Systems composed by a large number of heterogeneous interacting agents are ubiquitous in nature. These systems are often characterized by large scale emergent phenomena that involve the coordinated action of many agents; a few scattered examples are: the solidification of fluids, the spontaneous magnetization, the interaction of proteins inside a cell, the collective behavior of neurons in the brain, the evolution of ecosystems, large fluctuations of the stock market etc.

In many cases the collective behavior has an high level of complexity, e.g. the macroscopic behavior of the system must be described with a much rich vocabulary that the microscopic one. Often the large-scale movements become much slower that the small scale ones: a vivid example is the progressive slowing down of the flow of wax (or any other glass forming material) when the temperature decreases.

These examples belong to many different sciences, however similar statistical tools are needed to predict their large-scale behavior. This project aims to control their complex behavior in the case of physical systems (e.g. random magnets, glasses, electrons interacting in a random medium, surface growth). These physical systems are somewhat easier to be studied: many well-controlled experiments are possible, very large-scale simulations are feasible and many analytic tools may be used. In spite of their relative simplicity, the determination of the large scale behavior of these systems presents strong difficulties as far as there are many open fundamental questions: this project aims to answer to these questions. In the first two and half years very interesting steps forward have been done, among them the following:

  • We have been able to find a general method for computing the properties of very slow movements of the systems in such a way to separate the slow macroscopic dynamics from the fast microscopic dynamics (i.e. to build a systematic approach to the construction of adiabatic approximations).

  • Using renormalization group techniques, like real space renormalization, we have computed the critical exponents of spin glasses. This has been done for a given model system (i.e. the simplest non-trivial one: spins interacting in an one dimensional hierarchical lattice). The technique used is quite promising: an improved version of it should be able to give a solid answer to many questions concerning spin glasses.

  • A very well known approach to the glass transition (for structural glasses and some kind of spin glasses) is provided by mode coupling theory. We have been able to find out a general tool for deriving mode-coupling-like theories in the framework of static equilibrium theories by studying the properties of an appropriate statistical mechanical system composed by many replicas (or clones) of the original system. In this way we have been able to construct a static-dynamic mean field theory for these glassy transitions. If we go beyond mean field theory, the qualitative behavior of the system can be studied in great details, moreover the first corrections to mean-field theory have been computed in a quantitative way for the first time, determining in this way the range of validity of the mean field theory.

  • The discovery of quite diverse systems that behave in a similar way is often a crucial clue for understanding (an historical example is the concept of universality in second order phase transitions). We have studied the collective freezing (when the radius increases) of a hard sphere fluid in spaces of many dimensions (ranging from 2 to 9). We found the surprising results that the probability distribution of near contacts can be characterized by a critical exponent that is independents from the dimensions with very good approximation, while other properties become dimensional independent for dimensions greater than 4. We need a simple theory to explain these amazing findings and we believe that we are not far from deriving it.


  • In the mean field theory approximation glasses and some kind of spin glasses have two transitions: at higher temperature, a dynamical one (the so-called mode coupling transition) and, at a lower temperature, a real thermodynamic transition, called the Kautzman transition. The existence of two transitions is an artifact of the mean field approximation: in reality the dynamical transition marks a crossover from the high temperature region to a low temperature region where activated processes are dominant. However the remnants of the dynamical transition can be seen also in three dimensions (where the mode coupling theory has been developed) and the critical exponents associated to the dynamical transition can be studied experimentally and numerically.
    We have been able to fully characterize the critical properties of this phase transition, its universality class and its upper critical dimension (i.e. 8). We have found that there is a completely unexpected supersymmetry that simplifies the problem and allows a mapping on a simpler problem .
    We also understood in a clear way why the dynamical transition fades in finite dimensions. Moreover we have been able to compute the parameters of the effective Hamiltonian near the critical point starting from a microscopic theory. We found that, according to Ginsburg's criterion, the corrections are small unless very near to the critical temperature, thus explaining the phenomenological success of the mode coupling theory. The range of validity of mean-field theory depends on the microscopic details, e.g. it is much wider in superconductors than in superfluids.

  • We understood in a deep way the relations among the replica formalism and the slow dynamics of the system. We now know how to perform most of the dynamic computations (i.e. the evaluation of time dependent quantities) in a static framework using the appropriate Boltzmann-Gibbs distribution for a replicated system, where systems belonging to different replicas satisfy ad hoc constraints.
    In this way we are able to derive in a systematic way adiabatic approximations that can be used in analytic computations. The same adiabatic approximations can be also used for comparing the time dependence of the correlations measured in standard dynamical simulations with equilibrium simulations of the appropriate equilibrium systems. In the case of three dimensional spin glasses a detailed comparison of the two different numerical computations has been performed and the results obtained with the two different methods are very similar. We have thus been able to reproduce within a static approach the behavior of the dynamics correlations functions that measure dynamical heterogeneities.
    In this approach we have analytically computed the dynamical exponents in the mode coupling approach in terms of static equilibrium properties . We have checked that, when the results were already known, this computation reproduces known results. This approach has however a much wider range of application.
    The general formalism has been just derived and it will have many applications that are in progress at the present moment (e.g. we have been able to reproduce some of the results in a much simpler way).

  • The critical properties and the numerical value of the exponents of the hierarchical model for spin glasses have been studied in great details, using two different methodologies. The results are consistent and they should be compared with high precision numerical simulations that are in progress. Although these good quality results mark an important step forward, they do not have the precision that I wanted to reach. Further work should be dedicated to this problem in order to improve the accuracy of the analytic computations. In this context we are trying to find out the relations and the differences among the critical exponents hierarchical model and those of the usual one-dimensional long-range models.

  • We have done strong progresses in understanding the physics of systems with a random magnetic field (that can be experimentally realized as antiferromagnets in presence of an uniform magnetic field). In the ferromagnetic case we have finally clarified the reasons why these materials have a very slow dynamics in spite of the absence of any breaking of the replica symmetry at equilibrium. This result has been obtained by computing the properties of the system also in the metastable region, where one finds a multitude of different metastable systems, usually present only in the case of replica symmetry breaking.
    In the case of spin glasses with an external magnetic field we have performed very large-scale simulations and we obtained a very solid evidence of the existence of a transition in a finite dimensional model with short range, i.e. in 4 dimensions (the so called De Almeida-Thouless transition). A paper with the study of the 3 dimensional case is nearly finished.

  • Most of the simulations of glasses are usually done in 2-3 dimensions and in few cases in 4 dimensions. We have studied in great details the properties of the dynamical phase transition for an hard sphere liquid in higher dimensions (up to 9) where we have been able to study many properties: the critical density, the critical exponents and the shape of the cage. The results obtained display a completely unexpected regularity and some of the relevant results are independent from the dimensions, if written using the appropriate scaling variables. Also the probability distribution of near quasi-contacts at the jamming transition can be characterized by a critical exponent that is independent from the dimension with very good approximation .
    These results are very interesting: they suggest that many properties of glasses and of the jamming transition should be explained in a theory that has a very wide range of applications. The infinite dimensional case is particularly interesting because mean field theory should be exact in this limit. Therefore the appropriate formulation of mean field theory should be able to explain these results in a quantitative way. The renormalization group should be used only at a later stage to explain the residual dependence on the space dimensions. A first step in the direction of the construction of a microscopic theory valid in the infinite dimensional limit has been done very recently, generalizing and extending previous known results.
  • References: see the project output report.

    ERC advanced Grant 247328, Fundamental constituents of matter, FP7-IDEAS-ERC programme