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Scientific Report 2007-2009
Theoretical physics
T13. The glassy state
The glass is state of matter characterized by a very
large viscosity. Consequently relaxational processes in
a glass are extremely slow and the dynamics of the
glass constituents takes place on a broad spectrum of
timescales. The coexistence of fast and very slow processes
makes the glass dynamics very interesting from
the physical point of view, but also very difficult to treat
analytically.
Experimental facts and analytical theories about the
glass transition and the aging dynamics of glas-formers
have been reviewed in a recent book [1] written by a
member of our group. Thermodynamics of glasses poses
interesting questions still largely unanswered, e.g., the
existence of a thermodynamical phase transition to a
glass state, lowering the temperature or increasing the
density. Such a transition is predicted by mean field approximations
and is called a random first order transition
(RFOT), but its existence in finite dimensional systems
is still a matter of debate.
It is well known that the free-energy barriers between
states, that diverge in mean field approximation,
are large but non-diverging in finite-dimensional models.
Still, how much of the mean field scenario is maintained
in low-dimensional models is unclear. Recently we have
studied in Ref. [2] a one-dimensional version of the Derrida’s
Random Energy Model (REM). The REM, being
a long range model, has a clear RFOT. In our 1D model
we have introduced a length (proportional to the system
size, as in the Kac limit) such that interactions are REMlike
on smaller scales. We indeed find a limiting value
for this crossover length between the REM-like and the
1D behavior, but corrections with respect to the mean
field approximation are huge and would make hard to
find the crossover length in actual glassy models.
Our group has, as well, dedicated quite a large effort
in recent years on the study of glasses of hard spheres. A
system of monodisperse hard spheres is maybe the simplest
showing most of the glass phenomenology and can
be thus considered as a prototypical model. Moreover,
amorphous packings have attracted a lot of interest as
theoretical models for glasses, because for polydisperse
colloids and granular materials the crystalline state is not
obtained in experiments. We have reviewed in Ref. [3]
most of the recent results on systems of hard spheres
obtained with the replica method.
At a first sight it could look strange to use the replica
method, invented to average out the disorder, in systems
with no disorder at all. But a closer look will reveal that
a dense system of hard spheres is likely to be in one of
the many amorphous packing configurations. Even if the
original model has no disorder at all, the configurations
dominating the high density phase are very many as if
they were generated from a disordered Hamiltonian and
the replica method is a natural tool to deal with such
complexity.
Figure 1: The replicated potential for estimating the number
of states in a glassy phase.
In this context the replica method works more or less
as follows. Given a reference equilibrium configuration,
one can construct many replicated configurations, that
interact with a small coupling term with the reference
one. In the inset of Figure 1 we show with red dashed
circle the replicas of the central molecule, which are free
to evolve, but with a coupling ε with respect to the reference
configuration. The main question is what happens
when ε is sent to zero after the thermodynamical limit.
In this limit, under mean field approximations, one can
infer the existence of more than one state from the computation
of the so-called replicated potential (which is
shown with full lines in Figure 1). Actually in Figure 1
we are showing the entropy of configurations having a
certain overlap q with the reference configuration.
In Ref. [4] we have also extended our theory of
amorphous packings of hard spheres to binary mixtures
and more generally to multicomponent systems. The
theory is based on the assumption that amorphous
packings produced by typical experimental or numerical
protocols can be identified with the infinite pressure
limit of long-lived metastable glassy states. We test
this assumption against numerical and experimental
data and show that the theory correctly reproduces
the variation with mixture composition of structural
observables, such as the total packing fraction and the
partial coordination numbers.
References
1. L. Leuzzi et al., Thermodynamics of the glassy state,
Taylor & Francis (2007).
2. Franz et al., J. Phys. A41, 324011 (2008).
3. G. Parisi et al., J. Stat. Mech., P03026 (2009).
4. I. Biazzo et al., Phys. Rev. Lett. 102, 195701 (2009).
Authors
G. Parisi, E. Marinari, F. Ricci-Tersenghi, L. Leuzzi 3 , I.
Biazzo, F. Caltagirone
Sapienza Università di Roma 36 Dipartimento di Fisica