Generalized optimal protocols of Brownian motion in a parabolic potentia

TL;DR

This paper investigates how memory effects influence the optimal work and entropy production of a Brownian particle in a harmonic potential, revealing differences from Markovian models and highlighting the need for thermodynamic consistency.

Contribution

It extends the analysis of Brownian motion to include memory effects via a generalized Langevin equation, providing new insights into work and entropy production.

Findings

Memory effects reduce discontinuities in work in the underdamped regime.

External work can be done by the field in the generalized non-Markovian case.

Negative entropy production rates are observed, indicating the need for improved thermodynamic modeling.

Abstract

The generalized Langevin equation with an exponential kernel is used to analyze memory effects on the optimal work done by a Brownian particle in a heat bath and subjected to a harmonic moving potential. The generalized overdamping scenario is also investigated. Several facts emerge in these more precise descriptions using the same initial conditions of the Markovian which lead the particle to do mechanical work against the field. Compared with the results obtained with the latter, the memory fades the discontinuities observed in the highly underdamped regime, which suggests that this trait is a consequence of the Markov approximation as well as the dependence of the different dynamical susceptibilities with the external field. Unlike the overdamped Markovian, work is done by the external field in the analog generalized counterpart. A detailed calculation of the rate of entropy production gives negative values. It is mathematically correct because the dynamics deal with a reduced description of the degrees of freedom of the bath. The theory then requires improving the treatment of them to restore the second law and thus to get the results with the required thermodynamics consistency.