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

TL;DR

This work addresses minimizing entropy production $\Sigma(\tau)$ for a 1D underdamped Langevin system with a time-dependent parabolic potential, under fixed Gaussian initial and final distributions. It derives evolution equations for the mean $\boldsymbol{\mu}$ and covariance $\Xi$ under controls $k(t)$ and $r(t)$, formulates a constrained variational problem, and obtains Euler–Lagrange conditions that characterize optimal Gaussian protocols. A central finding is that, under continuous control, the endpoint covariance $\Xi^{fin}$ cannot be chosen arbitrarily due to limited degrees of freedom (e.g., $\mu_v$ must be constant), so the lower bound based on a Wasserstein-geodesic is not always achievable; allowing noncontinuous controls could, in principle, enlarge the reachable end-states. Overall, the paper extends previous overdamped results to underdamped systems, clarifies inertia-induced restrictions on optimal protocols, and offers numerical evidence that the continuous optimal path minimizes entropy production relative to nonoptimal trajectories, with discussion on practical implications and potential future control strategies.

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.

Figures (3)

• Figure 1: The set $[\Xi_{xx}(\tau),\Xi_{xv}(\tau),\Xi_{vv}(\tau)]$ when $\Xi^\mathrm{ini}$ is fixed and $k^\mathrm{ini}$ is varied from -0.344 to 1.569 incrementing by 0.001. We exclude the case of $k^\mathrm{ini}<-0.344$ and $k^\mathrm{ini}>1.569$ to prevent the drawing range from becoming excessively large.
• Figure 2: The time evolution of $\Xi$ for the optimal path and the nonoptimal path. For example, $\Xi_{xx}$ for the optimal path (purple line) and $\Xi_{xx}$ for the nonoptimal path (orange line) exhibit close values at $t=\tau(=1)$.
• Figure 3: The entropy production $\Sigma_\Xi(t)$ in Eq. \ref{['eq:Sigma Xi']} for the optimal and nonoptimal paths, and the lower bound Eq. \ref{['eq:lower bound comperison']} shown in the previous studies.