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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.
SUMMARY OF THE MAJOR ACHIEVEMENTS SINCE THE START OF THE PROJECT:
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.
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