Collaborator: A. Bovier,
C. Külske
Cooperation with:
I. Merola,
E. Presutti (Università di Roma ``Tor Vergata'', Italy),
M. Zahradník (Charles University, Prague, Czech Republic)
Supported by:
DFG Priority Program ,,Interagierende Stochastische Systeme von
hoher Komplexität`` (Interacting stochastic systems of high complexity)
Description:
Models of statistical mechanics with weak long-range interactions have been
introduced by V. Kac half a century ago to obtain a rigorous version of the
van der Waals mean field theory. Today we see these models anew as interesting
candidates to reach a better understanding of disordered spin systems and
in particular the relation between mean field theory and lattice models in the
context of disordered systems. In this project, whose funding through
the DFG ended with the termination of the Priority Program
``Interacting stochastic systems of high complexity'' in 2003, we have undertaken
a long-term
effort to investigate such models and to develop appropriate mathematical
tools for their analysis. An detailed review of the results obtained here
can be found in [1].
In a joint paper with Merola, Presutti, and Zahradník, we have addressed in
this context a classical question from statistical mechanics, that of the
Gibbs Phase Rule. ``In the abstract space of all potentials, phase transitions
are an exception''. This
statement by Ruelle in his classical textbook, [6], suggests the validity
of the Gibbs phase rule, but the notion must be accepted only very cautiously, as
a complete proof of the Gibbs rule would
require to show that in the space of the thermodynamically relevant parameters,
phase transitions occur on regular manifolds of positive co-dimension. But, as stated
again by Ruelle in a recent review on open problems in mathematical physics,
[7], the proof of such a statement must be regarded as one of the main
challenges in statistical mechanics.
In the Pirogov-Sinai regime where configurations can be described by contours which
satisfy Peierls conditions, the situation is definitely better, as the theory provides
tools for a very detailed knowledge on the structure of Gibbs measures in a region in
the relevant parameters space.
The traditional
Pirogov-Sinai theory is a low
temperature expansion which enables to control the entropic fluctuations from the
ground states, its natural setup being the lattice systems. But the theory is not
limited to such cases and it has been applied to a great variety of situations,
covering various types of phase transitions. One is the case of
Kac potentials, which are seen as a perturbation of mean field, where the small parameter
is the inverse interaction range of a Kac potential. According to van der Waals, the
theory becomes then well suited for investigating the liquid-vapor branch of the phase
diagram and, as shown in [5], its applications are not restricted to lattice
models, [3], [4], but continuum particle systems can be treated as well.
All the above cases have a common structure. There is a term in the Hamiltonian of
the form
- 
, where is an extensive quantity and
is
its conjugate variable: in the case of spins, is an external magnetic field and
the spin magnetization; for particles, is the chemical potential and
the particles number. Our main assumption is that at a value, say = 0, of the
intensive parameter there is phase coexistence with an order parameter, and that
defining contours in terms of the variable , the contours satisfy the Peierls
bounds with suitable coefficients.
Under this assumption the process described
in terms of the variable has the typical features of a
low temperature Ising
model.
We will thus have a class of ``plus'' measures where
is typically
positive (as well as its expectation) and a class of ``minus''
measures with
typically negative. We are talking of classes of plus and minus
measures and not just
of plus and minus measures, because we are not ruling out the
possibility of other
phase transitions, described by other order parameters.
These assumptions imply that at
= 0 there are two distinct classes of
DLR measures, the plus and minus ones, for
which the expected value of is positive, respectively negative, and which are
obtained by thermodynamic limits with plus, respectively minus,
boundary conditions.
Under this assumption (plus some technical conditions of super-stability type if the
variables are unbounded) we prove that there is a finite interval I of values of
, centered at = 0, where coexistence occurs only at = 0. More
precisely, if > 0 (or < 0) and in I, then any translational invariant DLR
measure has positive (negative) expectation, and both plus and minus boundary
conditions produce in the thermodynamic limit the same class of states.
References:
- A. BOVIER, C. KÜLSKE,
Coarse graining techniques
for (random) Kac models,
WIAS Preprint no. 894, 2003.
- A. BOVIER, I. MEROLA, E. PRESUTTI, M. ZAHRADNÍK,
On the Gibbs phase rule in the Pirogov-Sinai regime,
WIAS Preprint no. 844, 2003, to appear in: J. Statist. Phys.
- A. BOVIER, M. ZAHRADNÍK,
The low temperature phase of
Kac-Ising models,
J. Statist. Phys., 87 (1997), pp. 311-332.
- M. CASSANDRO, E. PRESUTTI,
Phase
transitions in Ising systems with long but finite range interactions,
Markov Process. Related
Fields, 2 (1996), pp. 241-262.
-
J.L. LEBOWITZ, A. MAZEL, E. PRESUTTI,
Liquid-vapor phase transitions for systems with finite-range
interactions,
J. Statist. Phys., 94 (1999), pp. 955-1025.
- D. RUELLE,
Rigorous Results in Statistical
Mechanics, Benjamin, 1969.
-
,
Some ill formulated problems on regular
and messy behavior in statistical mechanics and smooth dynamics
for which I would like the advice of Yacha Sinai,
J. Statist. Phys., 108 (2002), pp 723-728.
LaTeX typesetting by I. Bremer
2004-08-13