Stochastic Thermodynamics with Odd Controlling Parameters

Stochastic thermodynamics extends the notions and relations of classical thermodynamics to small systems with strong fluctuations. The definition of work and heat and the microscopically reversible condition are two key concepts in the current framework of stochastic thermodynamics. Here, we apply the 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 contraction also leads to a revision to the fluctuation theorems and nonequilibrium work relations. Through introducing an adjoint dynamics, we find that the total entropy production can be separated into three parts and two of them respectively satisfy the integral and the detailed fluctuation theorem. By revising the definition of heat and the microscopically reversible condition respectively, we can derive two sets of modified nonequilibrium work relations, including the Jarzynski equality, the detailed Crooks work relation, and the integral Crooks work relation. We take the strategy of shortcuts to isothermality as an example and give a more sophisticated explanation to the Jarzynski-like equality derived from shortcuts to isothermality.