We construct an exclusion process with Bernoulli product invariant measure and having, in the diffusive hydrodynamic scaling, a non symmetric diffusion matrix, that can be explicitly computed. The antisymmetric part does not affect the evolution of t
he density but it is relevant for the evolution of the current. In particular because of that, the Ficks law is violated in the diffusive limit. Switching on a weakly external field we obtain a symmetric mobility matrix that is related just to the symmetric part of the diffusion matrix by the Einstein relation. We show that this fact is typical within a class of generalized gradient models. We consider for simplicity the model in dimension $d=2$, but a similar behavior can be also obtained in higher dimensions.
We study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the
average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph.
We discuss a generalization of the conditions of validity of the interpolation method for the density of quenched free energy of mean field spin glasses. The condition is written just in terms of the $L^2$ metric structure of the Gaussian random vari
ables. As an example of application we deduce the existence of the thermodynamic limit for a GREM model with infinite branches for which the classic conditions of validity fail.
We give a generalization to a continuous setting of the classic Markov chain tree Theorem. In particular, we consider an irreducible diffusion process on a metric graph. The unique invariant measure has an atomic component on the vertices and an abso
lutely continuous part on the edges. We show that the corresponding density at $x$ can be represented by a normalized superposition of the weights associated to metric arborescences oriented toward the point $x$. The weight of each oriented metric arborescence is obtained by the exponential of integrals of the form $intfrac{b}{sigma^2}$ along the oriented edges time a weight for each node determined by the local orientation of the arborescence around the node time the inverse of the diffusion coefficient at $x$. The metric arborescences are obtained cutting the original metric graph along some edges.
The Box-Ball System (BBS) is a one-dimensional cellular automaton in ${0,1}^Z$ introduced by Takahashi and Satsuma cite{TS}, who also identified conserved sequences called emph{solitons}. Integers are called boxes and a ball configuration indicates t
he boxes occupied by balls. For each integer $kge1$, a $k$-soliton consists of $k$ boxes occupied by balls and $k$ empty boxes (not necessarily consecutive). Ferrari, Nguyen, Rolla and Wang cite{FNRW} define the $k$-slots of a configuration as the places where $k$-solitons can be inserted. Labeling the $k$-slots with integer numbers, they define the $k$-component of a configuration as the array ${zeta_k(j)}_{jin mathbb Z}$ of elements of $Z_{ge0}$ giving the number $zeta_k(j)$ of $k$-solitons appended to $k$-slot $jin mathbb Z$. They also show that if the Palm transform of a translation invariant distribution $mu$ has independent soliton components, then $mu$ is invariant for the automaton. We show that for each $lambdain[0,1/2)$ the Palm transform of a product Bernoulli measure with parameter $lambda$ has independent soliton components and that its $k$-component is a product measure of geometric random variables with parameter $1-q_k(lambda)$, an explicit function of $lambda$. The construction is used to describe a large family of invariant measures with independent components under the Palm transformation, including Markov measures.
We introduce and study a simple and natural class of solvable stochastic lattice gases. This is the class of emph{Strong Particles}. The name is due to the fact that when they try to jump to an occupied site they succeed pushing away a pile of partic
les. For this class of models we explicitly compute the transport coefficients. We also discuss some generalizations and the relations with other classes of solvable models.
We study the structure of stationary non equilibrium states for interacting particle systems from a microscopic viewpoint. In particular we discuss two different discrete geometric constructions. We apply both of them to determine non reversible tran
sition rates corresponding to a fixed invariant measure. The first one uses the equivalence of this problem with the construction of divergence free flows on the transition graph. Since divergence free flows are characterized by cyclic decompositions we can generate families of models from elementary cycles on the configuration space. The second construction is a functional discrete Hodge decomposition for translational covariant discrete vector fields. According to this, for example, the instantaneous current of any interacting particle system on a finite torus can be canonically decomposed in a gradient part, a circulation term and an harmonic component. All the three components are associated to functions on the configuration space. This decomposition is unique and constructive. The stationary condition can be interpreted as an orthogonality condition with respect to an harmonic discrete vector field and we use this decomposition to construct models having a fixed invariant measure.
We consider one dimensional weakly asymmetric boundary driven models of heat conduction. In the cases of a constant diffusion coefficient and of a quadratic mobility we compute the quasi-potential that is a non local functional obtained by the soluti
on of a variational problem. This is done using the dynamic variational approach of the macroscopic fluctuation theory cite{MFT}. The case of a concave mobility corresponds essentially to the exclusion model that has been discussed in cite{Lag,CPAM,BGLa,ED}. We consider here the convex case that includes for example the Kipnis-Marchioro-Presutti (KMP) model and its dual (KMPd) cite{KMP}. This extends to the weakly asymmetric regime the computations in cite{BGL}. We consider then, both microscopically and macroscopically, the limit of large external fields. Microscopically we discuss some possible totally asymmetric limits of the KMP model. In one case the totally asymmetric dynamics has a product invariant measure. Another possible limit dynamics has instead a non trivial invariant measure for which we give a duality representation. Macroscopically we show that the quasi-potentials of KMP and KMPd, that for any fixed external field are non local, become local in the limit. Moreover the dependence on one of the external reservoirs disappears. For models having strictly positive quadratic mobilities we obtain instead in the limit a non local functional having a structure similar to the one of the boundary driven asymmetric exclusion process.
We call emph{Alphabet model} a generalization to N types of particles of the classic ABC model. We have particles of different types stochastically evolving on a one dimensional lattice with an exchange dynamics. The rates of exchange are local but u
nder suitable conditions the dynamics is reversible with a Gibbsian like invariant measure with long range interactions. We discuss geometrically the conditions of reversibility on a ring that correspond to a gradient condition on the graph of configurations or equivalently to a divergence free condition on a graph structure associated to the types of particles. We show that much of the information on the interactions between particles can be encoded in associated emph{Tournaments} that are a special class of oriented directed graphs. In particular we show that the interactions of reversible models are corresponding to strongly connected tournaments. The possible minimizers of the energies are in correspondence with the Hamiltonian cycles of the tournaments. We can then determine how many and which are the possible minimizers of the energy looking at the structure of the associated tournament. As a byproduct we obtain a probabilistic proof of a classic Theorem of Camion cite{Camion} on the existence of Hamiltonian cycles for strongly connected tournaments. Using these results we obtain in the case of an equal number of k types of particles new representations of the Hamiltonians in terms of translation invariant $k$-body long range interactions. We show that when $k=3,4$ the minimizer of the energy is always unique up to translations. Starting from the case $k=5$ it is possible to have more than one minimizer. In particular it is possible to have minimizers for which particles of the same type are not joined together in single clusters.
Stationary non-equilibrium states describe steady flows through macroscopic systems. Although they represent the simplest generalization of equilibrium states, they exhibit a variety of new phenomena. Within a statistical mechanics approach, these st
ates have been the subject of several theoretical investigations, both analytic and numerical. The macroscopic fluctuation theory, based on a formula for the probability of joint space-time fluctuations of thermodynamic variables and currents, provides a unified macroscopic treatment of such states for driven diffusive systems. We give a detailed review of this theory including its main predictions and most relevant applications.
