Optimal protocols and universal time-energy bound in Brownian thermodynamics

Yoseline Rosales-Cabara,Laurent Mertz,Giovanni Manfredi,Paul-Antoine Hervieux,Cyriaque Genet,Gabriel Schnoering

doi:10.1103/physrevresearch.2.012012

Abstract

We propose an optimization strategy to control the dynamics of a stochastic system transferred from one thermal equilibrium to another and apply it experimentally to a Brownian particle in an optical trap under compression. Based on a variational principle that treats the transfer duration and the expended work on an equal footing, our strategy leads to a family of protocols that are either optimally cheap for a given duration or optimally fast for a given energetic cost. This approach unveils a universal relation $\Delta t\,\Delta W \ge (\Delta t\,\Delta W)_{\rm opt}$ between the transfer duration and the expended work. We verify experimentally that the lower bound is reached only with the optimized protocols.

Highlights

We propose an optimization strategy to control the dynamics of a stochastic system transferred from one thermal equilibrium to another and apply it experimentally to a Brownian particle in an optical trap under compression
Based on a variational principle that treats the transfer duration and the expended work on an equal footing, our strategy leads to a family of protocols that is either optimally cheap for a given duration or optimally fast for a given energetic cost
Recent work demonstrated the possibility to control the evolution of a small system, forcing, for instance, a nano- or microsystem to evolve from one equilibrium state to another much faster than the relaxation time expected from the energy difference between the two equilibria [6–8]

Summary

Rapid Communications

Λ 1 gives a protocol of low energetic cost but long duration, while λ 1 gives a fast protocol requiring a large amount of work This treatment leads to a universal relation t W ( t W )opt between the transfer duration t and the expended work W (in excess of the equilibrium freeenergy difference), where the lower bound depends exclusively on the initial and final states and is reached only under optimal control conditions. This result unveils a fundamental feature that underpins all optimization procedures in stochastic thermodynamics [13,14].

Published by the American Physical Society

Once written as

For t

Findings

The frontier can be drawn experimentally by measuring