Optimization of stochastic thermodynamic machines

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.