Known Girsanov and Bismut-Elworthy-Li identities are recast as explicit grid-free Monte Carlo algorithms for coupled Fokker-Planck and Hamilton-Jacobi-Bellman systems, with a neural-network demonstration on a Schrödinger bridge problem.
On the efficiency of heat engines at the micro-scale and below
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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--Amp\`ere--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.
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On the numerical integration of the Fokker-Planck equation driven by a mechanical force and the Bismut-Elworthy-Li formula
Known Girsanov and Bismut-Elworthy-Li identities are recast as explicit grid-free Monte Carlo algorithms for coupled Fokker-Planck and Hamilton-Jacobi-Bellman systems, with a neural-network demonstration on a Schrödinger bridge problem.