Stochastic thermodynamics of system with continuous space of states

TL;DR

This paper develops a continuous-space stochastic thermodynamics framework, analyzing entropy production, time reversal, and noise effects, including a special noise type for isolated systems, contrasting with classical invariance results.

Contribution

It introduces a continuous-time formulation of stochastic thermodynamics for systems with continuous states, highlighting the role of dissipation and noise in entropy production.

Findings

Entropy production is a bilinear form in dissipation current components.

A special noise can keep energy constant yet produce entropy.

Contrasts with Liouville invariance in isolated systems.

Abstract

We analyze the stochastic thermodynamics of systems with continuous space of states. The evolution equation, the rate of entropy production, and other results are obtained by a continuous time limit of a discrete time formulation. We point out the role of time reversal and of the dissipation part of the probability current on the production of entropy. We show that the rate of entropy production is a bilinear form in the components of the dissipation probability current with coefficients being the components of the precision matrix related to the Gaussian noise. We have also analyzed a type of noise that makes the energy function to be strictly constant along the stochastic trajectory, being appropriate to describe an isolated system. This type of noise leads to nonzero entropy production and thus to an increase of entropy in the system. This result contrasts with the invariance of the entropy predicted by the Liouville equation, which also describes an isolated system.