# Optimization of stochastic thermodynamic machines

## TL;DR

This paper optimizes the performance of stochastic thermodynamic machines by deriving optimal external potentials that minimize irreversible work and entropy production, providing explicit bounds and illustrative examples.

## Contribution

It introduces a method to optimize stochastic thermodynamic machines using Fokker-Planck equations, variational techniques, and characteristics, deriving bounds on work, power, and efficiency.

## Key findings

- Optimal external potentials minimize irreversible work and entropy production.
- Explicit bounds for work output, power, and efficiency are derived.
- Examples demonstrate the application of optimal protocols.

## Abstract

The study of stochastic thermodynamic machines is one of the main topics in nonequilibrium thermodynamics. In this study, within the framework of Fokker-Planck equation, and using the method of characteristics of partial differential equation as well as the variational method, performance of stochastic thermodynamic machines is optimized according to the external potential, with the irreversible work $W_{irr}$, or the total entropy production $\Delta S_{\rm tot}$ equivalently, reaching its lower bound. Properties of the optimal thermodynamic machines are discussed, with explicit expressions of upper bounds of work output $W$, power $P$, and energy efficiency $\eta$ are presented. To illustrate the results obtained, typical examples with optimal protocols (external potentials) are also presented.