On the efficiency of heat engines at the micro-scale and below

Abstract

We investigate the thermodynamic efficiency of sub-micro-scale heat engines operating under the conditions described by over-damped stochastic thermodynamics. We prove that at maximum power the efficiency obeys for constant isotropic mobility the universal law $\eta =2{\eta }_C/(4-{\eta }_C)$ where ${\eta }_C$ is the efficiency of an ideal Carnot cycle. The corresponding power optimizing protocol is specified by the solution of an optimal mass transport problem. Such solution can be determined explicitly using well known Monge--Ampère--Kantorovich reconstruction algorithms. Furthermore, we show that the same law describes the efficiency of heat engines operating at maximum work over short time periods. Finally, we illustrate the straightforward extension of these results to cases when the mobility is anisotropic and temperature dependent.

5 pages; revised version including the derivation of the efficiency and of the corresponding optimal protocols in the presence of anisotropic temperature dependent mobility

References in corpus (5)

• Efficiency at maximum power: An analytically solvable model for stochastic heat engines
• An all-optical nanomechanical heat engine
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• Adiabatic processes realized with a trapped Brownian particle
• On how nanomechanical systems can minimize dissipation

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• Extracting maximum power from active colloidal heat engines
• Thermodynamics at the microscale: from effective heating to the Brownian Carnot engine
• Thermal brachistochrone for harmonically confined Brownian particles
• Maximum efficiency of low-dissipation refrigerators at arbitrary cooling power
• Optimal Control of Underdamped Systems: An Analytic Approach
• Thermodynamics of a Periodically Driven Qubit
• Minimum time connection between non-equilibrium steady states: the Brownian gyrator
• Classical uncertainty relations and entropy production in non-equilibrium statistical mechanics
• On the numerical integration of the Fokker-Planck equation driven by a mechanical force and the Bismut-Elworthy-Li formula