Hidden temperature in the KMP model
In the Kipnis Marchioro Presutti (KMP) model a positive energy $ζ_i$ is associated with each vertex $i$ of a finite graph with a boundary. When a Poisson clock rings at an edge $ij$ with energies $ζ_i,ζ_j$, those values are substituted by $U(ζ_i+ζ_j)$ and $(1-U)(ζ_i+ζ_j)$, respectively, where $U$ is a uniform random variable in $(0,1)$. A value $T_j\ge 0$ is fixed at each boundary vertex $j$. The dynamics is defined in such way that the resulting Markov process $ζ(t)$, satisfies that $ζ_j(t)$ is exponential with mean $T_j$, for each boundary vertex $j$, for all $t$. We show that the invariant measure is the distribution of a vector $ζ$ with coordinates $ζ_i=T_i X_i$, where $X_i$ 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.