Diffusion approximations to the chemical master equation only have a consistent stochastic thermodynamics at chemical equilibrium

TL;DR

This paper examines the extent to which diffusion approximations of the Chemical Master Equation preserve stochastic thermodynamics, finding they are only consistent at equilibrium but can approximate thermodynamics away from equilibrium.

Contribution

It demonstrates that diffusion approximations only have consistent stochastic thermodynamics at equilibrium, providing insights into their limitations and applicability.

Findings

Stochastic thermodynamics is only consistent at chemical equilibrium.

Diffusion approximations can approximate thermodynamics away from equilibrium.

The study clarifies the conditions under which diffusion models inherit thermodynamic properties.

Abstract

The stochastic thermodynamics of a dilute, well-stirred mixture of chemically-reacting species is built on the stochastic trajectories of reaction events obtained from the Chemical Master Equation. However, when the molecular populations are large, the discrete Chemical Master Equation can be approximated with a continuous diffusion process, like the Chemical Langevin Equation or Low Noise Approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the Chemical Master Equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochastic thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the Chemical Master Equation.