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11:15-13:00:
Charles-Edouard
Pfister
On
the Nature of Isotherms at First Order Phase Transitions.
14:15-16:00: Rachel Brouwer
Self-destructive percolation
Abstracts:
Charles-Edouard
Pfister
On the Nature of Isotherms at
First Order Phase Transitions.
The
first theory of condensation originated with the celebrated equation of state
of van der Waals.
(p+a*v^{-2})(v-b)=RT.
When
complemented with the Maxwell Construction (``equal area rule'') it leads to
isotherms describing general characteristics of the liquid-vapor equilibrium.
The isotherms obtained with the van der Waals-Maxwell Theory have a very simple
analytic structure: they are analytic in a pure phase and have analytic
continuations along the liquid and gas branches, through the transition points.
These analytic continuations were originally interpreted as describing the
pressure of metastable states.
The theoretical question of knowing whether the predictions of the van der Waals
Theory can be derived from first principles of Statistical Mechanics remained an
open important problem during a large part of the twentieth century. The first
rigorous result was the study of Isakov [3] on the Ising model, which confirmed
the impossibility of an analytic continuation at low enough temperatures. It was
later generalized in [4].
I shall present recent results about this question, which have been obtained by
Sacha Friedli and myself. In [1] we consider lattice models ($d\geq 2$) with
arbitrary finite state space, and finite-range interactions which have two
ground states. Under the only assumption that the Peierls Condition is satisfied
for the ground states and that the temperature is sufficiently low, we prove
that the pressure has no analytic continuation at first order phase transition
points. In [2] we consider Ising models with Kac potentials Jγ(x)=γ^dφ(γ
x), in the limit when γ tends to 0. Our analysis exhibits a crossover
between the non-analytic behaviour of finite range models (γ>0) and
analyticity in the mean field limit for γ to 0.
The
first lecture will be devoted
to the history of the problem,
to a precise formulation of the results, as well as an exposition of
Pirogov-Sinai Theory, which is the framework in which they are established.
The second
lecture
will be devoted to a detailed proof
of Isakov's theorem and its generalization [1].
In the last lecture the results concerning the Kac limit γ to 0 will be presented [2]. I shall conclude with a discussion of important open problems.
[1] Friedli S.,
Pfister C.-E., On the Singularity of the Free Energy at First Order Phase
Transition, to appear in Commun. Math. Phys.
[2]
Friedli S., Pfister C.-E., Non-Analyticity and the van der Waals limit,
to appear in J. Stat. Phys.
[3] Isakov S.N.,
Nonanalytic Features of the First Order Phase Transition in the Ising Model},
Commun. Math. Phys. 95, 427-443, (1984).
[4]
Isakov S.N., Phase Diagrams and Singularity at the Point of a Phase
Transition of the First Kind in Lattice Gas Models, Teoreticheskaya i Matematicheskaya
Fizika, 71, 426-440, (1987).
Rachel Brouwer
Consider ordinary site percolation on an infinite graph in which the sites, independent of each other, are occupied with probability p and vacant with probability 1-p. Now suppose that, by some catastrophe, all sites which are in an infinite occupied cluster become vacant. Finally, each site that is vacant after the catastrophe, becomes occupied with probability d, independent of the other sites. This model can be used to represent forest fires, as a limiting case of various models. One would expect that if p is larger (but very close to) the critical value, that after the removal of the infinite cluster, a very small value of d is needed to create an infinite cluster in the final configuration. On the binary tree this is indeed the case. However, we conjecture that on the square lattice, this intuition is not true. We will show that the conjecture, if it is true, has some remarkable consequences.