Local Equilibrium in Inhomogeneous Stochastic Models of Heat Transport

Abstract

Abstract. We extend the duality of Kipnis Marchioro and Presutti [KMP82] to inhomogeneous lattice gas systems where either the components have different degrees of freedom or the rate of interaction depends on the spatial location. Then the dual process is applied to prove local equilibrium in the hydrodynamic limit for some inhomogeneous high dimensional systems and in the nonequilibrium steady state for one dimensional systems with arbitrary inhomogeneity.

Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations

“…when Bethe ansatz is not available, still the connection between the non-equilibrium system coupled to reservoirs and the absorbing dual turns out to be very useful to obtain macroscopic properties such as the hydrodynamic limit, fluctuations, propagation of chaos and local equilibrium (see e.g. [13,15,18]).…”

Heat conduction and the nonequilibrium stationary states of stochastic energy exchange processes

“…This property, which is known as the gradient property [11], is central to the model's simplicity and the fact that its transport coefficient is given in terms of the current's average value with respect to local thermal equilibria. Further models of heat transport similar to the KMP process in that they share the gradient property have been considered in recent years; see references [5,[12][13][14][15][16]. Of particular interest for our sake are so-called Brownian energy processes (BEP) [5], which have been extensively studied in the framework of duality [17]; see also references [18] and [19].…”

“…We thus focus more specifically on the study of systems with uniform and alternating shape parameters, which all have linear temperature profiles. An example of such a system was recently considered in reference [15], corresponding to shape parameters alternating between 1 and 1/2. There, its dual was shown to be a symmetric simple exclusion process with alternating jump rates [23,24]; see also references [25,26] for disordered cases.…”