system.)

   In this model one observe both a transition from satisfiability to unsatisfiability and and the transition between a Replica Symmetric structure and a structure where the symmetry is broken. It is extremely interesting to study in details the relations among these two transitions. This can be done both using analytic techniques and numerical ones.

   At last we quote our work on neural networks. We have systems of neurons and synapses. These systems, in a sense, are "doubly" complex: first, the typical synaptic matrix renders the neural dynamics glassy. Second: the neural dynamics reacts on the much larger set of synaptic degrees of freedom at another time scale. The double dynamics, while much more difficult to analyze, is at the same time more interesting and promising. The challenge is the development of a dynamical mean-field theory for the collective features of the double dynamics, to render the search of relevant parameter regions more effective.

Strongly Correlated Quantum Systems

Senior Researchers: Giovanni Bachelet, Claudio Castellani, Ferdinando de Pasquale, Carlo Di Castro, Luciano Pietronero.

3 Ph.D. students, 5 Postdoctoral fellows, and 6 coworkers.

The renewed interest in strongly correlated quantum systems has been mostly triggered by the discovery of high temperature superconductivity in cuprates, which are doped Mott insulators under many aspects. However, there is a growing experimental evidence that the electronic correlation plays a main role in various other novel and old materials like the colossal magnetoresistance manganites, the vanadium oxide V2O3, and the low density 2d electron systems, which show a metal-insulator transition challenging the expectations based on the Anderson localization. Correlation rules the physical properties of all low-dimensional conductors like quasi-1d inorganic and organic metals, 1d quantum wire, and edge states in fractional quantum Hall systems. Correlation is also relevant in C60 based materials, at least in determining their conducting or insulating character and possibly by cooperating to induce superconductivity. In this respect the combination of strong correlation within non-adiabatic channels of interactions with phonons appears to be a very promising concept to understand the new superconducting properties, with Tc=52 K, observed by Batlogg et al. in FET hole-doped materials.

1. New Concepts from Novel and Old Materials.

   It is by now clear that most of the above systems are approaching a breakdown of the conceptual scheme introduced by Landau more than forty years ago, i.e. the Fermi Liquid theory based on the adiabatic switching on of the interaction and on the existence of quasi-particles. We are facing new properties emerging from correlation and non-adiabatic interactions, as realizations of complexity in quantum many-body systems. A main aspect, common to complexity, is often the absence of any length and/or energy scale. Sometimes one faces the opposite, but indeed similar, limit of too many scales. While quasi-1d systems find their new conceptual framework in the so called Luttinger Liquid theory, strongly correlated 2d and 3d systems, and cuprates in particular, have not yet found a proper theoretical framework which should be more than a ad hoc (and a posteriori) description of some of their properties. Indeed, it is quite urgent to identify the robust features of these new materials with the aim of putting constrains on the multitude of different mechanisms proposed for instance for high Tc superconductivity. This is urgent from a theoretical point of view and, more effectively, from a practical point of view because of the huge relevance of many of these materials for applications.

2. Kinetic Energy, Correlation and Instabilities.

   The success of Fermi liquid theory in accounting for the metallic properties of many electron systems is essentially based on the kinetic energy of the electrons being the main energy term to be minimized. In this framework, screening processes embed the largest part of the interaction into the quasi-particle parameters leaving only a usually weak residual effective interaction. However, it became clear already several decades ago with Mott, that strong e-e interactions may in some cases disrupt the metallic state promoting the formation of an insulator. In the Mott insulator it is the interaction term that dominates and the minimization of the interaction energy (instead of the kinetic one) is more relevant. In more recent times the discovery of the novel materials has put in evidence that this competition between kinetic and interaction energy terms can produce a rich phenomenology and eventually lead to a disruption of the FL behavior. The breakdown of Fermi Liquid theory is now an issue even in the field of heavy fermion systems, which in the past were the most quoted example of the success and of ductility of the Landau quasi-particle concepts. Heavy fermions can indeed be considered at the border of the applicability of FL theory, since the smallness of the quasi-particle kinetic energy leaves them on the verge of interaction-driven instabilities. Recently, several of these instabilities have been detected. Quite remarkably, close to the instability even the metallic state acquires unusual properties strongly violating the FL theory. From a general point of view, many of the strongly correlated systems share the property that the kinetic energy is substantially reduced by correlations opening the way to various instabilities of the metallic state, like Mott-insulating behavior, non-FL behavior, non-adiabatic interactions, polaron formation, charge, magnetic, and/or orbital ordering. This tendency reflects in the rich and complex phase diagrams of all the compounds mentioned above in which many different phases may be stabilized by varying external parameters like temperature, doping or pressure. Remarkably, a more or less unconventional superconductivity often appears in the phase diagram.

3. Theoretical Issues.

   A key common element of cuprate superconductors and fullerene doped materials is the extremely low density of charge carriers (electrons or holes), up to twenty times less than in a normal metal. This situation decreases drastically the Fermi energy which becomes comparable to the Coulomb interaction and to the phonon frequencies. It is therefore natural to expect much more complex phenomenology with respect to the standard Fermi liquid in which electrons are strongly correlated and interact with phonons or other possible superconducting mediators in a strongly non-adiabatic way.

   From the above discussion and examples one sees that the interactions can indeed disrupt the FL metallic state (or can lead it to the verge of disruption like for heavy fermions). Actually, two ways of breaking a FL state are already known since long time: the formation of a superconductor and the formation of a Mott insulator. Remarkably the examples above have shown that there are other less traditional possibilities. In particular the heavy fermions near zero-temperature second-order instabilities (the so-called quantum critical points, QCP) suggest that the proximity to a second-order phase transition is a source of FL violation: the low-energy critical fluctuations always present near a critical point can couple to the quasi-particles providing a mechanism for strong "residual" effective interactions. In this case the quasi-particles no longer are weakly interacting well-defined excitations and the systems may acquire non-FL properties. Moreover, the strong interactions mediated by the critical fluctuations can also provide a quite effective mechanism for the formation of Cooper pairs, eventually condensing into a phase-coherent superconducting state. This scheme may be of obvious relevance also for the cuprates since it let normal and superconducting properties stem from a single mechanism: The interactions favor the instability and, before the system becomes unstable is looses its FL properties and possibly becomes superconducting. Another line of thought favored by the discovery of HTSC materials is related to Anderson's idea that two-dimensional strongly correlated materials simply cannot be FL irrespective to the proximity to a QCP. This point of view borrows the Luttinger-liquid concept from the physics of one-dimensional materials and applies it to the (3D, but nearly two-dimensional) cuprates.

   Summarizing, under the action of e-e interactions, the FL metallic state can be disrupted in rather traditional or more unconventional ways. In the first case the system may

   • (a) become superconducting either by forming a standard BCS state with tightly phase-locked Cooper pairs or, less conventionally, by condensing pre-formed (phase-incoherent) pairs. A second traditional possibility is that, at least at some specific densities, the system

   • (b) form a Mott insulator (customarily characterized by some parasitic magnetic ordering, which does not cause but takes further advantage from the insulating state).

     The two unconventional ways which recently emerged as possible opponents of the FL state are

   • (c) the proximity to a QCP and

   • (d) the direct formation of a non-FL phase.

   A further source of non-FL behavior may arise from the presence of a sizable electron-phonon (e-ph) coupling. This ingredient, is indeed substantial in many of the real strongly correlated materials mentioned above. A substantial e-ph coupling besides inducing an effective e-e interaction (and possibly superconductivity), can promote the formation of polaronic states, where the electrons are heavily dressed by phonons. Such states are present in many regions of the phase diagram of the manganites, and polaronic features are also present in the normal state of the cuprates.

   Strong modifications of the classical Fermi liquid properties related to the electron-phonon interaction can arise also in the so-called non-adiabatic regime, which is qualitatively and quantitatively different to the polaronic one. A primary role in this regime, identified by one of the groups in the area, is played by the dynamically interference of electron and phonon degrees of freedom when the energy scale of electrons and phonons are comparable. This situation, characteristic of cuprates and also of fullerene compounds, leads to the breakdown of the Migdal's theorem and open new non-adiabatic channels of interactions which, in the presence of strong electron-electron correlation, effectively favor the superconducting pairing.

   What emerges from the discussion in this section is that several mechanisms can be active in the various materials to produce a non-FL behavior, possibly driving the system superconducting at high temperature. The separate understanding of the various emerging properties of strong correlation is "per se" quite relevant and interesting. However, to cast them in a unified scheme and to determine the interplay of these different mechanisms is the crucial issue both theoretically and practically. This long-term perspective will be tackled both within our specific research activity, and by promoting dedicated programs and workshops.

4. High Temperature Superconductors and Superconductivity in Novel Materials.

   The investigation of the cuprates (or more generally of superconductivity in novel materials) will provide the "backbone" of the activity of the Center on "Strongly correlated quantum systems". One reason for this is that cuprates are the most challenging from the theoretical point of view since their phase diagram contains a Mott insulating, a superconducting and an anomalous normal state phase. Thus they surely realize the mechanisms (a) and (b) for disrupting the FL, while the possible realization of a metallic non-FL state via (c) or (d) is at present a matter of strong and hot debate. Thus they are a crossroad of non-FL mechanisms, which is an ideal playground to test ideas or to elaborate new concepts. A second reason is that the study of the cuprates has already been a main activity for some of the Investigators in Rome. Therefore there is a well grounded competence in this specific area. In particular the broad spectrum of technical know-how possessed by the Investigators, like field-theoretical techniques, many-body approaches, renormalization group, techniques to handle disordered electron systems, numerical techniques (Monte Carlo, Lanczos exact diagonalization, Density-Matrix Renormalization Group), has already been (or can easily be) applied to this topic. Therefore, while the amplitude of the FL vs non-FL issue requires a long-term perspective, at the same time this robust and well established wealth of competences should guarantee an immediate and high-profile research and organization activity. Last but not least, due to the evident relevance for applications, the field of high-temperature superconductors (or more generally of superconductivity in novel materials) is among the most important fields of the international research in condensed matter and with the highest level of financial support.

   • The Stripe-Quantum-Critical-Point.

     A major achievement of one of the groups in Rome in the field of strongly correlated electron systems is the proposal of a QCP near the optimal doping of the cuprates. This proposal refers to the conceptual framework (c) described above and is based on a second order phase transition at zero temperature, between a uniform metal and a stripe phase with local charge and magnetic order. According to this proposal this Stripe Quantum Critical Point, which is eventually masked by the occurring of superconductivity, rules the physics of the cuprates at intermediate doping so that the phase diagram of most of the cuprates is partitioned into an under-, an optimally, and an over doped region corresponding respectively to the nearly ordered (but for the presence of intrinsic disorder), the quantum critical , and the quantum disordered regions of the QCP. The presence of the QCP entails large charge and spin fluctuations mediating a singular effective interaction, thus accounting for both the non-Fermi liquid properties of the normal state and the strong pairing mechanism for superconductivity. Specifically, this scenario is made possible by the strong correlation present in the cuprates, which favor the phase separation in highly doped metallic regions and AF insulating regions.

     Besides the derivation of specific observable consequences of the Stripe-QCP scenario (isotopic effect, spectroscopic features, and so on) an urgent general point that we plan to investigate is the identification of the control parameters (strength of the e-e interaction, e-ph coupling, lattice constants, and so on) ruling the time scale of stripe fluctuations (to distinguish between static and dynamical stripes). This has a relevance for the experimental observation of stripes, since dynamical charge fluctuations can only be detected by probes having sufficiently fast time resolution.

     The overall goal is to elaborate a detailed phase diagram where, besides temperature and doping, the additional ingredient(s) are identified determining the position of the QCP. Depending on their specific details, the different classes of cuprates can be located in different regions of the phase diagram, accounting for their partially different physical properties. In particular there might exist materials with such parameters, that a truly ordered phase is absent at any doping, but close enough to the QCP so that the critical fluctuations still play their relevant role.

   • Pseudo-gap, phase fluctuations, and disorder in under-doped cuprates.

     A relevant phenomenological feature of under doped cuprates is the opening of pseudo-gaps both in the charge and spin degrees of freedom below a crossover temperature T* (larger than the superconducting temperature Tc), which eventually merges with Tc around optimal doping. Angular Resolved Photo-emission (ARPES) experiments show that this gap is first formed around the (pi,0) points (and symmetry related) of the Brillouin zone, while leaving finite arcs of the Fermi surface gapless. This highly unconventional phenomenology motivated our recent research on a partial breakdown of a FL arising from the formation of local Cooper pairs in restricted regions of the k-space. This line of research based on the coexistence in k-space of FL and non-FL states is promising and worth being pursued as a long-term project. In this framework we are planning to adopt non-perturbative tools like Dynamical Mean-Field theory (DMFT) or Quantum Monte Carlo to investigate this novel aspect of the FL vs non-FL issue.

     A further relevant issue in the cuprates, particularly in the under doped regime, is the validity of the BCS theory for the superconducting ordered phase (even in the low-temperature limit) and the possible occurrence of a crossover to the Bose-Einstein condensation picture. In particular the role of the quasi-particles and of the collective modes in depleting the superfluid density together with their role in heat and charge transport are at present a matter of lively debate. We have planned a future activity on both topics. This includes the consideration of the correlation effects in determining the doping dependence of the superfluid density and of the effects of disorder in the transport properties of the quasi-particles and their possible localization.

   • The non-adiabatic superconductivity.

     Within the FL theory, the e-ph coupling is taken into account in the framework of the so-called Migdal theorem, which relies on the assumption that the Fermi energy is by far the largest energy scale, in particular much larger than any characteristic phonon energy. The non-adiabatic Fermi liquid concept stems from the observation that in cuprates and in fullerenes as well the electron and phonon relevant energies can be of the same order of magnitude leading to a non-adiabatic interplay between electrons and phonons. In this situation the basilar simplifications based on the Born-Oppenheimer approximation do not hold true and the interference between the mutual response of the electron and phonon dynamics has to be explicitly taken into account. The diagrammatic Feynman theory and the Green's function formalism permit a particular powerful and simple way to formalize this concept in term of a graphical representation and to identify in the ``vertex'' diagrams the interference processes neglected in the Born-Oppenheimer approximation. The interplay between electron and phonon degrees of freedom in this regime induces complex and not trivial effects on many electronic and phononic properties so that the simple idea of Landau-Fermi quasi-particles is unable to describe properly this situation and a new concept of non-adiabatic metal is necessary.

     The non-adiabatic Fermi liquid can be considered as a description of the electron-phonon coupled system complementary to the Migdal-Eliashberg and polaronic regimes. In the last years our group, having established the basis of the theory, has started a program with the aim of identifying specific properties of the non-adiabatic regime that could be verified experimentally. We also aim to extend and refine the theory in order to include realistic band structure and the interplay between electron-phonon interaction and the strong electronic correlation. The recent discovery of superconductivity at 52 K in hole-doped fullerene definitely points out to the failure of the adiabatic Migdal-Eliashberg theory in describing superconducting and normal state properties in fullerene compounds. The analysis of detailed experimental data of these new class of high temperature superconductors within the framework of the non-adiabatic theory appears therefore of primary relevance and is an urgent research in one of the recent most important topics.

     Most of the other approaches take the complementary point of view to focus mainly on correlations. Actually the e-e correlation enhances the non-adiabatic effects. It is important therefore to explore the different paths and to consider their possible convergence or incompatibility. A further step forward the comprehension of the non-adiabatic regime has been achieved by classifying the physical interpretation of the electron-phonon vertex correction. Such a non-adiabatic contribution has a fundamental importance in the theory and its comprehension in terms of physical processes may help to extend the theory beyond the perturbative approach.

     It is worth noting that the analysis of non-adiabaticity, and specifically the inclusion of vertex corrections, is of relevance also in the context of pairing induced by large e-e interactions. In particular the occurrence of nearly singular interactions near the QCP calls for the inclusion of these effects. This illustrates the synergetic aspects of these researches.

   • Electron-Phonon correlations from the intermediate to the strong coupling regime.

     The crossover to the strong-coupling regime for the e-ph coupling is a specific topic, which has been studied within the Holstein model. For this model the phase diagram and the conduction properties have been derived in the adiabatic and large dimensionality limit. Both weak and strong coupling regime have been analytically studied together with the transition from electronic to polaronic behavior both in the high temperature disordered phase and in the low temperature charge-ordered phase. Future research will consider non-adiabatic corrections and the competition of charge-order with superconductivity away from half-filling. Preliminary results on the latter point have been obtained in the attractive Hubbard model case. The competition between two different local ordering phenomena can be also studied from an alternative point of view, which more generically involves quantum-coherence properties of spontaneously ordered phases and takes into account the specific changes of quantum-coherence close to a critical point. In a spin system local quantum coherence is defined as a statistical average of the scalar product of a local spin vector with a unitary vector associated to a local direction in the spin space. The physical interpretation is in terms of the probability of a given spin configuration at statistical equilibrium. In the electronic system, spins are replaced by pseudo-spin operators. The local phase will determine the relative weight between charge and superconducting local ordering. The study of quantum coherence in a spontaneously ordered phase is of relevant interest also in the perspective of quantum computation.

5. Numerical Approaches to Strong Correlation.

   In the study of strongly correlated systems it is useful to complement the theoretical analysis with numerical simulations. This is particularly important in the light of the above mentioned complexity of strongly correlated systems: The lack in these systems of relevant length- and energy scales, as well as natural small parameters makes the standard perturbative approaches difficult to control. The quantum Monte Carlo (QMC) methods offer a favorable trade-off between accuracy and size/complexity of the system (as compared to exact diagonalization techniques, limited to extremely small system size, and to the density-matrix renormalization group method, whose extension to more than one spatial dimension implies a severe loss of reliability and efficiency).

   The main drawback of the QMC method, deeply rooted in its stochastic nature, is the so-called sign problem, which has precluded so far the implementation of general-purpose, exact, and stable algorithms for Fermionic ground states (as well as for excited states). The most commonly used remedy against the sign problem has been the fixed-node (FN) approximation, in which the nodes of the ground state are assumed to be the ones of a suitable trial function.

   The FN approximation gives rigorous upper bounds to the ground-state energy, which are often very strict. Furthermore, there exist methods that locally release the nodal constraint, making the role of the chosen trial function less crucial. Various ``hot'' topics will be tackled within this scheme.

   A first topic is the phase separation in the Hubbard model. We already emphasized above the role of strong correlation in suppressing the kinetic energy, thereby opening the way to electronic instabilities. In particular the electronic instability towards phase separation is attracting much interest since it has been pointed out that it may be related to the high Tc superconductivity. Despite intense theoretical and numerical effort devoted to the subject, the behavior of relevant models for this problem remains still debated. In particular, while various realizations of mean-field approaches based on Hartree-Fock decoupling schemes and a density-matrix renormalization group approach indicate that the ground state of the Hubbard model or of the t-J model in the physically relevant region of small J is characterized by charge ordering in stripes, QMC simulations do not reveal compelling evidence for phase separation. More systematic QMC simulations, including better assessments of finite size effects, and use of modulated trial functions, possibly with better optimization schemes, are necessary to make conclusive statements on the physical properties of these strongly correlated models. Furthermore, inclusion of additional realistic interactions may be relevant for the understanding of real materials.

   A second topic, which is worth analyzing by QMC methods, concerns the ground state properties of the 2d electron gas. Electrons (or holes) can be confined into effectively two--dimensional systems, for instance in Si MOSFET's and III-V semiconductor heterostructures, over a density range extending down to the freezing transition. Far from being a mere playground for testing many--body theories and numerical simulations, strongly correlated two--dimensional electronic systems offer an extremely rich and interesting phenomenology, including a previously unexpected metal--insulator transition.

   Using standard FN QMC simulations, we have recently shown that the 2d electron gas undergoes a first order polarization transition, as the density decreases, shortly before crystallization takes place.

   We plan to extend this study to higher densities, where the transition is induced by transverse magnetic fields. Extensive simulation data will be collected for the spin susceptibility, the momentum distribution, the pair correlation functions, and the exchange-correlation energy. Building on previous work on the 3d electron gas, analytic interpolation formulas, including the exactly known limits and fitting the QMC data, will be provided for pair correlations and exchange-correlation energies as functions of the density and of the polarization.

   Less standard QMC simulations will be attempted to calculate current-current correlations and to describe the effects of disorder on the above physical properties.

   A further issue within the QMC method concerns the derivatives of the fixed node energies. In some sense, QMC simulations produce a relatively limited amount of physical information, being often limited to the calculation of the total energy. While there are remarkable exceptions, an extension of the range of physical properties routinely accessible by QMC calculations remains a significant methodological objective. Using a new simulation algorithm, we are exploring the feasibility of the systematic calculation of analytic derivatives of the total energy, say with respect to external fields. This is relatively straightforward for bosonic ground states, but poses nontrivial technical problems in conjunction with the FN approximation: indeed, previous work in this field has always involved further approximations on top of the FN one.

6. Parallel Activities in Quantum Strong Correlation and Interfacing Topics to Complexity.

   The research activity on strongly correlated systems of the permanent Investigators of the Center will mainly concern the topics illustrated above with the intellectual focus on superconductivity and the non-FL behavior. However, we plan to promote several parallel activities with programs and workshops. This is of particular importance, since the physics of strongly correlated quantum systems is so rich that a cascade of interesting ``secondary'' issues can be related to those illustrated above. Among several others, we mention the meso- and nano-scopic systems, where correlation has definitely been recognized to be important, and the phenomenology of extreme type-II super-conductors, where the physics of vortices displays a rich overlap with the Activities of the Center. Moreover the high complexity of the field of strongly correlated quantum systems calls for stimulating interactions and for multidisciplinary approaches. The contact with adjacent fields with distinct but related problems surely provides a useful cross-fertilization for concepts and techniques. For instance, already inside the Center, the slow-dynamical phenomena and the effects of topological disorder are a common subject shared by the three Activities.

   Finally, a specific attention will also be devoted to new achievements of Material Science, particularly regarding novel materials and devices. In this particularly active field, the Center should play a relevant role in creating a rapidly reacting environment, where new instances and novelties are promptly discussed and elaborated.

Chaos, Fractals and Non-Equilibrium Critical Phenomena

Senior researchers: Enzo Marinari, Giorgio Parisi, Luciano Pietronero and Angelo Vulpiani.

3 PhD students, 6 Postdoctoral fellows and 6 coworkers.

1. Turbulence.

   A satisfactory theory of the small scale statistics of fully developed turbulence is one of the most challenging problem in theoretical physics with clear interest for many applicative issues, e.g. geophysics and engineering. Up to now the use of the dynamical systems approach to turbulence has been limited to low dimensional systems. For fluids under particular constraints, e.g. the Rayleigh-Benard convection in small cells, the success of a dynamical systems approach was undisputed. On the contrary fully developed turbulence, which implies spatial as well as temporal disorder, cannot be reduced to a low-dimensional system, and thus a large part of the theory of dynamical systems, in particular regarding bifurcation structures and symbolic dynamics, becomes basically inapplicable.

   Our main project is to extend to the treatment of the experimental signals and realistic models, the dynamical systems approach and some techniques introduced in information theory.

2. Transport and Diffusion.

   Transport processes play a crucial role in many geophysical flows with obvious interest for atmospheric and oceanic problems. The most natural framework for investigating such phenomena is to adopt a Lagrangian viewpoint in which the particles are advected by a given Eulerian velocity field. Despite its apparent simplicity the problem of connecting the Eulerian property of the velocity field to the Lagrangian properties of the trajectories is a very difficult task. In the last years the situation has become even more complex by the recognition of the ubiquity of Lagrangian chaos (chaotic advection). Even very simple Eulerian fields can generate very complex Lagrangian trajectories which are indistinguishable from those obtained in a turbulent flow.

   One of the main issues is: what does the knowledge of structure of the velocity field tell us about the diffusion of fluid particles? In particular, whether the diffusion is anomalous or not and, if it is normal, how to compute the diffusion coefficients.

   Up to now the scenario is not completely clear: in particular the effects of intermittency corrections on the Lagrangian statistics of advected particles are not well understood. Even in a context of simple phenomenological models, e.g. multi-fractal one, it is not trivial to generalize the anomalous scalings of the Richardson law for relative dispersion, which is valid for the K41 limit (i.e. neglecting intermittency).

   In addition in many cases, e.g. closed basins and open systems, the diffusion coefficients are not able to give a complete description of many interesting phenomena, as the spreading of pollutant initially confined in a small region.

   In non ideal cases it is necessary to take into account the effects of the boundaries and the non large ratio between the size of the domain and the typical length of the Eulerian field. Recently we treated the non asymptotic transport properties in terms of methods and techniques of the dynamical systems theory (i.e. finite size Lyapunov exponent and exit time approach). One of our goals is now to improve the method for the treatment of geophysical data.

3. Reacting Flows.

   Other phenomena strictly related to the transport are those in the advection-reaction-diffusion (ARD) systems. They have received in the last decades a rather natural attention due to their relevance for spatially extended ecological communities, mixing in reacting flows, environmental processes in atmosphere such as ozone reactions. The analytical treatment of these systems is not trivial at all: even in the limit of a turbulent velocity field there are no simplifications, due to the well known problem of the closure. Up to now there exists a huge literature about the pure reaction-diffusion equation, i.e. without the advection term. About the ARD only the limit cases (even if non trivial) have been studied, e.g the velocity field is assumed to be a random process in space and/or time.

   Our main aims are:

   • (i) the understanding of the relevance of the reaction part for the scaling law of the reactants at small and large scales;
   • (ii) to study the combined effects of the two parts, advection and reaction, of the dynamics (which can be chaotic or regular) on the complete problem;
   • (iii) the investigation of the possibility of a super-fast front propagation (i.e. the front position moves faster than linear in time).

4. Chaos and Complexity.

   Even in simple chaotic dynamical systems, the leading Lyapunov exponent is not sufficient to estimate the predictability time if one is interested in finite resolution. This fact is due the saturation of the error on the fast components of the system which therefore do not contribute to the exponential growth of the error at large values. A rather similar problem is present in information theory, where a generalization of the Shannon (or Kolmogorov-Sinai) entropy, the so called epsilon-entropy, has been introduced in order to characterize the information content of a signal observed with a given finite resolution.

   Recently we proposed to use a generalization of the Lyapunov exponent, the FSLE, which is based on the natural concept of error growing rate at finite error size. The method had been successfully applied to geophysical models, to describe the coherent dynamics of globally coupled maps showing macroscopic chaos, to systems with an uncertainty in the evolution law, and to the problem of transport in closed basins.

   The characterization of systems with a given resolution is particularly relevant if many different space and time scales are involved, as e.g. in fully developed turbulence. We are interested in a systematic classification of signal behaviors (without referring to any specific model) as stochastic or deterministic at a certain scale of the resolution according to the behavior of the epsilon-entropy and of the FSLE at varying the scale. In this framework we also intend to study systems with discrete states, e.g. cellular automata, and their continuous limit.

5. Granular Matter.

   In the past few years granular materials have become a fast growing field of research as witnessed by the creation of a brand new section of Physical Review E, European Journal of Physics E, and a new Journal entirely devoted to the subject.

   Granular materials are seldom in thermodynamic equilibrium, due to the presence of dissipative forces, and no ergodic principle has been stated until now for them. The construction of a coherent statistical mechanics for these systems is still matter is one of the aim of the first activity (some steps in this direction have already been done, e.g. the definition of a time dependent temperature).

   The present activity in this area is mainly focused on two particular complementary directions: the study of fundamental models for the so called ``granular gases'' and the study of the response properties in dense granular media as well as the possibility of defining a set of statistical ensembles suitable to describe their phenomenology.

   In granular gases inelastic particles are involved in fluid-like rapid dynamics, therefore the hydrodynamics approach seems to be the natural one. It is always assumed that the hydrodynamic fields for density, flow velocity and energy fluctuation (also called "granular temperature") are well defined and are subject to local balance laws. The results of numerical simulations, however, compared to the granular hydrodynamics predictions, show a disagreement in various aspects. One of them is the failing of the Navier-Stokes approximation: strong density instabilities like (density gradients growing on time scales faster than typical hydrodynamics scales) or inelastic collapse (the local divergence of collision rate so that an infinite number of collisions occurs in a finite time) have been observed in a cooling granular assembly, that is a granular gas losing his starting kinetic energy because of dissipative collisions.

   The scientific goals we intend to pursue can be summarized as follows:

   • (i) We intend to do more extensive calculations on the granular gas model with particular emphasis on the two dimensional case, where geometric packing effects come into play in a less trivial way.
   • (ii) In addition we plan to derive an 'equation of state' for granular matter connecting the energy dissipation to the density. Besides, to understand the emergence of non-Maxwellian behavior it is necessary to study the slow approach to the macroscopic steady state by employing a self-consistent kinetic approach. In this direction it is unavoidable to construct new ideal (numerical) experiments to understand how the classical thermodynamics concepts must be adapted to this class of open systems.
   • (iii) Another important point is the investigation of the limits of the Boltzmann ``Molecular Chaos'' assumption. It's an assumption of lack of correlations between colliding particles and cannot be accepted at high inelasticities, when a great number of collisions occur between highly clusterized particles trapped in a sort of solid phase, as experiments indicate.
   • (iv) Granular media are usually considered non-thermal systems. This means that, unless perturbed in some way (e.g. driving energy into the system), they can not explore spontaneously their phase space, remaining trapped in one of the numerous metastable configurations. One has then to look at the dynamics of a granular system always as a response to some perturbations and in general the response will depend in a non-trivial way on the rheological properties of the medium, on the boundaries, on the driving procedure and, last but not the least, on the past history of the system. Though some general aspects of this phenomenology have already been clarified, a deep understanding is still lacking and the construction of a thermodynamics for these systems is an open problem. In this respect we will investigate the possibility to define a set of statistical ensembles suitable to describe the phenomenology of granular matter and more generally of glassy systems, (see also the first activity for a complementary approach).

6. Fractals and Non-Equilibrium Critical Phenomena.

   In the last decades there has been a growing interest in the understanding a variety of scale invariant and critical phenomena occurring in nature. In scale invariant phenomena, events and information spread over a wide range of length and time scales, so that no matter what is the size of the scale considered one always observes surprisingly rich structures. New types of collective behavior arise and their understanding represents one of the most challenging areas in modern statistical physics. The fact that certain structures exhibit fractal and complex properties does not tell us why this happens. A crucial point to understand is therefore the origin of the general scale-invariance of natural phenomena. This would correspond to the understanding of the origin of fractal structures from the knowledge of the microscopic physical processes at the basis of these phenomena. Fractal geometry represents a mathematical framework which allows one the quantitative study of scale-invariant and complex systems. The application of these methods in various field leads to a re-analysis of known data that results in a revamping of long standing points of view. This widespread interdisciplinary corresponds to the fact that these new ideas allow us to look at natural phenomena in a radically new and original way, eventually leading to unifying concepts independently of the detailed structures of systems. The general properties of fractal structures have been investigated in detail in the past few years and the research is currently devoted understand the following problems:
   • (i) Structure formation in presence of long-range interactions represents one of the current most challenging issues in statistical mechanics. This topic is relevant for astrophysics (dynamics of galaxies interacting through gravity), plastic deformation (motion of interacting dislocations) in crystalline materials (motion of interacting dislocations), and vortices in a turbulent flow, despite the clear differences in the length scales involved. Nevertheless, it has been observed that galaxies and dislocations both form fractal structures.
   • (ii) Fractal growth models, based on a stochastic growth process in which the probability is defined through Laplace equation (e.g. Diffusion Limited Aggregation (DLA) and Dielectric Breakdown Model (DMB)) are currently investigated analytically and numerically. A related class of phenomena under active investigation is the driven dynamics of systems with quenched disorder. Example range from fluid invasion in porous media (invasion percolation), to interface depinning in a random environment (magnetic domain walls, flux lines or dislocations), and the fracture of brittle materials.
   • (iii) An interesting example of fractal growth is provided by network formation. In several different cases (river networks, veins, telephone, electricity or Internet webs) one observes structures that exhibit fractal properties and whose properties should optimize some a priori unknown cost function.

   The scientific goals we intend to pursue can be summarized as follows:

   • (i) We intend to investigate deeply the basic characteristics of gravitational instability. Namely, to confirm if gravity (or elasticity for dislocations) is the main ingredient for the formation of fractal structures. Next we will study viscoplastic deformation of single crystals, integrating numerically the over-damped dynamics of interacting dislocations, including as well the possibility of mutual annihilation and multiplication due to Frank-Read sources. We will then compare the structures obtained by numerical simulations of dislocation dynamics with electron micro-graphs of deformed metals, which show interesting fractal patterns. A similar analysis will be performed in the context of galaxies. We will also investigate the analogies and differences between vortices in a turbulent flow, dislocations in a crystals and galaxies. Finally we will develop statistical data analysis tools aimed at probing the emergence of scaling laws during the process of galaxy formation. To this end, methods and ideas from the field of statistical fluid turbulence, such as Extended self-similarity, will be applied.
   • (ii) While past activity in the field has been mainly devoted to understand theoretically fractal structures and non-equilibrium systems in relatively simple model systems, we aim now to strengthen the connections between models and experiments. This task will be pursued by analyzing available experimental data and by designing and studying realistic models. In particular, we will study the hysteretic behavior of soft magnetic material dominated by domain wall displacements, and the connection between hysteresis and relaxation in a weakly interacting collection of ferromagnetic grains. In addition, we want to understand the correlation between damage and acoustic emission in dynamic fracture by analyzing numerical simulations of lattice models and experiments on selected materials. Finally, in the context of fracture and earthquakes, we would like to investigate detectable correlations between precursor smaller seismic/rupture events and large catastrofic events, with the aim of developing forecasting methods.
   • (iii) In the framework of Internet networks, we intend to find a way to minimize the total number of steps one has to travel between two random points in the network by suitable drawn shortcuts (that would play the role of information highways) and characterize through multi-fractal analysis the traffic on the network. Finally we will study the structure of traffic due to Web requests/responses, collecting existing data and performing new direct measures. We want then to use a cluster of dedicated systems simulating the behavior of a number of concurrent Web users.

We will describe here the high risk (hopefully) high pay-off main seed activities that the Center will start.

A few general words to start with. In first we notice that we will only discuss here about seed activities that are at least partially under developments, where contacts with industries, institutions, groups of people have already been taken. Clearly there are many other possible initiatives that we hope will be developed in the Center life time span: we will come back to a discussion of the quick and effective response mechanism discussed in the instructions, that we highly appreciate and try to implement effectively.

It is also important to notice that seed activities, as defined in the call for proposals for INFM DRC's, are crucial in a research like the one that will be the basis of our Center. We believe we can produce theoretical results about the physical behavior of complex systems that are typically of large, immediate impact on a number of applications. Performing this second step, and leading our results to successful application, will be for us of overwhelming importance.

At last we notice that our research is interdisciplinary in nature, as we hope that for example our description of the Center activities clarifies: cross-fertilization from the different research activities will be satisfactory only when seed activities will have given the expected fruits.

We will discuss now few of our most promising activities (ranging from information technology to VLSI design).

On the Internet

The physical connectivity of the Internet network and the hyper-link structure between the web pages represent examples of complex structures whose properties belong to the area of scale invariant complex systems.

For the case of Internet one can for example consider the physical connections between users and providers. One of our projects, for example, will be based on modeling them as branches of a world spanning tree [I1]. These results have important scientific and technological implications, since we may describe Internet, as seen by a single user, as a stochastic Cayley tree which accounts for both qualitative and quantitative properties.

This question is not only of a scientific relevance, but it also addresses a very important technological question. Namely, whether it is possible to define a cost function to be minimized on the web in order to improve the net properties. This should allow both to plan future wiring of developing countries and to improve the quality of the net connection for countries already connected.

Since one is interested in minimizing the total number of steps between two random points of the network, a natural solution could be provided in the framework of the so-called small-world networks model [I2], recently introduced by Watts and Strogatz, where the geometrical properties of a net result from the coexistence of a local structure and random long-range connections.

Furthermore, since several investigations [I3-I5] put in evidence that also web pages follow a scale-free distribution, we believe that also the exploration of the web with the aim to design more powerful and effective search engines can be fruitfully achieved with the methods of statistical mechanics. At the moment those ensembles are only characterized by means of some statistical properties, as the average connectivity, the inward connection, the outward degree. We are planning to measure also the degree of connectivity of the web communities.

We have different contacts with industries that are interested in different developments of this research. For example a private company Internet venture is funding in this period one of our young coworkers with a six month fellowship for studying the content connectivity of the Internet tree: in this program we are developing simple searchers for getting an experimental handle over the real physical data of the Internet world. We intend to push this seed activity to try to make a real practical use of the information we will gather and (hopefully) understand.

Physics and Finance

Effective Global Optimization

Effective procedures for performing global optimization is of tantamount importance, and directly in the realm of Statistical Mechanical applications, and, more specifically, of the numerical simulations of complex system with a complex phase space (that when composed by many individual elements develop high barriers among different approximate solutions).

Annealing is the prototype of such a range of applications: a carefully thought cooling schedule allows to determine with a reasonable computing effort a good solution of a complex problem.

We have proposed and analyzed Optimized Monte Carlo Methods [GO1-GO2] (like for example the Tempering updating scheme), where one succeeds to build an effective numerical updating scheme that allows to get reliable statistical informations about very complex systems.

Methods like Tempering can be used as a global optimization scheme: one can look for solutions very close to the real minimum of the energy of the system. There are advantages over the annealing: for example the schedule is built in (in annealing determining an effective scheduling is frequently a cumbersome task).

We believe it will be worth to dedicate some of the resources of the Center to this kind of seed activity: with these resources we will try to solve some interesting problem of resource allocations, in collaboration with interested industries and/or institutions.

Vehicular Traffic Improvement

The study of vehicular traffic is a field where concepts and methods of statistical physics have a natural application. From the simple behavior of a large number of interacting individual agents (drivers) complex collective phenomena emerge, typical of systems out of equilibrium: phase-transitions, criticality, metastable states, etc.

In the past few years this connection has started to be explored and interest in the field is rapidly growing [VT1]. One line of research that will be investigated deals with the problem of traffic flow along a single highway. Approaches to this problem are both of "microscopic" (cellular automata) and of "coarse-grained" type (fluid dynamical description).

Another line of research will be the modelization of flow in road networks. This topic has received comparatively less attention in the past. We plan to extend the simple Biham-Middleton-Levine [VT2] model for road networks in order to incorporate more realistic features.

On a broader perspective, from the point of view of applications, this may be of help for the design of new roads or for an improved use of existing infrastructure. In this regard, contact with local traffic authorities have already occurred. We plan to proceed together with, among other, the administrative local authorities, to obtain practical and useful results.

Earthquakes

The goals of proposed research are:

(a) Detailed analysis of the earthquake catalogs from the point of view statistical physics.

(b) To develop and test new models for earthquake phenomena based on the new concepts of statistical physics like fractal structures, self-organization, criticality. The essential properties of earthquake dynamicst that should be reproduced by the theoretical models. The main properties under consideration are: (i) The Gutenberg-Richter law; (ii) The presence of aftershocks satisfying the Omori's law or its generalizations;(iii) Precursors effects; (iv) Space-time correlations.

(c) To define a scientific framework for the concepts at the basis of earthquake prediction research.

Extensive contacts and collaborations with national and international experts in the field of earthquakes have already been developed.

Other Seeds (Yet to Come)

A few words to go back to the quick and effective response issue. In a Center like the one we propose this is a crucial concept, and we will guarantee that our managing techniques will guarantee the flexibility and the control that will allow it. We work on many different frontier issues: we have in mind for example all the strong correlated research, discussed at length in a different section of this proposal. It is clear that these activities can, in different moments, give rise opportunities of important applications. In these cases we will try to have a structure ready to catch the occasion.

References

• [I1] G. Caldarelli, R. Marchetti and L. Pietronero, Europhys. Lett. 52 (2000) 386.
• [I2] D. J. Watts and S. H. Strogatz, Nature 393 (1998) 440.
• [I3] R. Albert, H. Jeong and A. L. Barabasi, Nature 401 (1999) 130.
• [I4] A. L. Barabasi and R. Albert, Science 286 (1999) 509.
• [I5] B. A. Huberman and L. A. Adamic, Nature 401 (1999) 131.
• [F1] E. Aurell, R. Baviera, O. Hammarlid, M. Serva and A. Vulpiani, cond-mat/9910212.
• [F2] R. Baviera, D. Vergni and A. Vulpiani, cond-mat/9903144.
• [GO1] E. Marinari and G. Parisi, Europhys. Lett. 19 (1992) 451.
• [GO2] E. Marinari, in Advances in Computer Simulation, edited by J. Kertesz and I. Kondor (Springer 1998), p. 50.
• [VT1] D. Chowdhury, L. Santen and A. Schadschneider, Phys. Rep. 329 (2000)199.
• [VT2] O. Biham, A. A. Middleton and D. Levine, Phys. Rev. A 46 (1992) 6124.

From "bottom" to "top", we will try to organize our activities in such a way to have an impact already on the Laurea students (and below), by putting them in direct contact with advanced research. On one side advanced introductory seminars will be presented during standard courses, on the other one dedicated tools will be developed to introduce young people, also high school students, to the beauty of research in physics. We plan to organize different initiatives in this directions: for example to develop computer programs with nice graphical interfaces which simulate physical interesting phenomena and to record some of the main conferences (and may be also small movies): we plan to dedicate some 2 year contracts to this specific goal in order to have produce products at an high professional level.

Doctoral students will be a crucial resource of the Center: we dedicate to doctoral fellowships (three per year in the first three years) a small part of our budget. These positions will be added to the positions we will get from other funding channels (the University "La Sapienza" first). We hope to increase our capacity of attracting PhD from outside Rome.
We note that our groups have lot of experience in the sector. We normally follow PhD students from our University, from different Universities in Italy, and we run, for example joint programs with Ecole Normale Superieure in Paris (programme de cotutele), where the student has two tutors, one French and one Italian, and he is working half time in France and half time in Italy.

An other activity that we plan to organize is a Doctoral School for students coming from outside Rome (we want to put efforts toward a global improvement of physics doctoral programs). This Doctoral School should held monographic courses for PhD students on subjects related to the research activity of the Center.
We plan to organize two sessions of two weeks of courses, one in June and the second in September, as an activity based in our Center. The Center will contribute to the stay and travel expenses of the students (and to support the speakers coming from out of Rome). These courses should be complementary to the other initiatives that are actually taken by the INFM in this direction.

In total we should have four courses every year, of 20 hours each, on different subjects in Statistical Mechanics (the subjects will change every year); some seminar by experimentalists is also planned as a necessary complement. The courses will be residential, and will bring young future researchers to interact in the Center for extended periods of time.
We will work on having these courses accepted as part of the requirements for the most part of Italian physics doctorate courses, but we also plan to have a large part of the attendance coming from abroad. A similar initiative, with a slightly different format has been done very successfully for more that ten years in France (in Bretagne) and it was attracting many students from different countries. Actually there is no doctorate school with these characteristics in Europe and our initiative will presumably fill an hole.

The training of postdoctoral fellows is one of the most important part of the scientific activity: in the first years after the PhD physicists start to perform research activities in an independent way and the presence of an stimulating environment may have a crucial effect on their future career.
In our proposal the presence of postdoctoral fellows is a crucial part of the Training and of the Research activities of a Center. Postdoctoral students coming from other universities and other countries will give an extremely important contribution to the research in the Center; they will also play a crucial role in strengthening our relations with the research done outside the Center and establish strong collaboration links.
We are sure that the activities we will run will be differentiated enough to make it possible to interact successfully with a large number of postdoctoral fellows (we are thinking about a total number close to 15: there are 10 senior investigators signing this proposal and at least 8 more junior and non junior faculty members that will work full time in the program). We have experience at that already now we interact with many postdoctoral fellow, that come to our Department with Italian MURST funds, with EEC funding, and with other sources of money (for example direct Spanish funding, French CNRS funding, Argentinean fellowships and similar sources).
We already receive many (a few tenths) requests from many young physicists to come to Rome after the PhD: in spite of the fact that many of them are very brilliant and promising, we have to dismiss most of them, due to the lack of flexible fundings.
However we believe the existence of the Center will be crucial not only for the new funding, but also and maybe mainly for the synergies it will be able to create; the visibility ad the coordination linked to the Center will be an important factor for the success of the career of many young, promising researchers (and on the way back such young people will give a crucial visibility and success to the Center).

One of the most important tasks of the center would be to organize "programs" and workshops on many hot subjects both in the area of the main activities and of the seed activities. These programs and workshops would fulfill many roles:

The Center, although established in Rome, should also be considered as national facility as far as the organization of the doctorate school, programs and workshops. We plan to integrate the whole Italian community working in the area of interest of the Center in the these activities, both for proposing the subjects and for the organization of these tasks.

This can be easily done because many of us have long standing collaborations with (and non-INFM) INFM scientists working in this field in many cities of Italy (for example in Firenze, L'Aquila, Messina, Napoli, Pisa, Trento, Trieste). These collaborations will be certainly be facilitated and enhanced by the establishment of the center.

The groups proposing the Center are also very well connected with the research done at an international level. We have many collaborations with scientist abroad and these collaborations has also been at the origine of the fact that we have been (or are) part of a few European networks. Moreover many of us had important positions in the organization of the research outside Italy (scientific Committee of laboratories or bigger institutions, editorial boards, members of the advisory committee of important international scientific meetings...).

The Center itself plans to establish links of various nature with other centers or laboratories working on similar projects in the world. We have asked for letters of support and to propose possible collaborations to the directors or to some of the key scientists of many scientific centers. We get a very positive response from all the people to which we have written, i.e. from professor Comptet (University of Orsay and Institute Henri Poicaré), professor Herz and professor Hoyer (Nordita), Professor Moffat (Newton Institute), Professor Gross (Santa Barbara), Professor Newman (Santa Fe).

All of them where considering with high interest the establishment of the center in Rome and look forward the possibility of some form of collaboration. Some of them enter in more details on the merit of the proposal and present interesting observations.

In particular Professor Comptet makes a very penetrating remark "It is also worth pointing out the breath of the proposal. The center is not limited to some fashionable but narrow subject. It has the ambition to gather people coming form different fields and to enhance synergies and interactions between them. One may expect very fruitful exchanges and confrontations. Indeed it is often in unexpected directions that existing theoretical tools find their best applications. (...) This future center holds many of the winning cards to become a major reference point in Europe."
This is one of the main points which we had in mind when we proposed this Center. Interdisciplinary, cross-fertilization form one branch to an other branch of physics have been some of the major forces that have reshaped theoretical physics in the last thirty years. The theory of second order phase transitions (for which Kenneth Wilson got the Nobel prize and to which some of us gave an important contribution) is an a very nice example of this fact.

The three main activities of the proposed Center have many points in common both at the technical level and on a more general grounds.

We will describe here our plans for managing and administering the Center. We will try to detail as precisely as possible the procedures we have plane to implement, since we believe that a careful managing is one of the keys needed for the success of our enterprise.

The Director of the Center will have the responsibility of running the Center, under the guidelines of the Advisory Scientific Committee (ASC), that will have a very prevailing role. Such an Advisory Scientific Committee will draw the main Scientific lines that the Center will follow, will verify that such lines have been properly implemented, will judge the validity of the different Center programs (research lines, workshop, educational activities, outreach). We will demand to the members of the ASC an active involvement in the Center: we believe that their contribution will be one of the crucial keys of the Center success.

The ASC will have 12 members: at least 6 of them will not be from National (i.e. Italian) institutions, but will be selected from Universities, Research Centers and institutions abroad. It will be selected as follows: the Director of the Center will propose a list of potential members of the ASC to the INFM Scientific Council. The INFM Scientific Council will nominate the ASC after modifying, integrating, approving the list.

The ASC will meet once every year. It will receive from the Director with due advance before the scheduled meeting all the details needed for an accurate and fair evaluation of the Center. After each meeting the ASC will prepare a written report that will be send to the INFM President and Scientific Council for evaluation. This report will have a large importance in the analysis, development, follow up of the Center different programs.

The ASC members will be renewed starting from the third year of life of the Center: we have in mind that more or less one third of the members of the ASC will be renewed starting from the third year. This should take care of the physiological need for rest and change of people, and should allow to keep the ASC very efficient.

The Center Director, that will have the responsibility of the daily running of the Center, according to the lines indicated by the ASC, will be assisted by an Executive Board (EB). The EB will be composed by the Director and by four more scientists of the Center, nominated by the Director. Three of them will be delegated to follow the three main activities of the Center (one scientist per activity). The members of the EB will be nominate for two renewable years.

Together with the Center Director the EB will work on implementing the guidelines given by the ASC. The Director will be supported by the EB in selecting the individual responsible for the programs ran by the Center.

Normally the Center programs will be lead by two responsible scientists, one from the Center and one external.

Activities of the Center will be reported on the Web with frequent updates about the different programs. The Center will produce an annual report that will assist the ASC in the task of preparing the annual review of the Center achievements. At the end of each project there will be a report that will summarize the activities of the project: different kinds of projects will probably have to use different formats, depending on the specific task. All the reports will also be submitted to the INFM Scientific Council, that will be able in this way to stay in closer touch with the Center developments.

Seed activities, because of their special structure, will get a strong level of care and attention: they will also need special mechanisms to be selected, evaluated, supported and valued (we will discuss later on about general mechanisms for evaluation and quality control in the Center). As soon as the possibility of a seed activity will spur the Director, together with the EB, will take responsibility for deciding if an action has to be taken: she or he will get all the needed information, and in case of a positive decision will give to the seed activity all the tools needed for a start. All seed activities will have to report in detail, for example every 6 months, about their progresses: on the basis of these reports the Director and the EB will decide about the way in which the seed activity has to progress. Eventually the ASC will get all the informations needed to express an informed advice and decision about the seed activity. We believe this mechanism guarantees on one side a fast reaction of the Center to new opportunities, while preserving the guarantee of high quality given by the a posteriori control of the ASC.

The main administration of the Center will be run by the INFM Unit of Roma La Sapienza. There will be obviously a need for added person-power (there is in this moment one person taking care about a very large unit like the one of Roma La Sapienza, and it is unthinkable to add all the work of such a Center on a single person). We believe one full time person will work inside the INFM Unit structure to take care of the Center: heavy programs demanding more assistance will use more part time personnel where needed.

Now we give a few more details about the procedures, criteria and mechanisms that will be used to select and evaluating projects. Our Center will be involved in research in theoretical physics, and that will involve some peculiarities in the selection mechanisms. Obviously there will be fundamental research (organized in the three activities we have discussed) that will proceed at its own pace: the ASC will be able to report about the achievements of these researches, and to suggest mechanisms to effectively improve its efficiency. The Center programs will have to be selected with good care, and the mechanism used during the process of selection will have to be tuned with care, since they will be very important: also here we consider the role of the ASC crucial. The large amount of feedback we will require from the projects will be a guarantee of the possibility of a serious review of the work done. To summarize, we believe we will be focusing fast and on the right spots because we will have mechanisms to perform fast choices (thanks to the Director supported by the EB): we believe we will be able to have an effective mechanism of evaluation and redirection of our efforts (when needed) thanks to the crucial role of the ASC.

Our administration will also have to cowork with other cofunding institutions (University, MURST, EEC, ESF, private industry): we are confident we will be able to set up viable mechanisms to make this interaction working smoothly. The same kind of problems will exist for activities like educational programs and outreach: again, we believe that a strong interaction with existing INFM administrative structures, together with a high level of administrative competence in the Center, will allow to solve all technical problems.

The Center will be hosted in Department of Physics of the University of Rome La Sapienza.

When the Center will be created it will be possible to allocate right ahead at least 300 square meters of space in the Physics Depatment: this spaces will be clearly labelled with the Center name and with the INFM logo. This will allow us to start to be effective and visible from the first moment. We will also have extra space to host the programs.

Indeed the University of Rome and the Physics Department are strongly supporting this initiative.

The Head of the Physics Department, professor Francesco Guerra, has taken precise commitments about space and other forms of technical support. Among the letters of support there is his letter (also at http://chimera.roma1.infn.it/CDE/LETTERE/Department_Head.html). We quote from his letter:

1. The Department will be able to provide at least a space of the order of 300 square meters inside the Department (in addition to the office space normal allocated to faculty members) to the research groups that are building the Center.
2. New office space (located at less than one kilometer from the "Citta' Universitaria") will be allocated to the Center for organizing the "programs" that will be one of the Center focus: if for example 15 visitors will join a 3 months workshop they will be able to use this office space. This office space (located close to the "Citta' Universitaria") could be increased in the next future (...) and the Center will be able to expand, if needed, its target.
3. We will also, obviously, guarantee office space to all the people that will be hired in connection to the Center (from INFM, our Department or others).
4. We will also be able to commit, as needed, work by some of our technicians (...) and support by our computer center, that takes care about networking, geographical connections and is able to host, for example, among other, clusters of parallel computer.
5. Our Department will do everything to make the Center a big success.

Our university also strongly support this proposal. The Dean of our university, professor D'Ascenzo (see the statement of Dean at http://chimera.roma1.infn.it/CDE/LETTERE/dean.jpg ) has written that "the University of Rome will give to the Center all the space it needs. In particular, beyond the space given by the Department of Physics, it will make available other space in prestigious buildings near the main Campus".

It is also possible that eventually we will be able to move some of our activities in the Centro Studi e Ricerche , located in the historical buildings of Via Panisperna, which will operate in collaboration with INFM and INFN. Indeed the Parliament has approved (in 1999) a law according to which the historical building of Via Panisperna (where Fermi and the other scientists of the via Panisperna gang used to work) will be dedicated again to host research activities. However it is not clear how fast the Centro will start its activities. We are following the problem very closely: two of us are in the Administrative Council of the Centro. In any case we have organized the structure of the SMaC Center independently from this possibility.

Other international institutions outside Italy are interested to the collaboration with our Center. Among the letters of support there are the letters of professor Comptet (University of Orsay and Institute Henri Poicaré), professor Hertz and professor Hoyer (Nordita), Professor Moffat (Newton Institute), Professor Gross (Santa Barbara), Professor Newman (Santa Fe).

We have no doubts that the Centro Interdisciplinare dell'Accademia dei Lincei is interested to co-organize courses for high school teachers in physics. One of us is in the Scientific committee of the Centro Interdisciplinare and this point was already discussed with the President, professor Carra'.

Most of us are involved in different types of projects and receive funding from various sources. The scientific programs of these projects are within the scope of the center and the funds allocated to these programs can be considered matching funds for the center. At the present moment the main sources of money are

1. Fondi grandi Attrezzature (big apparatus) from the University of Rome (coordinator Parisi).
2. The PRA HOP (Coordinator Tartaglia), Physics of deep supercooled systems. The hopping regime between the critical and the glass temperature.
3. The PRA-TURBO (Coordinator Vulpiani).
4. (National Coordinator Pietronero)
5. COFIN 2000:
   1. (National Coordinator Pietronero) Statistical Physics and Condensed Matter Theory.
   2. (National Coordinator Parisi) Complex Problems in Statistical Mechanics and Field Theory: a Theoretical Study Based on Analytical and Computational Approaches
   3. (Coordinator Coniglio, local coordinator Tartaglia) Connectivity and slow dynamics in complex systems.
   4. (Coordinator Viliani, local coordinator Sciortino) Vibrations and relaxations in disordered condensed systems. Experimental, theoretical and numerical studies.
6. MAE project on: "Breakdown and Critical Geophysical Phenomena" within the Convention of scientific collabotration between Italy and Russia. Italian Team: Group of L. Pietronero, Russian Team: Russian Academy of Sciences: International Institute of Earthquake Prediction and Mathematical Geophysics 1999-2000. Possible extension to 2001.
7. The European Network (General coordinator) on: Fractal Structures and Self-Organization (see http://pil.phys.uniroma1.it/eec1.html).
8. The European Science Foundation network SPHYNX (local coordinator Parisi)

These matching funds may be used for different aims (depending on the source): computer facilities like work stations or parallel computers, contributions to programs and workshops, phD fellowships and so on.

As theoretician working in a field where large scale computer simulations play an important role, the main facilities are computers. Apart from a large number of personal workstation, we have four parallel clusters of four Alpha clusters and we are in the process of assembling a parallel cluster with 10 Pentium based biprocessor computers (which will expand up to 40 computers). We have also access to some of the parallel computers of the INFN APE project. We also have (and plan to develop inside the center scope) collaboration with scientists from other european countries, that allow us to receive (even substantial) allocations on computer time on foreign supercomputers. We plan to extend this activity involving different research lines in the Center.

In the past we have been able to gather funds from different sources, both at the local level, at the national level and the European level.

1. In the last four years people we have been three times recipients of Fondi grandi Attrezzature (big apparatus) from the University of Rome.
2. Each of the three activities has obtain money form the Italian COFIN at least twice: eight different projects approved in total. We have received fundings also from two INFM PRA.
3. The total number of european network in which we have been involved (counting past and present) is five. Beyond the networks the European community has financed five PhD fellowships (foreign PhD coming to Rome). We have also be part of an exchange program among Argentina European community. The European Science Foundation is financing the SPHYNX (Statistical Physics of Glassy and Non-equilibrium Systems) network, proposed by Parisi, Sourlas and Sherrington.

In the future the establishment of the Center and the related synergies will help us in going on with this capacity of attracting funds at different levels.

We plan to go on with fund raising from different sources. In particular we have already started discussions at the European level in order to organize two networks for the next round of proposals in the fields of research covered by the Center: one of the two proposed networks shall have the coordinator from Rome (Parisi).

Let us be precise on the way in which we understand the person-months commitments. We have funding from different sources, and, if the Center will be approved, in the next five years, we will continue looking for non-INFM funding to use in the Center: for doing that we will use person-months in different programs (MURST, EEC and others). We cannot know right now how successfull we will be and how many person-months we will be able to allocate. What we mean by indicating a 60 months in 5 years commitment for all investigators involved in the Center is that all the funding we will receive in these 5 years will be dedicated to activities connected to the Center, not that we will not allocate person-months to request additional funding (that would be absurd, and against the idea of a Center that also works hard to receive matching funds from external sources).

Let us discuss the details of our budget. We will see that, again, the main issues are education, workshops, the doctoral school, and the Center programs.

Part of the budget is allocated directly to our three different activities. We have allocated to the three different activities an average sum of 400 ML per year. A small part of this amount will be dedicated to hardware purchase (computers), and to maintenance and perishable material (the most part of these items will be acquired through independent sources of funding). The most part of this amount will be dedicate to running the programs (as we already discussed) and to inviting long term visitors. We also plan to organize a Conference to start the Center and one after five years. Part of the funding needed for these two Conferences will come from University funding. Activities will probably have also to cover some parts of administrative costs (when organizing programs or running complex activities) from their budget.

We have given only the total amount of money that we plan to dedicate to cold start the seed activities, and we have not divided the sum among the different seed activities. We believe that this flexibility is needed to assign the money to the seed activities that will really need it (for details about the importance of an effective mechanism for implementing both fast reactions and quality control see also the chapter on management). On general grounds the amount of money requested for seed activities is quite low. Indeed we are asking only 50 ML per year: the reason for which we plan to use a relatively small amount of money for seed activities is mainly that they should fund them self mainly by a direct relation with the outside world. Among others the money allocated will typically be used for organizing workshops related to the seed activity, and for giving short-term fellowships to investigate detailed issues related to the seed activity.

Education and human resources is the main budget entry. We will fund at least three PhD scholarships per year, for the first three years (at a rate of 25 ML per scholarship). We plan to hire of the order of 7 postdoctoral fellows per year, at a rate (including traveling) of 60 ML per year. We are confident that at least five additional postdoctoral positions per year will be funded from other sources. Because of the scheduled timings of the Center creation we demand for the first year only 4 PostDoctoral fellowships (we believe that in April it would be difficult to find more than 4 good PostDoctoral fellows). The amount allocated (here and in the other entries of the budget) is meant in average, and includes an expected devaluation of the order of 1.5 percent per year.

Always in the education budget the organization of the Statistical Mechanics School will require 75 ML per year, that will be used to contribute to the stay and travel expenses of the students (and to support the speakers coming from out of Rome).

The funds we have destinated to the outreach will help in enhancing the visibility of our Center and the quality of its impact. We plan among other to organize classes for formation for high school teachers (quite likely together with the Accademia dei Lincei), and to set up software for having seminars and colloquia organized in the Center on line on the net (we plan to have a set of video-transparencies on-line shows). We also see the possibility of setting up a CD with effective didactic programs, problems and games.

The funding for Administration includes one full time person that will be running the Center together with the INFM Roma Unit (this job could be for example based on a 5 year contract). We will also hire on part-time basis the personnel needed for specific issues related to workshop and program organization. Perishable material will be charged to individual activities.

When discussing about deliverables and milestones we like to start from a simple, basic, introductory issue: our research is a theoretical one, and we expect to produce a constant flux of publications on high prestige international reviews (e.g J. Phys. A, Nucl. Phys. B, Europhys. Lett., Phys. Rev. Lett.). Before publication preprints will be sent to the Los Alamos archive (with a mirror in Italy at Sissa).

The evaluation commissions, independently from a direct scientific analysis of the obtained results, may get information of the judgment of the scientific community on our work by monitoring the flux of these publications and making an analysis of the impact factor of the reviews used (which are based on the anonymous referees system). We plan to present our results to international meetings in a systematic way. Obviously it will be crucial that all the different problems studied will bring to important results and will be published on international reviews. It will be also interesting (for the evaluation commission) to monitor the presence of our papers on the citation index.

Let us go now in some more detail. We believe our milestones should be set yearly: the work of the ASC will be crucial in this respect. The ASC will indeed yearly analyze and evaluate our scientific production, programs, schools, workshops and all of our activities. We see indeed the ASC as the monitoring, controlling body, under the direct influence of the INFM SC.

As far as publications are concerned we clearly suggest one will look at their number, at the level of the journals where they will be published (impact factor or more sophisticated patterns of measurements), and to the number of citations they will receive.

Our School should be evaluated from the number and the quality of both students and lecturers: we will probably ask students to fill forms that can be used to check the final level of satisfaction.

Conferences will also be evaluated from the level of the speakers, and from the interest of the topics discussed.

Evaluation of programs will be crucial, since the programs will be a crucial part of the Center activities. We believe that the ASC will have to give to this issue a lot of attention, since its reports will be crucial in allowing the SC of INFM to give an informed assessment. The problem with evaluating programs will be that they will be formed by different activities (like workshops, residential meetings, schools, conferences): the two organizers of each program will be asked to write a summarizing report, that will be the starting point of the assessment.

Seed activities, as we have already discussed, will require a special track both for being started (they will frequently need a sudden start) and for being evaluated. Here deliverables can be very specific, depending on the seed activity: sometimes we will produce computer codes (it would be nice, in the vehicular traffic activity, to have a code making the crossing of Piazza Venezia fast and easy), sometimes set of procedures or applied knowledge, coming directly from first line theoretical developments.

So, to be very clear, a first milestone could be that after the first year already the doctoral school is in place, workshops are already running, programs are already functioning. These milestones will repeat every year, demanding a higher level of performances.

Ultimate success or failure of the Center will depend very much, we believe, on how we will be able to exploit the synergies on which we are now betting. One should take what has been done from the Center, subtract what would have probably been done without it, and check what is left. We hope and believe it will be a lot, but, again, this is a bet. In more practical terms that will mean to check if in terms of published work, of education, of helping young researchers to mature, of outreach, of seed activities, of having people meeting and working together, the interaction of the groups that are founding the Center will have given the large amount of added value we expect. Again, we are proposing to create this Center because the INFM call arrived in a situation where we were constantly talking about something of this kind. Somehow we were feeling we were over a critical threshold (one could not develop a Center without putting together differentiated forces) but at the same time with a consistent intellectual path (Statistical Mechanics and Complexity, as we say in the title). We believe that checking if this value will be present in the work we will have done in five years will be indeed the sign of ultimate success or, sadly, of ultimate failure.

INFM B1

Born in Rome, 4/8/1948. He graduated in Rome in 1970, under the the supervision of N. Cabibbo. He has worked as a researcher at the Laboratori Nazionali di Frascati from 1971 to 1981. In this period he has been on leave of absence from Frascati at the Columbia University (1973-1974), at the Institute des Hautes Etudes Scientifiques (IHES) (1976-1977) and at the Ecole Normale Superieure, Paris (1977-1978).

He became full professor at Rome University in 1981. From 1981 to 1992 full professor of Theoretical Physics at Tor Vergata University. He is now professor of Quantum Theories at the University of Rome I, La Sapienza. He received the Feltrinelli Prize for physics from the Academia dei Lincei in 1986, the Boltzmann medal in 1992, the Italgas Prize in 1993, the Dirac Medal and Prize in 1999. In 1992 fellow of the Accademia dei Lincei; also fellow of the French Academy from 1993 and of the Accademia dei Quaranta from 2000.

He is (or he has been) member of the scientific committees of the IHES, of the Ecole Normale Superieure, of the Scuola Normale, of the Human Frontiers Science Program Organization, of scientific committee of the INFM and of the French National Research Panel and head of the Italian delegation at the IUPAP.

Research Activity

He has written about 350 scientific publications on reviews and about 50 contributions to congresses or schools. His main activity has been in the field of elementary particles, theory of phase transitions and statistical mechanics, mathematical physics and string theory,disorded systems (spin glasses and complex systems), neural networks, theoretical immunology, computers and very large scale simulations of QCD, non equilibrium statistical physics.

He has also written three books: Statistical Field Theory, (Addison Wesley, New York, 1988), Spin glass theory and beyond (Word Scientific, Singapore, 1988), in collaboration with M. Mezard and M. A. Virasoro and Field Theory, Disorder and Simulations (Word Scientific, Singapore, 1992).

Some of the Research Interests and Achievements

INFM B1

INFM B1

Personal Data

• Place and date of Birth: Roma, May 3rd, 1955.
  
• Citizenship: Italian.
  
• e-mail: giovanni.bachelet@roma1.infn.it
  

Academic Activity

• 1979: Laurea in Fisica cum Laude at the University of Rome "La Sapienza".
  
• 1979-1981: Post-Doctoral member of the Technical Staff, Bell Labs,
  Murray Hill, NJ (USA).
  
• 1982: visiting scientist at Scuola Normale Superiore, Pisa, Italy.
  
• 1982-1984: research contract at Max-Planck-Institut FKF, Stuttgart,
  Germany.
  
• 1984-1988: permanent CNR research staff member at the "Centro Stati
  Aggregati" of Povo, Trento, Italy.
  
• 1988-1991: Associate Professor of Condensed Matter Physics at the
  University of Trento, Italy
  
• 1991-2000: Associate Professor of Condensed Matter Physics at the
  University of Roma "La Sapienza".
  
• 2000: I just became Full Professor of Condensed Matter Physics at the
  University of Roma "La Sapienza".
  

Fields of Research

Organizational Activity

Schools, Conferences and Workshops

Selected Publications

1. G. B. Bachelet, P. Gori-Giorgi and F. Sacchetti,
   Analytic Structure Factors and Pair-Correlation Functions
   for the Unpolarized Homogeneous Electron Gas,
   Phys. Rev. B 61 (2000) 7353.
   
2. G. B. Bachelet, A. C. Cosentini, M. Capone and L. Guidoni,
   Phase Separation in the 2D Hubbard Model: a Fixed-Node Quantum
   Monte Carlo Study,
   Phys. Rev. B 58 (1998) R14685.
   
3. G. B. Bachelet, A. Malatesta and S. Fahy,
   Variational Quantum Monte Carlo Calculations of the
   Cohesive Properties of Cubic Boron Nitride,
   Phys. Rev. B 56 (1997) 12201.
   
4. P. Alippi, G. B. Bachelet and P. La Rocca,
   Alkali-Metal Plasmons, Pseudo-Potentials, and Optical Sum Rules,
   Phys. Rev. B 55 (1997) 13835.
   
5. G. B. Bachelet, A. Filippetti, D. Vanderbilt,
   W. Zhong and Y. Cai,
   Chemical Hardness, Linear Response, and
   Pseudo-Potential Transferability,
   Phys. Rev. B 52 (1995) 11793.
   
6. G. B. Bachelet, P. Focher, A. Lastri and M. Covi,
   Pseudo-Potentials and Physical Ions,
   Phys. Rev. B 44 (1991) 8486.
   
7. G. B. Bachelet, D. M. Ceperley and M. Chiocchetti
   Novel Pseudo-Hamiltonian for Quantum Monte Carlo,
   Phys. Rev. Lett. 62 (1989) 2088.
   
8. G. B. Bachelet and N. E. Christensen
   Relativistic and Core Relaxation Effects on the Energy Bands
   of Gallium Arsenide and Germanium,
   Phys. Rev. B 31 (1985) 879.
   
9. G. B. Bachelet, M. Scheffler and J. P. Vigneron,
   Tractable Approach for Calculating Lattice Distortions
   around Simple Defects in Semiconductors:
   Application to the Single Ge Donor in GaP,
   Phys. Rev. Lett. 49 (1982) 1765.
   
10. G. B. Bachelet, D. R. Hamann and M. Schlüter ,
    Pseudopotentials that Work: from Hydrogen to Plutonium,
    Phys. Rev. B 26 (1982) 4199.
    

INFM B1

Personal Data

• Place and Date of Birth: Roma, February 3rd, 1948
  
• Citizenship: Italian.
  
• e-mail: claudio.castellani@roma1.infn.it
  

Academic Activity

• July 1972: Laurea in Fisica cum Laude, Universita' di Roma "La Sapienza"
  
• 1973 - 1974: Temporary Collaborator at the Laboratori di Fisica
  Nucleare Applicata del CNEN-Casaccia and Laboratori INFN-Frascati
  
• 1975 - 1976: Research fellow at the Department. of Physics of the
  University of Rome "La Sapienza".
  
• 1977 - 1985: Research collaborator at the Dept. of Physics of the
  University of Rome "La Sapienza".
  
• 1976 - 1982: Assistant Professor of Mathematical Methods in
  Physics at the University of L'Aquila (Italy).
  
• 1983 - 1985: Associate Professor of Mathematical Methods in
  Physics at the University of L'Aquila (Italy).
  
• 1986 - 1994: Associate Professor of Physics of Metals at the
  University of Rome "La Sapienza".
  
• Since 1994: Full Professor of Physics of Metals at the University
  of Rome "La Sapienza".
  
• Present Position: Professor of Low-Temperature Physics at the
  University of Rome "La Sapienza".
  

Fields of Research

Schools, Conferences and Workshops

Selected Publications

1. M. Fabrizio and C. Castellani,
   Anderson Localization in Bipartite Lattices,
   Nucl. Phys. B 583 (2000) 542.
   
2. A. Perali, C. Castellani, C. Di Castro, M. Grilli,
   E. Piegari and A. Varlamov,
   Two-Gap Model for Under-doped Cuprate Super-conductors,
   Phys. Rev. B 62 (2000) R9295.
   
3. M. Leadbeater, R. Raimondi, P. Schwab and C. Castellani,
   Non Linear Conductivity and Quantum Interference in
   Disordered Metals.
   Eur. Phys. J. B (Rapid Note) 7 (1999) 175.
   
4. A. Gamba, M. Grilli and C. Castellani,
   Renormalization Group Analysis of the Quantum Non-Linear Sigma Model with
   a Damping Term,
   Nucl. Phys. 556 (1999) 463.
   
5. C. Castellani, C. Di Castro and P. A. Lee,
   Metallic Phase and Metal-Insulator Transition in Two-Dimensional
   Electronic Systems,
   Phys. Rev. B 57 (1998) R9381.
   
6. W. Metzner, C. Castellani and C. Di Castro,
   Fermi Systems with Strong Forward Scattering,
   Adv. in Phys. 47 (1998) 317.
   
7. C. Castellani, C. Di Castro and M. Grilli,
   Non-Fermi Liquid Behavior and d-Wave Super-conductivity near the
   Charge-Density-Wave Quantum Critical Point,
   Z. Phys. B 103 (1997) 137.
   
8. C. Castellani, C. Di Castro, F. Pistolesi and G. Strinati,
   Infrared Behavior of Interacting Bosons at Zero Temperature,
   Phys. Rev. Lett. 79 (1997) 1612.
   
9. A. Perali, C. Castellani, C. Di Castro and M. Grilli,
   d-Wave Super-conductivity near Charge Instabilities,
   Phys. Rev. B 54 (1696) 16216.
   
10. C. Castellani, C. Di Castro and M. Grilli,
    Singular Quasiparticle Scattering in the Proximity of
    Charge Instabilities,
    Phys. Rev. Lett. 75 (1995) 4650.
    

INFM B1

Personal Data

Place and date of birth: Rome July 6th 1939.

Present position:
Full Professor of Optics at the University of Rome "La Sapienza".

Academic Curriculum

• 1962: Graduated in Physics, laurea in Fisica, at the University of Rome
  "La Sapienza".
  
  
• 1966: Assistant Professor of Physics at the Engineering Faculty of the
  University of Rome "La Sapienza".
  
• 1981: Associated Professor of "Fundamentals of Quantum Mechanics" at
  the Engineering Faculty of the University of Rome "La Sapienza".
  
• 1986: Full Professor of Molecular Physics at the University of
  Rome "La Sapienza".
  
  
• 1996: Full Professor of Optics at the Engineering Faculty of the
  University of Rome "La Sapienza".
  

Short periods of research activity abroad in Soviet Union
J. I. N. R. Dubna, Grenoble C. N. R. S. , Spain University of Iles Baleares,
Varsavia Institute of Chemical Physics of Academy of Science.

Field of Research

• - Many body theory, super-fluidity and super-conductivity,
  itinerant electrons Magnetism (1962-1969);
  
• - Critical phenomena, phase transitions and the field
  renormalization theory y applied to condensed matter (1969-1975);
  
  
• - Nonlinear relaxation in the decay from an unstable
  state. Transient behavior of laser radiation statistics, spontaneous
  order growth in magnetic systems and spin glasses (1975 up to now);
  
• - Electron phonon interaction in strong and intermediate
  coupling regime. Quantum coherence phenomena.
  

Schools, Conferences and Workshops

Has lectured for many years the postgraduated school of physics at
the University of Rome and in some International Schools. Has been
invited at workshops and international conferences.

Organizing Activity

He organized the scientific activity of a C. N. R. group (Proprieta'
Collettive). He has been the coordinator of the postgraduate school
in physics at the University of L'Aquila.

Selected Publications

1. S. Ciuchi and F. de Pasquale,
   The charge ordered state from weak to strong coupling,
   Phys. Rev. B 59, 5431(1999).
   
2. S. Ciuchi, F. de Pasquale, S. Fratini and D. Feinberg,
   Dynamical mean-field theory of the small polaron,
   Phys. Rev. B 56, 4494 (1997).
   
3. S. Ciuchi, F. de Pasquale and D. Feinberg,
   Exact solution of the small polaron problem in infinite dimensions,
   Europhys. Lett. 30, 151 (1995).
   
4. S. Ciuchi, F. de Pasquale, C. Masciovecchio and D. Feinberg,
   Super-conductivity and density waves in high dimensions,
   Europhys. Lett. 24, 575 (1993).
   
5. S. Ciuchi and F. de Pasquale,
   Non linear relaxation and ergodicity breakdown
   in random anisotropy spin glasses,
   Nucl. Phys. B 300 [FS22] (1988).
   

INFM B1

Personal Data

• Place and date of Birth: Roma, August 14th, 1937.
  
• Citizenship: Italian.
  
• e-mail: carlo.dicastro@roma1.infn.it
  

Academic Activity

• 1961: Laurea in Fisica cum Laude at the University of Rome "La Sapienza".
  
• 1964: Ph. D. in Mathematical Physics - Birmingham University (UK)
  
• 1970: "Libero docente in Fisica Teorica", University of Rome "La
  Sapienza".
  
• Has been attached to the Physics Department of the University of
  Rome since graduation (INFN researcher up to 1969, associate
  professor(1969-1976), full professor since 1978), except for a period
  of military service (1964-65), several short periods of research
  activity abroad (Rice University Houston, CNRS Grenoble, Queen Mary
  College London, Central Research Institute Budapest, MIT Cambridge,
  Gakushuin University Tokyo, ETH Zurich, KTH Stockholm, . . . ) and two
  years (1976-1978) as full professor in Condensed Matter Physics at the
  University of L'Aquila, where he has also served as Dean of the
  Faculty of Sciences.
  
• Present position: Professor of Statistical Mechanics at the
  University of Rome "La Sapienza"; "Socio corrispondente" of the
  "Accademia dei Lincei"
  

Fields of Research

• Many-body theory, super-fluidity and super-conductivity (1964-1969);
  
• Static and dynamic critical phenomena - Phase transition and the
  renormalization group applied to condensed-matter physics (1969-80)
  (together with G. Jona Lasinio he was one of the initiators of this
  last topic).
  
• Renormalization group in quantum systems - Strongly
  correlated electron systems and Mott insulator. Disordered systems and
  Anderson localization. Interaction effects in disordered electronic
  systems: the metal-insulator transition and transport
  phenomena(1980-89).
  
• High Tc super-conductivity and strongly
  correlated electron systems. Non-Fermi liquid metallic phases
  (1988-2000).
  

Organizational Activity

• He has organized and served as a member of the Program and
  Advisory Committee of several international conferences.
  
• Has been the coordinator of national and Europeans research
  projects and of the postgraduate school in physics at the University
  of Rome for several years.
  
• Member of the IUPAP C10 commission (1996-1999). Editor of volumes
  of proceedings.
  

Schools, Conferences and Workshops

Selected Publications

1. L. Benfatto, S. Caprara and C. Di Castro,
   Gap and Pseudo-gap Evolution within the Charge-Ordering Scenario
   for Super-conducting Cuprates,
   Eur. Phys. J. B 17 (2000) 95.
   
2. A. Perali, C. Castellani, C. Di Castro, M. Grilli,
   E. Piegari and A. Varlamov,
   Two-Gap Model for Under-Doped Cuprate Super-Conductors,
   Phys. Rev. B 62 (2000) R9295.
   
3. G. Seibold, F. Becca, F. Bucci, C. Castellani,
   C. Di Castro and M. Grilli,
   Spectral Properties of Incommensurate Charge-Density Wave Systems,
   Eur. Phys. J. B 13 (2000) 87.
   
4. S. Caprara, M. Sulpizi, A. Bianconi, C. Di Castro and M. Grilli,
   Single-Particle Properties of a Model for Coexisting Charge
   and Spin Quasi-Critical Fluctuations Coupled to Electrons,
   Phys. Rev ()B 59 (1999) 14980.
   
5. S. De Palo, C. Castellani, C. Di Castro and B. K. Chakraverty,
   Effective Action for Super-conductivity and BCS-Bose Crossover,
   Phys. Rev. B 60 (1999) 564.
   
6. C. Castellani and C. Di Castro,
   Breakdown of Fermi Liquid in Correlated Electron Systems,
   Physica A 263 (1999) 197.
   
7. C. Castellani, C. Di Castro and P. A. Lee,
   Metallic Phase and Metal-Insulator Transition in Two-Dimensional
   Electronic Systems,
   Phys. Rev. B 57 (1998) R9381.
   
8. C. Castellani, C. Di Castro, F. Pistolesi and G. Strinati,
   Infrared Behavior of Interacting Bosons at Zero Temperature,
   Phys. Rev. Lett. 79 (1997) 1612.
   
9. C. Castellani, C. Di Castro and M. Grilli,
   Singular Quasiparticle Scattering in the Proximity of
   Charge Instabilities,
   Phys. Rev. Lett. 75 (1995) 4650.
   
10. C. Castellani, C. Di Castro and W. Metzner,
    Dimensional Crossover from Fermi to Luttinger Liquid,
    Phys. Rev. Lett. 72 (1994) 316.
    

INFM C

Personal Data

• Born in Avellino (Italy), 7/7/1957.
  
• Citizenship: Italian.
  
• Laurea in Fisica cum Laude at the University of Rome "La Sapienza", in 1981.
  

Academic Activity

• Visiting Scientist at CEA Saclay, for two years, from September 1982 to September 1984.
  
• 1985: Assistant Professor in Theoretical Physics at Rome University Tor Vergata.
  
• He has been one of the proposers of the supercomputing INFN APE project.
  
• 1988: Associate Professor of Quantum Mechanics at Rome University Tor Vergata.
  
• 1988: prize from the Accademia dei Lincei as best physicist under 35.
  
• 1992-1994: also visiting faculty at the Physics Department of
  Syracuse University (N. Y. , U. S. A. ), and Associate Director for
  Physicsof NPAC- Northeastern Parallel Architecture
  Center, directed from Prof. Geoffrey Fox.
  
• 1992: Fourth Award for Essaysfrom the Gravity Research
  Foundation (USA) for his joint work with Mark Bowick, Quantum
  Gravity, Random Geometry and Critical Phenomena , Gen. Rel. Grav. 24
  (1992) 1209.
  
• 1994: Full Professor of Theoretical Physics at Cagliari University.
  
• 1999: Full Professor of Physics at the Science Faculty of the Rome University La Sapienza.
  
• He has been Invited Professor of the Institut Universitaire de France (at the University Paris 11 ) for one month during the year 1999-2000, and for one month in the same year he has been an invited member of the Theoretical Physics Group of CEA Saclay.
  

Teaching

• 1999-2000:Disordered Systems for fourth year
  Laureastudents;
  Mechanics for second year
  Laurea students (in Informatics);
  
  
• 1995-1999: Theoretical Physics (i. e. QED and QFT) for
  fourth year Laureastudents .
  
• 1995-1999: PhD Course of Theoretical Physics.
  
• 1989-1995: As an associate professor in Tor Vergata
  University he kept six times the Theoretical Physics course.
  
• He has directed many Laurea and PhD students.
  He has been in PhD Thesis committee in different European countries.
  

Various Activities

• He has organized about six international conferences and coedited
  three volumes of proceedings.
  
• He has visited for periods of medium duration and given seminars
  in a large numbers of theoretical physics centers and departments in
  all the world. In the last years he has given, among other, plenary
  invited talks and lectures at the "1996 Sakharov Conference", at the
  "1996 Budapest Summer School", in Saclay (1998), at the Bari
  Statistical Mechanics Conference (1998), at FSU (Florida, 1999), at
  CECAM in Lyon (1999), at Orsay (2000).
  
• He has been and he is participating to projects funded by
  MURST, EEC, INFM, INFN.
  
• He is member of PQE2000 italian parallel computing initiative,
  and of the parallel computingboard of CEA-Saclay
  fundamental research division.
  
• He is member of the editorial board of Journal of Physics A
  (Math. Gen. ). He acts as a referee for Nucl. Phys. B, Phys. Rev. ,
  Phys. Rev. Lett. , Europhys. Lett. , J. Phys. , Phys. Lett. , J. Physique.
  

Research Interests

• He has published more than 180 scientific works, of which more than
  three quarters on international journals with referee.
  
• Field Theory.
  
• Lattice Gauge Theories.
  
• Numerical Simulations.
  
• Critical Phenomena. Phase Transitions.
  
• Statistical Mechanics of Disordered Systems.
  
• Supercomputing. Dedicated Computers.
  
• Random Surfaces. Matrix Models. Euclidean Quantum Gravity.
  
• Heteropolymeric Chains. Protein Folding.
  
• Algorithms: Tempering, Multi Grid, Cluster Algorithms.
  
• Random Field Ising Model, Spin Glasses, Slow Dynamics.
  
• Replica Symmetry Breaking. Ultrametricity.
  
• Ground State Calculations.
  

Selected Publications

1. A. Billoire and E. Marinari,
   Evidences Against Temperature Chaos in Spin Glasses,
   J. Phys. A 33 (2000) L265.
   
2. E. Marinari, G. Parisi, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo
   and F. Zuliani,
   Replica Symmetry Breaking in Short Range Spin Glasses: a Review
   of the Theoretical Foundations and of the Numerical Evidence,
   J. Stat. Phys. 98 (2000) 973.
   
3. E. Marinari, V. Martin-Mayor and A. Pagnani,
   Spin Glass Ordering in Diluted Magnetic Semiconductors,
   Phys. Rev. B 62 (2000) 4999.
   
4. E. Marinari, A. Pagnani and G. Parisi,
   Critical Exponents of the KPZ Equation via Multi-Surface
   Coding Numerical Simulations,
   J. Phys. A 33 (2000) 8181.
   
5. E. Marinari,
   Numerical Evidence for Continuity of Mean Field and
   Finite Dimensional Spin Glasses,
   Phys. Rev. Lett. 82 (1999) 434.
   
6. E. Marinari and F. Zuliani,
   Numerical Simulations of the 4D EA Spin Glass,
   J. Phys. A 32 (1999) 7447.
   
7. A. Chessa, E. Marinari and A. Vespignani,
   Energy Constrained Sandpile Models,
   Phys. Rev. Lett. 80 (1998) 4217.
   
8. E. Marinari,
   Optimized Monte Carlo Methods, in Advances in Computer Simulation,
   edited by J. Kertesz and Imre Kondor (Springer-Verlag, Berlin 1998), p. 50.
   
9. G. Iori and E. Marinari,
   On the Stability of the Mean-Field Spin Glass Broken Phase under
   Non-Hamiltonian Perturbations,
   J. Phys. A 30 (1997) 4489.
   
10. B. Coluzzi, E. Marinari and J. Ruiz-Lorenzo,
    New Evidence for Super-Roughening,
    J. Phys. A 30 (1997) 3771.
    

More detailed informations are available from the WEB page:

http://chimera.roma1.infn.it/ENZO/

INFM B1

• 1971 Nov. Laurea in Physics, Univ. of Roma (110+L)
  
• 1972-73 CNR fellowship, Univ of Roma and Orsay (F)
  
• 1974-75 Xerox Webster research Center, N. Y. USA; Associate Scientist.
  
• 1975-83 Brown Boveri Research Center, Baden (CH). Theoretical Physics Group
  - Member of the Staff.
  
• 1983-87 Full Professor of Condensed Matter Theory. University of Groningen,
  The Netherlands.
  
• 1983-88 Professor of Solid State Physics, Department of Physics, University
  of Roma La Sapienza.
  
• 1992-95 Head of the Theory Group (including all areas of theoretical
  physics), Univ. of Roma La Sapienza.
  1995- Director of the INFM Unit, Univ. of Roma La Sapienza. The Unit
  consists of all the scientists (about 200), theoretical and experimental,
  in the area of Condensed Matter.
  

Short-Medium Leaves

• 1982 Summer Institute of Theoretical Physics, Univ. of S. Barbara, USA
  
• 1982 Fall Lyman Lab. Of Physics, Harvard Univ. , Boston USA
  
• 1998 ICTP, Trieste
  

Teaching

• 1983-87 Course on Advanced Solid State Physics, Univ. of Groningen, The
  Netherlands.
  
• 1987- Course on Solid State Physics, Univ. of Roma La Sapienza
  
• 1987- Graduate Courses on Statistical Physics and High Tc Super-conductivity
  
• Supervisor of about 35 research Thesis at various levels, Ph D and Laureas,
  in Groningen and Roma.
  

Publications

Various

• Member of the editorial Board of: Physica A; Il Nuovo Cimento D: Int. J. of
  Fractals; Europhysics Letters (1994-97).
  
• Member of the IUPAP Committee on the Structure and Dynamics of Condensed
  Matter (1987-93)
  
• Fellow of the American Physical Society (1990-)
  
• Consultant: IBM Zurich laboratories (1992) and IBM T. J. Watson
  Laboratories, Yorktown Heights USA (1994)
  
• Member of the SISSA Evaluation Committee, SISSA (1994-)
  
• Member of the Int. Eval. Committee of the Bogoliubov Lab. of Theoretical
  Physics, Dubna, Russia (1995-)
  
• Mc Minn Lecturer (1999), Vanderbilt University, Nashville, USA.
  
• Organizer of about 12 International Conferences in the areas of Statistical
  Physics, Fractals and self-organization and Condensed Matter, mostly High
  Tc Super-conductivity.
  
• Invited or Plenary Speaker at over 80 International Conferences in various
  fields.
  

Composition of the research group and collaborators:
The group consists of about 12 people in Roma (mostly junior) plus a large
number of former students or collaborators at different Institutions in
Italy and abroad (for a total of about 30) with whom we are in active
interaction. For more details please consult the WEB page.

Organizing Activities

• General Coordinator of the European Network on: Fractal Structures and
  Self-Organization. This Network involves 11 Teams from 8 European countries
  and it is probably the largest single project in Statistical Physics at the
  international level.
  
• Director of various other Research Projects from the European Community,
  the Ministry for Science and Research and the INFM (PRA and PAIS). The
  main subjects of these projects are Statistical Physics and High Tc
  Super-conductivity.
  

Selected Publications

1. M. Joyce, P. W. Anderson, M. Montuori, L. Pietronero and F. Sylos
   Labini,
   Fractal Cosmology in an Open Universe
   Europhys. Lett. 50 (2000) 416.
   
2. C. Castellano, M. Marsili and L. Pietronero,
   Non-Perturbative Renormalization of the KPZ Growth Dynamics,
   Phys. Rev. Lett. 80 (1998) 4830.
   
3. F. Sylos Labini, M. Montuori and L. Pietronero,
   Scale Invariance of Galaxy Clustering,
   Phys. Rep. 293 (1998) 61.
   
4. A. Erzan, L. Pietronero and A. Vespignani,
   The Fixed Scale Transformation Approach to Fractal Growth,
   Rev. Mod. Phys. 67 (1995) 545.
   
5. C. Grimaldi, L. Pietronero and S. Strassler,
   Non-Adiabatic Super-conductivity: Electron Phonon Interaction
   beyond Migdal's Theorem,
   Phys. Rev. Lett. 75 (1995) 1158.
   
6. L. Pietronero, A. Vespignani and S. Zapperi,
   Renormalization Study of Self-Organized
   Criticality in Sandpile Models,
   Phys. Rev. Lett. 72 (1994) 1690.
   
7. P. Coleman and L. Pietronero,
   The Fractal Structure of the Universe,
   Phys. Rep. 213 (1992) 311.
   
8. G. Parisi and L. Pietronero,
   Theory of the Depinning Transition in Charge Density Waves,
   Europhys. Lett. 16 (1991) 321.
   
9. M. Di Stasio, K. A. Muller and L. Pietronero,
   Non-homogeneous Charge Distribution in Layered High Tc
   Super-conductors,
   Phys. Rev. Lett. 64 (1990) 2827.
   
10. L. Niemeyer, L. Pietronero and H. J. Wiesmann,
    Fractal Dimension of Dielectric Breakdown,
    Phys. Rev. Lett. 52 (1984) 1033.
    

INFM B1

Born in Rome, Italy Feb. 6, 1942.

Chronology

1. Jul. 24, 1968 Laurea cum laude in
Physics, University of Rome, presenting the thesis Study through
Fadeev equations of the final - state interaction in the annihilation
in three pions of the system antiproton - neutron supervised by
Prof. G. Jona-Lasinio.

2. Feb. 1, 1969 - Apr. 4, 1983 Assistant
Professor of Physics, Faculty of Engineering of the University o f
Rome.

3. Apr. 1, 1971 - Dec. 31, 1972 Visiting scientist at the
Department of Nuclear Engineering, Massachusetts Institute of
Technology of Cambridge (U. S. A. ) with a fellowship of the National
Research Council.

4. Sep. 1, 1974 - Oct. 31, 1974 Lecturer at the
Department of Physics of the Catholic University of Leuven (Belgium).

5. Apr. 5, 1983 - Oct. 31, 1991 Associate Professor of Physics,
Faculty of Engineering and Department of Physics, University of Rome
La Sapienza.

6. Oct. 1, 1987 - Dec. 31, 1987 Associate Researcher
at the Centre National de la Recherche Scientifique a t the Centre de
Physique Moleculaire Optique et Hertzienne de l'Universite' de
Bordeaux I (France).

7. May 20, 1988 - Apr. 8, 1994 Member of the
Physics Committee of the National Research Council.

8. Nov. 1,
1991 Professor of Physics, Faculty of Science and Department of
Physics, University of Rome La Sapienza.

9. Mar. 16, 1999 Member of the Scientific Advisory Committee of the
National Research Council.

10. Mar. 1, 2000 - Aug. 31, 2000 Visiting Professor at the
Department of Nuclear Engineering, Massachusetts Institute of
Technology of Cambridge (U. S. A. ).

Scientific Activity

The scientific activity, in the field of statistical physics,
developed along the following lines:

1. Static and dynamic properties of multicomponent liquid systems
   close to critical points.
   
2. Statistical properties of radiation scattering.
   
3. Non-equilibrium dynamics of systems close to an instability.
   
4. Aggregation kinetics in colloidal suspensions.
   
5. Dynamic properties of complex liquids at criticality and percolation.
   
6. Kinetic glass transition in supercooled liquids.
   

Selected Publications

1. F. Sciortino and P. Tartaglia,
   Extension of the Fluctuation-Dissipation Theorem
   to the Physical Aging of a Model Glass-Forming Liquid,
   Phys. Rev. Lett. (2000).
   
2. C. Donati, F. Sciortino and P. Tartaglia,
   Role of the Unstable Directions in the Equilibrium and
   Aging Dynamics of Supercooled Liquids,
   Phys. Rev. Lett. 85 (2000) 1464 (2000).
   
3. F. Mallamace, P. Gambadauro, N. Micali, P. Tartaglia,
   C. Liao, and S. -H. Chen,
   Kinetic Glass Transition in a Micellar System with Short-Range
   Attractive Interaction,
   Phys. Rev. Lett. 84 (2000) 5431.
   
4. W. Kob, F. Sciortino and P. Tartaglia,
   Aging as Dynamics in Configuration Space,
   Europhys. Lett. 49 (2000) 590.
   
5. L. Fabbian, W. Goetze, F. Sciortino, F. Thiery and P. Tartaglia,
   Ideal Glass-Glass Transitions and Logarithmic Decay
   of Correlations in a Simple System,
   Phys. Rev. E 59 (1999) R1347.
   
6. F. Sciortino, W. Kob and P. Tartaglia,
   Inherent Structure Entropy of Supercooled Liquids,
   Phys. Rev. Lett. 83 (1999) 3214.
   
7. F. Sciortino and P. Tartaglia,
   Harmonic Dynamics in Supercooled Liquids: The Case of Water,
   Phys. Rev. Lett. 78 (1997) 2385.
   
8. P. Gallo, F. Sciortino, P. Tartaglia and S. -H. Chen,
   Slow Dynamics in Supercooled Water,
   Phys. Rev. Lett. 76 (1996) 2730.
   
9. C. Cametti, F. Sciortino, P. Tartaglia, S. -H. Chen and J. Rouch,
   Complex Electrical Conductivity of Water-in-Oil Microemulsions,
   Phys. Rev. Lett. 75 (1995) 569.
   
10. F. Sciortino and P. Tartaglia,
    Structure Factor Scaling During Irreversible Aggregation,
    Phys. Rev. Lett. 74 (1995) 282.
    

INFM B1

• Born in 1954.
  
• Graduate in Physics (Roma) 1977.
  
• CNR fellowship 1978-1981.
  
• Assistant Professor (Roma)
  1981-1988.
  
• Associate Professor of Theoretical Physics (L'Aquila)
  1988-1991.
  
• Associate Professor of Mathematical Methods for Physics (Roma "La
  Sapienza" ) 1991-2000.
  
• Full Professor of Theoretical Physics (Roma "La Sapienza")
  2000-now.
  

He spent some periods at Niels Bohr Institute Copenhagen, NORDITA
Conpenhagen, University of California San Diego, University of
Marseille, University of Bruxelles, University of Stockholm, University
of Lausanne.

Main scientific interest: Statistical Mechanics, Turbulence, Chaos,
Transport and Mixing.

About 150 papers on international journals, 3 books (2 in English
and 1 in italian) and 40 contributions to conference proceedings.

About 2100 citations in the period 1981-1997.

About 70 invited talks at international conferences.

Referee of Phys. Rev. Lett. , Phys. Rev. E, Journal of Phys. A, Chaos,
Europhys. Lett. , Phys. of Fluids, Physica D, Physica A.

Principal investigator of the INFM project PRA-Turbo (1997-2000).
National coordinator of the EEC Network "Intermittency in Fully
Developed Turbulence" (1998-2001).

He has organized 10 international conferences, workshops and schools.

Selected Publications

1. M. Abel, L. Biferale, M. Cencini, M. Falcioni, D. Vergni and A.
   Vulpiani,
   An Exit-Time Approach to Epsilon-Entropy,
   Phys. Rev. Lett. 84 (2000) 6002.
   
2. T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani,
   Dynamical Systems Approach to Turbulence
   (Cambridge University Press, Cambridge U. K. 1998).
   
3. A. Puglisi, V. Loreto, U. Marini Bettolo Marconi, A. Petri and A.
   Vulpiani,
   Clustering and Non-Gaussian Behavior in Granular Matter,
   Phys. Rev. Lett. 81 (1998) 3848.
   
4. E. Aurell, G. Boffetta, A. Crisanti, G. Paladin and A. Vulpiani,
   Growth of Non-infinitesimal Perturbations in Turbulence,
   Phys. Rev. Lett. 77 (1996) 1262.
   
5. A. Crisanti, G. Paladin and A. Vulpiani,
   Products of Random Matrices in Statistical Physics
   (Springer-Verlag, Berlin 1993).
   
6. A. Crisanti, M. Falcioni, G. Paladin and A. Vulpiani,
   Lagrangian Chaos: Transport, Mixing and Diffusion in Fluids,
   La Rivista del Nuovo Cimento 14 (1991) 1.
   
7. M. Falcioni, G. Paladin and A. Vulpiani,
   Regular and Chaotic Motion of Particles in a Two
   Dimensional Fluid,
   J. Phys. A 21 (1988) 3451.
   
8. G. Paladin and A. Vulpiani,
   Anomalous Scaling Laws in Multi-fractal Objects,
   Phys. Rep. 156 (1987) 147.
   
9. R. Benzi, G. Paladin, G. Parisi and A. Vulpiani,
   On the Multi-fractal Nature of Fully Developed Turbulence and
   chaotic systems,
   J. Phys. A 17 (1984) 3521.
   
10. R. Benzi, G. Parisi, A. Sutera and A. Vulpiani,
    Stochastic Resonances in Climatic Change,
    Tellus 34 (1982) 10.