The open harmonic process: non-equilibrium steady state, pressure, density large deviation and additivity principle

Gioia Carinci, Chiara Franceschini, Rouven Frassek, Cristian Giardinà, Frank Redig

Published 2023-07-27Version 1

We consider the boundary driven harmonic model, i.e. the Markov process associated to the open integrable XXX chain with non-compact spins. Via integrability we show that it is possible to caracterise the stationary measure as a mixture of product measures. We identify the law of the mixture in terms of the Dirichlet process for all spin values $s>0$. This further develops the results of [25] where the moments and the stationary state were obtained for all $s>0$ in closed-form and extends the result of [8] where the mixture structure was unveiled for $s=1/2$ with another method. Next, by using the explicit knowledge of the non-equilibrium steady state we establish formulas predicted by Macroscopic Fluctuation Theory for several quantities of interest: the pressure (by Varadhan's lemma), the density large deviation function (by contraction principle), the additivity principle (by using the Markov property of the mixing law). To our knowledge, the results presented in this paper constitute the first rigorous derivation of these macroscopic properties for models of energy transport with unbounded state space, starting from the microscopic structure of the non-equilibrium steady state.