# Stochastic thermodynamics with odd controlling parameters

## TL;DR

This paper explores the impact of odd controlling parameters on stochastic thermodynamics, revealing inconsistencies in traditional definitions and deriving modified fluctuation theorems for small fluctuating systems.

## Contribution

It introduces revised definitions of work and heat for systems with odd controlling parameters, leading to new fluctuation relations and a deeper understanding of entropy production.

## Key findings

- Incompatibility between heat definition and microscopic reversibility for odd parameters
- Modified fluctuation theorems including Jarzynski and Crooks relations
- Decomposition of entropy production into three parts satisfying fluctuation theorems

## Abstract

Stochastic thermodynamics extends the notions and relations of classical thermodynamics to small systems that experience strong fluctuations. The definitions of work and heat and the microscopically reversible condition are two key concepts in the current framework of stochastic thermodynamics. Herein, we apply stochastic thermodynamics to small systems with odd controlling parameters and find that the definition of heat and the microscopically reversible condition are incompatible. Such a contradiction also leads to a revision to the fluctuation theorems and nonequilibrium work relations. By introducing adjoint dynamics, we find that the total entropy production can be separated into three parts, with two of them satisfying the integral fluctuation theorem. Revising the definitions of work and heat and the microscopically reversible condition allows us to derive two sets of modified nonequilibrium work relations, including the Jarzynski equality, the detailed Crooks work relation, and the integral Crooks work relation. We consider the strategy of shortcuts to isothermality as an example and give a more sophisticated explanation for the Jarzynski-like equality derived from shortcuts to isothermality.