Taming the time evolution in overdamped systems: shortcuts elaborated from fast-forward and time-reversed protocols

TL;DR

This paper develops a reverse-engineering method to design external driving protocols that can control the evolution of overdamped Brownian systems, enabling rapid, inverted, or accelerated transitions between states.

Contribution

It introduces a systematic approach to construct smooth, physically admissible protocols for connecting arbitrary states in short times, extending beyond existing explicit solutions.

Findings

Derived analytical protocols for state transitions in overdamped systems.

Demonstrated ability to invert the arrow of time in Brownian dynamics.

Provided a framework for optimizing and generalizing control protocols.

Abstract

Using a reverse-engineering approach on the time-distorted solution in a reference potential, we work out the external driving potential to be applied to a Brownian system in order to slow or accelerate the dynamics, or even to invert the arrow of time. By welding a direct and time-reversed evolution towards a well chosen common intermediate state, we derive analytically a smooth protocol to connect two \emph{arbitrary} states in an arbitrarily short amount of time. Not only does the reverse-engineering approach proposed in this Letter contain the current -- rather limited -- catalogue of explicit protocols but it also provides a systematic strategy to build the connection between arbitrary states with a physically admissible driving. Optimization and further generalizations are also discussed.