Entropic fluctuations in thermally driven harmonic networks

TL;DR

This paper analyzes entropy production in harmonic oscillator networks driven by heat reservoirs, establishing fluctuation theorems and providing methods to compute large deviation rate functions using spectral data.

Contribution

It introduces a framework for entropy fluctuations in non-equilibrium harmonic networks, proving fluctuation relations and linking cumulant generating functions to spectral properties.

Findings

Established a large deviation principle for entropy production.

Proved the Gallavotti--Cohen fluctuation theorem for the network.

Provided a spectral method for computing cumulant generating functions.

Abstract

We consider a general network of harmonic oscillators driven out of thermal equilibrium by coupling to several heat reservoirs at different temperatures. The action of the reservoirs is implemented by Langevin forces. Assuming the existence and uniqueness of the steady state of the resulting process, we construct a canonical entropy production functional which satisfies the Gallavotti--Cohen fluctuation theorem, i.e., a global large deviation principle with a rate function I(s) obeying the Gallavotti--Cohen fluctuation relation I(-s)-I(s)=s for all s. We also consider perturbations of our functional by quadratic boundary terms and prove that they satisfy extended fluctuation relations, i.e., a global large deviation principle with a rate function that typically differs from I(s) outside a finite interval. This applies to various physically relevant functionals and, in particular, to the heat dissipation rate of the network. Our approach relies on the properties of the maximal solution of a one-parameter family of algebraic matrix Riccati equations. It turns out that the limiting cumulant generating functions of our functional and its perturbations can be computed in terms of spectral data of a Hamiltonian matrix depending on the harmonic potential of the network and the parameters of the Langevin reservoirs. This approach is well adapted to both analytical and numerical investigations.