Lower Bound of Entropy Production in an Underdamped Langevin System with Normal Distributions

TL;DR

This paper derives the minimum entropy production bounds for underdamped Langevin systems with normal distributions, providing optimal control protocols and highlighting limitations in covariance matrix transitions.

Contribution

It extends the understanding of entropy production bounds from overdamped to underdamped Langevin systems with normal distributions, including optimal control strategies.

Findings

Derived conditions for minimum entropy production in underdamped systems.

Identified limitations in covariance matrix transitions due to control constraints.

Extended existing overdamped system results to underdamped dynamics.

Abstract

We study the lower bound of the entropy production in a one-dimensional underdamped Langevin system constrained by a time-dependent parabolic potential. We focus on minimizing the entropy production during transitions from a given initial distribution to a given final distribution taking a given finite time. We derive the conditions for achieving the minimum entropy production for the processes with normal distributions, using the evolution equations of the mean and covariance matrix to determine the optimal control protocols for stiffness and center of the potential. Our findings reveal that not all covariance matrices can be given as the initial and final conditions due to the limitations of the control protocol. This study extends existing knowledge of the overdamped systems to the underdamped systems.