Thermodynamics of Glassy Systems: Glasses, Spin Glasses and Optimization

After a long, self-standing dominance of Newtonian reductionism, at
the beginning of the nineteenth century, with the birth of
thermodynamics, a new approach began to develop to the study of
nature. Indeed, the prediction of the behaviour of a macroscopic
material through the knowledge of all the Newtonian trajectories of
the molecules composing it was an impossible task because of the
insurmountable difficulty of computing them for any initial condition
in a time shorter than the age of the universe. Besides this, the
computation itself was realized as unfruitful and senseless since,
whatever the initial conditions of all the trajectories can be, a
macroscopic system always behaves in the same way if subject to the
same external physical conditions. Another conceptual approach was
requested to overcome such a stalemate: physical phenomena began to be
analyzed in the viewpoint of statistical mechanics.

Not only were the techniques different, but also the questions to be
answered changed. The (hopeless) hunt for the behaviour of every
single microscopic component of a system was substituted by the
analysis of the statistical ensemble of the configurations in which
the system can find itself and by the computation of the average
values of the relevant, thermodynamic, observables on such an
ensemble.

A similar kind of (r)evolution, with a new change of paradigm, is now
occurring in physics once again: the shift from statistical mechanics
to the theory of complex systems. In a hopefully effective attempt to
give a concise explanation, we like the definition of complex systems
as those systems ``whose behaviour crucially depends on its details''
\cite{PCM02}.
To study such systems, the standard tools of statistical mechanics are
not sufficient anymore and new techniques need to be developed. In
statistical mechanics, for large systems, what matters is the average
of the significant observables and the theory tries to predict its
value. For what concerns complex systems, instead, since they are
very sensitive to the initial conditions, the maximum information we
can get is the probability distribution of all possible behaviours.
The main issue is a change of paradigm in predicting the physical
properties of a system. Indeed, in this kind of approach, it makes no
sense to look at the behaviour of a particular system. Rather, the aim
becomes to obtain the general features of the statistical class of
systems to which the probed one belongs. The class being
characterized by a given probability distribution of possible
realizations, where a single realization (i.e., one system)
unpredictably depends on the initial conditions. The approach to
complex problems is therefore typically a probabilistic one. Hence,
the ``complexity'' of the system can be viewed, as the information we
would need (but we are not able to obtain) in order to give a complete
explanation of the behaviour of one system. The more a system is
sensitive to its initial condition, the higher is its complexity and
the more difficult it becomes to carefully outline a typical
behaviour.

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In our thesis we have considered, in different ways and with different
degrees of approximation, a number of topics related to the theory of
complex systems. We have been analyzing aspects of glasses, spin
glasses and of the optimization problem known as {\em K-SAT problem}.
Depending on the specific case we have applied different statistical
mechanics techniques, both analytical and numerical.

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The study of glass forming materials has been the object of active
scientific interest for over half a century, and yet, in spite of the
huge amount of accumulated knowledge there is still much to be
understood on the nature of the glass phase and on its formation
\cite{McKenna89,ASci95,ANMcKMcMMAPR00,Donth01}.
In particular, the quest for a satisfying, comprehensive, realistic
theoretical model for the glass is still far from being accomplished,
even if, in the theoretical field, many interesting and fascinating
steps forward have been made in the last two decades.

In order to study the mechanisms of glassy dynamics we have been
following the approach of kinetically facilitated models. These
models usually display a trivial static behaviour, showing, however,
the characteristic slowing down of the dynamics occurring in glasses
with its main characteristic: aging. Though the imposed separation
between statics and dynamics takes us away from real systems, it
allows us, anyway, to clarify which properties of the glass can be
understood as consequence of the dynamics alone. The outcome is
encouraging and, actually, all the basic qualitative features of glass
formers can be reproduced by means of such models. Furthermore, using
these models, it is also possible to verify the generalized
thermodynamic theory of out of equilibrium phenomena, formulated by
Nieuwenhuizen~\cite{NPRL97_1,NJPA98,NPRL98,NPRE00}, that could encode
into the thermodynamic language the slow aging relaxation of the
complex systems here considered.

A basic concept to enable such translation is the {\em effective
temperature}, as it has appeared in literature in many different ways
since Tools formulation, in 1946. To analyze the robustness of such a
concept, we have faced the study of a model of harmonic oscillators
and spherical spins (HOSS model) evolving on two separated time
scales, with a very simple statics and a facilitated, exactly
solvable, Monte Carlo dynamics (chapter $3$). We considered two
different dynamical versions: the first leading to an Arrhenius law
for the relaxation time to equilibrium (strong glass HOSS model), the
second one displaying a Vogel-Fulcher-Tammann-Hesse (VFTH) law
(fragile glass HOSS model) and a Kauzmann transition, taking place at
the VFTH temperature $T_0$, with true discontinuity of specific heat.
Our goal was to check whether the chance existed of inserting the
effective temperature, $T_e$, into the construction of a consistent
out of equilibrium thermodynamic theory. Indeed, the possibility of
computing an exact solution for the dynamics allows for a precise
formulation of the two temperature picture, including $T_e$ besides
the heat bath temperature $T$. Even though the physics of our model
is simple, we have formulated general aspects of the results by
analyzing them in the thermodynamic language. To a certain extent it
is possible to obtain a thermodynamic theory for the aging regime, but
this did not turn out to be a universal feature valid in every model
for all allowed aging regimes.

Another main motivation for the analysis of such a model was to get
more insight in the various aspects of the glassy dynamics exploiting
the analytical solubility of our model. Thanks to its simplicity,
every detected feature of the glassy behaviour can be directly
connected with given elements of the model. Certain properties can
be even switched on and off tuning the model parameters or
implementing the facilitated dynamics in alternative ways. The model
we considered was intimately based on time scale separation between
fast and slow processes. A direct consequence of this time scale
separation is that we encountered a both mathematically and
physically well defined configurational entropy, being a function of
the dynamic variables of the model.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We have, then, carried out the inherent structure approach on the HOSS
model (Chap. $4$). Decreasing the temperature, the free energy local
minima do not split into smaller local minima, just like, for
instance, in the $p$-spin model in zero magnetic field \cite{CSJPI95},
because every allowed configuration of harmonic oscillators of the
HOSS model is and remains an inherent structure at any temperature.
Consequently we were able to set a one-to-one correspondence between
the minima of the free energy and those of the potential energy
(i.e. the inherent structures). Because of this exact correspondence
the dynamics through inherent structures should have been a valid
symbolic dynamics for the real system, i.e. at a finite heat-bath
temperature $T$.

Taking a system in equilibrium at an effective temperature $T_e^{\rm
(is)}$, such that the configurations visited by the system at
equilibrium are the same as those out of equilibrium at temperature
$T$, we defined an effective temperature through the matching of the
equilibrium and the out of equilibrium internal energy of the
inherent structures (section 4.3.1). For the strong glass model this
effective temperature coincides with the finite temperature $T_e$ at
low temperature. On the contrary, for the fragile glass HOSS model,
we found that this effective temperature was quite different from the
effective temperature that we were able to identify in the finite $T$
dynamics. Even proposing a new definition, following a quasi static
approach (section 4.3.2), and, thus, exploiting the solvability of
the model, the so defined effective temperature was analytically
different from the finite $T$ dynamics effective temperature. The
inherent structure scheme for the study of dynamics can, then, only
be view as an approximation (even if in particular cases a rather
good one) of the realistic dynamics of the system. As a consequence,
also the derivation of out of equilibrium thermodynamic quantities
(e.g. the configurational entropy) obtained making use of this
approach could suffer of a systematic deviation from the exact
result.


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In chapter $5$ we have considered the disordered Backgammon model
(DBG) in which each state is associated with a positive random energy,
obtained from a distribution $g(\epsilon)$. When the distribution
$g(\epsilon)=\delta(\epsilon-1)$ is chosen, the model is the standard
Backgammon (BG) model~\cite{RPRL95}. The DBG model displays slow
relaxation due to the presence of entropic barriers. The relaxation
at $T=0$ of occupation probabilities $P_k$, $k=0, \ldots, N$ is
exactly the same as the original BG model, and, in particular,
independent of the disorder distribution $g(\epsilon)$. On the contrary,
the relaxation of other observables, such as the energy, displays an
asymptotic relaxation which depends on the statistical properties of
$g(\epsilon)\sim \epsilon^\nu$ in the limit $\epsilon\to 0$. In the asymptotic
long-time regime, relaxation takes place by diffusing particles among
states with the lowest values of $\epsilon$. Therefore the asymptotic
decay of the energy as well as the one of other observables only
depends on the exponent $\nu$ which describes the limiting behavior
$g(\epsilon)\to\epsilon^{\nu}$.


The DBG model offers a scenario where there are two energy sectors
separated by the energy scale $\epsilon^*$ which have very different
physical properties. In the out of equilibrium regime entropic
barriers are typically higher than the time dependent barrier at the
threshold level $\epsilon^*$ and an effective temperature can be defined
(it comes out to be $T_e\sim (\epsilon^\star)^{\nu+2}$). For
$\epsilon>\epsilon^*$ barriers are lower and equilibrium is rapidly achieved.


A method has been proposed to numerically determine the threshold
energy scale $\epsilon^*$ by computing the general probability
distribution of proposed energy changes in the Monte Carlo dynamics.
Preliminary investigations in other glassy models show that this
distribution provides a general way to determine the threshold scale
$\epsilon^*$. Moreover it gives interesting information about
fluctuations in the aging state although future work is still needed
to understand better its full implications in our understanding of the
aging regime.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In chapter $6$ we have studied the thermodynamics of a tiling model
built by Wang tiles, i.e. a system model with no built-in quenched
disorder (unlike the disordered Backgammon model) but a disorder
caused by a complex geometry.

For this kind of system we have found evidence of a phase transition
from a completely disordered phase, in which the tiles on the plane
are completely uncorrelated between each other, to a phase in which
they begin to present an organized, also if very complicated,
structure and we have numerically identified an order parameter.

Below the critical point the answer of the system to an external
perturbation and the time autocorrelation function depend on the
history of the system. The existence of such aging leads to a
violation of the fluctuation-dissipation theorem in the low
temperature phase. From the behaviour of the response function
vs. the correlation function is not clear whether the model belongs to
the class of systems showing domain growth or it is rather more
similar to a spin glass in magnetic field. Indeed, for very long
times (small values of the correlation function) the
fluctuation-dissipation ratio goes eventually to zero and it cannot be
excluded that the dynamics evolves through domains growth
\cite{BPRL98}, even though in our case the nature of the domains of
tiles should still be theoretically understood. Nevertheless for a
very large interval of time the Fluctuation-Dissipation Ratio is
continuously and slowly decreasing to zero like in a spin glass model
\cite{MPRRJPA98,PRREPJB99}. Moreover, this is also the prediction we get by
linking the dynamical results with the static ones, in the hypothesis
of {\em stochastic stability}, making quite difficult the distinction
between a domain growth dynamics, where the response function is
constant in the aging regime, and a more complicated behaviour where
even the response function also relaxes, though extremely slowly in
comparison with the aging relaxation of the correlation function.

The last two chapters have been dedicated to the study of two models
exhibiting a spin glass phase from the static viewpoint instead that
from a dynamic one.

In chapter 7 we probed a {\em spin glass mean field} model, the Random
Blume-Emery-Griffiths-Capel model, a spin glass lattice gas whose
equilibrium low temperature phase is stable in the full replica
symmetry breaking limit, i.e. it is a true spin glass phase (as
opposed to the glass like phase displayed by the disordered mean-field
models with discontinuous order parameter and dynamic transition).
The interest of this model comes from the fact that it undergoes both
a second order phase transition (for values of thermodynamic
parameters above a certain critical point) and a true first order
thermodynamic phase transition (below the critical point). To
calculate the stable full replica symmetry breaking solution a new
technique has been developed to solve the differential equations for
the functional order parameters that we can obtain making use of the
variational approach of Sommers and
Dupont~\cite{SDJPC84,CRPRE02,CLCM}.

Like in any gas-liquid transition a latent heat is involved in the
transformation between paramagnet and spin glass below the tricritical
point. Moreover, for a certain range of parameters (between the
spinodal lines), no pure phase is achievable, not even as a metastable
one, and the two phases coexist.

The last chapter is dedicated to the 3-SAT problem, an optimization
problem constituted by an ordered string of $N$ boolean variables
(bits) that have to satisfy a certain number of conditions. Such
conditions are made of three boolean numbers, each one associated
with one of three indices referring to the position in the string of
a randomly chosen bit.

We performed the study of the RSB solution of the 3-SAT problem in the
limit of many clauses, mapping it on a slightly diluted spin glass
model with long-range random quenched interactions. The mapping to a
statistical mechanics model was carried out introducing an artificial
temperature and taking, in the end, the limit $T\to 0$, to recover the
original model. The structure of the solutions to the problem is of
the full RSB kind: in order to get a stable solution the replica
symmetry has to be broken in a continuous way, similarly to the SK
model (in external magnetic field). The full RSB structure holds down
to the interesting limit of zero temperature.

No phase transition is expected in the UNSAT phase, other than the
SAT-UNSAT transition at zero temperature, driven by the ratio $\alpha$
between the number of clauses and the number of bits and occurring at
$\alpha=\alpha_c\simeq 4.2$ Therefore we expect the same full RSB
structure of solutions, as found for the over-constrained case
($\alpha \gg \alpha_c$), to hold also in the critical region.

We presented the precise procedure to get the full RSB solution of a
general class of models that, besides the over-constrained 3-SAT
model, includes the SK model, the $p$-spin model and, more generally,
models with any combination of $p$ interacting terms. As a
consequence, the numerical code developed to solve the present model
can be applied to the whole class of models without any relevant
change, providing an efficient tool for the analysis of the structure
of the solutions of a large number of spin models interacting via
quenched random couplings.

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