Optimizing the relaxation route with optimal control

We look into the minimization of the connection time between nonequilibrium steady states. As a prototypical example of an intrinsically nonequilibrium system, a driven granular gas is considered. For time-independent driving, its natural time scale for relaxation is characterized 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) minimize 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 analyzed 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 nonequilibrium 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.

Automated summary

By utilizing control theory and information theory, this study shows that the use of bang-bang protocols can minimize the connection time in a granular gas system, surpassing the natural relaxation time scale. Experimental results demonstrate excellent agreement with theoretical predictions, indicating the applicability of this approach to a wide range of driven nonequilibrium systems under certain conditions.