Hidden temperature in the KMP model

Abstract

In the Kipnis Marchioro Presutti (KMP) model a positive energy ζi is associated with each vertex i of a finite graph with a boundary. When a Poisson clock rings at an edge ij with energies ζi,ζj, those values are substituted by U(ζi+ζj) and (1−U)(ζi+ζj), respectively, where U is a uniform random variable in (0,1). A value Tj≥0 is fixed at each boundary vertex j. The dynamics is defined in such way that the resulting Markov process ζ(t), satisfies that ζj(t) is exponential with mean Tj, for each boundary vertex j, for all t. We show that the invariant measure is the distribution of a vector ζ with coordinates ζi=TiXi, where Xi are iid exponential(1) random variables, the law of T is the invariant measure for an opinion random averaging/gossip model with the same boundary conditions of ζ, and the vectors X and T are independent. The result confirms a conjecture based on the large deviations of the model. When the graph is one-dimensional, we bound the correlations of the invariant measure and perform the hydrostatic limit. We show that the empirical measure of a configuration chosen with the invariant measure converges to the linear interpolation of the boundary values.