Hidden temperature in the KMP model

In the KMP model a positive energy $\zeta_i$ is associated with each site $i\in\{0,\frac1n,\dots,\frac{n-1}{n},1\}$. When a Poisson clock rings at the bond $ij$ with energies $\zeta_i,\zeta_j$, those values are substituted by $U(\zeta_i+\zeta_j)$ and $(1-U)(\zeta_i+\zeta_j)$, respectively, where $U$ is a uniform random variable in $(0,1)$. The boundary dynamics is defined in such way that the resulting Markov process $\zeta(t)$, with boundary conditions $T_0,T_{1}$, satisfies that $\zeta_b(t)$ is exponential with mean $T_b$, for $b=0,1$, for all $t$. We show that the invariant measure is the distribution of a vector $\zeta$ with coordinates $\zeta_i=T_iX_i$, where $X_{k/n}$ are iid exponential$(1)$ random variables, the law of $T$ is the invariant measure for an opinion model with the same boundary conditions, and $X,T$ are independent. The result confirms a conjecture based on the large deviations of the model. The discrete derivative of the opinion model behaves as a neural spiking process, which is also analysed. The hydrostatic limit shows that the empirical measure of a configuration chosen with the invariant measure converges to the lineal interpolation of $T_0$ and $T_{1}$.