Balint A Toth

Improved Lower Bound on the Thermodynamic Pressure of the Spin 1/2 Heisenberg Ferromagnet*

We introduce a new stochastic representation of the partition function of the spin 1/2 Heisenberg ferromagnet. We express some of the relevant thermodynamic quantities in terms of expectations of functionals of so-called random stirrings onℤd. By use of this representation, we improve the lower bound on the pressure given by Conlon and Solovej inLett. Math. Phys.23, 223–231 (1991).

Onsager Relations and Eulerian Hydrodynamic Limit for Systems with Several Conservation Laws

We present the derivation of the hydrodynamic limit under Eulerian scaling for a general class of one-dimensional interacting particle systems with two or more conservation laws. Following Yaus relative entropy method it turns out that in case of more than one conservation laws, in order that the system exhibit hydrodynamic behaviour, some particular identities reminiscent of Onsagers reciprocity relations must hold. We check validity of these identities whenever a stationary measure with product structure exists. It also follows that, as a general rule, the equilibrium thermodynamic entropy (as function of the densities of the conserved variables) is a globally convex Lax entropy of the hyperbolic systems of conservation laws arising as hydrodynamic limit. As concrete examples we also present a number of models modeling deposition (or domain growth) phenomena. The Onsager relations arising in the context of hydrodynamic limits under hyperbolic scaling seem to be novel. The fact that equilibrium thermodynamic entropy is Lax entropy for the arising Euler equations was noticed earlier in the context of Hamiltonian systems with weak noise, see ref. 7.

Persistent random walks in random environment

SummaryWeak convergence of a class of functionals of PRWRE is proved. As a consequence CLT is obtained for the normed trajectory.

Derivation of the Leroux System as the Hydrodynamic Limit of a Two-Component Lattice Gas

The long time behavior of a couple of interacting asymmetric exclusion processes of opposite velocities is investigated in one space dimension. We do not allow two particles at the same site, and a collision effect (exchange) takes place when particles of opposite velocities meet at neighboring sites. There are two conserved quantities, and the model admits hyperbolic (Euler) scaling; the hydrodynamic limit results in the classical Leroux system of conservation laws, even beyond the appearance of shocks. Actually, we prove convergence to the set of entropy solutions, the question of uniqueness is left open. To control rapid oscillations of Lax entropies via logarithmic Sobolev inequality estimates, the symmetric part of the process is speeded up in a suitable way, thus a slowly vanishing viscosity is obtained at the macroscopic level. Following [4, 5], the stochastic version of Tartar–Murat theory of compensated compactness is extended to two-component stochastic models.

Towards a Unified Dynamical Theory of the Brownian Particle in an Ideal Gas

We consider the trajectoryQM(t) of a Brownian particle of massM in an ideal gas of identical particles of mass 1 and of density 1 in equilibrium at inverse temperature 1 (the dynamics is uniform motion plus elastic collisions with the Brownian particle). Our theory, in dimension one, describes a variety of limiting processes — containing the Wiener process and the Ornstein-Uhlenbeck process — forA−1/2QM(A)(At) depending on the asymptotic behaviour ofM(A). Part of the theory is hypothetical while another part relies upon known results. We also prove that, ifA1/2+ε≪M(A)≪A, thenA−1/2QM(A) (At) converges to a Wiener process whose variance is known from papers of Sinai-Soloveichik and of the present authors.

A lower bound for the critical probability of the square lattice site percolation

SummaryWe prove by elementary combinatorial considerations that the critical probability of the square lattice site percolation is larger than 0.503478.

Persistent random walks in a one-dimensional random environment

Central limit theorems are obtained for persistent random walks in a onedimensional random environment. They also imply the central limit theorem for the motion of a test particle in an infinite equilibrium system of point particles where the free motion of particles is combined with a random collision mechanism and the velocities can take on three possible values.

Bounds for the limiting variance of the “heavy particle” inR1

AbstractWe consider a one-dimensional system consisting of a tagged particle of massM surrounded by a gas of unit-mass hard-point particles in thermal equilibrium. Denoting byQt the displacement of the tagged particle, we give lower and upper bounds — independent ofM — for. It results from the proof that the correct nontrivial norming ofQt — if any — is

Diffusivity bounds for 1D Brownian polymers

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