Optimal finite-time processes in weakly driven overdamped Brownian motion

TL;DR

This paper develops a Hamiltonian approach to optimize finite-time thermodynamic processes in weakly driven overdamped Brownian motion, revealing that optimal protocols often involve jumps and are time-reversal symmetric.

Contribution

It introduces a Hamiltonian linear response framework to derive optimal protocols, connecting them with known solutions and uncovering new properties like time-reversal symmetry.

Findings

Optimal protocols often include jumps at process boundaries.

Fast-but-weak protocols are time-reversal symmetric.

The approach recovers known solutions and reveals new optimal features.

Abstract

The complete physical understanding of the optimization of the thermodynamic work still is an important open problem in stochastic thermodynamics. We address this issue using the Hamiltonian approach of linear response theory in finite time and weak processes. We derive the Euler-Lagrange equation associated and discuss its main features, illustrating them using the paradigmatic example of driven Brownian motion in overdamped regime. We show that the optimal protocols obtained either coincide, in the appropriate limit, with the exact solutions by stochastic thermodynamics or can be even identical to them, presenting the well-known jumps. However, our approach reveals that jumps at the extremities of the process are a good optimization strategy in the regime of fast but weak processes for any driven system. Additionally, we show that fast-but-weak optimal protocols are time-reversal symmetric, a property that has until now remained hidden in the exact solutions far from equilibrium.