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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: Henk
van Beijeren
Thermodynamic formalism for dlute Lorentz gases.
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).
Henk
van Beijeren
Thermodynamic formalism for dlute Lorentz gases.
Ruelle's thermodynamic formalism assigns a dynamical partition function to a chaotic system by raising its expansion factor along unstable manifolds to a power 1-\beta and averaging over all initial points on the relevant shell in phase space.
For a closed system its logarithm over t, usually called the topological pressure, yields the Kolmogorov-Sinai entropy as a derivative at \beta=1, and is called the topological entropy for \beta=0. For open systems the thermodynamic pressure at \beta=1 also gives the average escape rate from the system. For a dilute disordered Lorentz gas at equilibrium( that is, a system of fixed hard spherical scatterers with one light particle moving elastically among them) the thermodynamic pressure may be calculated explicitly, yielding results in agreement with previous calculations. For \beta-values different from unity the topological pressure for large enough systms always becomes dominated by orbits confined either to the direct neighborhood of a periodic orbit or to a small subspace with a higher than average collision rate. For example the topological entropy with increasing system size soon is determined exclusively by orbits confined to a very small subsystem of the total system. In the presence of a driving field combined with a gaussian thermostat the calculation of the dynamic partition function involves a simple transfer matrix formalism.
The same holds for a system with open boundaries. In the latter case it is helpful mapping the problem to a random flight model with escape through the boundaries.