Optimising the relaxation route with optimal control

TL;DR

This paper demonstrates that bang-bang control protocols can optimally minimize the connection time between non-equilibrium steady states in driven granular gases, outperforming natural relaxation times and classical speed limits.

Contribution

It introduces a control-theoretic approach to identify optimal bang-bang protocols for rapid state transitions in non-equilibrium systems, validated through numerical simulations.

Findings

Bang-bang protocols shorten relaxation time below classical speed limits.

Numerical simulations confirm theoretical predictions.

Optimal protocols are applicable to a broad class of driven non-equilibrium systems.

Abstract

We look into the minimisation of the connection time between non-equilibrium steady states. As a prototypical example of an intrinsically non-equilibrium system, a driven granular gas is considered. For time-independent driving, its natural time scale for relaxation is characterised from an empirical -- the relaxation function -- and a theoretical -- the recently derived classical speed limits -- point of view. Using control theory, we find that bang-bang protocols -- comprising two steps, heating with the largest possible value of the driving and cooling with zero driving -- minimise the connecting time. The bang-bang time is shorter than both the empirical relaxation time and the classical speed limit: in this sense, the natural time scale for relaxation is beaten. Information theory quantities stemming from the Fisher information are also analysed over these optimal protocols. The implementation of the bang-bang processes in numerical simulations of the dynamics of the granular gas show an excellent agreement with the theoretical predictions. Moreover, general implications of our results are discussed for a wide class of driven non-equilibrium systems. Specifically, we show that analogous bang-bang protocols, with a number of bangs equal to the number of relevant physical variables, give the minimum connecting time under quite general conditions.