Engineered Swift Equilibration for Brownian objects: from underdamped to overdamped dynamics

TL;DR

This paper introduces a general framework for rapidly transforming underdamped Brownian particles between equilibrium states using engineered potentials, extending existing overdamped protocols to a broader range of friction conditions.

Contribution

It develops explicit inverse-engineered protocols for swift state transformations of Brownian particles, applicable across different friction regimes and encompassing known overdamped solutions.

Findings

Protocols depend on key dimensionless parameters.

Framework extends overdamped solutions to underdamped regimes.

Protocols enable rapid, controlled state changes in Brownian systems.

Abstract

We propose a general framework to study transformations that drive an underdamped Brownian particle in contact with a thermal bath from an equilibrium state to a new one in an arbitrarily short time. To this end, we make use of a time and space-dependent potential, that plays a dual role: confine the particle, and manipulate the system. In the special case of an isothermal compression or decompression of a harmonically trapped particle, we derive explicit protocols that perform this quick transformation, following an inverse engineering method. We focus on the properties of these protocols, which crucially depend on two key dimensionless numbers that characterize the relative values of the three timescales of the problem, associated with friction, oscillations in the confinement and duration of the protocol. In particular, we show that our protocols encompass the known overdamped version of this problem and extend it to any friction for decompression and to a large range of frictions for compression.