Optimising the relaxation route with optimal control

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.

25 pages, 14 figures, final version accepted In Physical Review Research. (Major revision with respect to v1 with extensive changes, title modified to "Optimising the relaxation route with optimal control".)

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Cited by in corpus (16)

• Driving rapidly while remaining in control: classical shortcuts from Hamiltonian to stochastic dynamics
• Control of Active Brownian Particles: An exact solution
• Taming the time evolution in overdamped systems: shortcuts elaborated from fast-forward and time-reversed protocols
• Thermal brachistochrone for harmonically confined Brownian particles
• Optimal time-entropy bounds and speed limits for Brownian thermal shortcuts
• Non-equilibrium memory effects: granular fluids and beyond
• Optimal control of uniformly heated granular fluids in linear response
• Optimal Control in Soft and Active Matter
• Shortcuts To Adiabaticity for Lévy processes in harmonic traps
• Minimum time connection between non-equilibrium steady states: the Brownian gyrator
• Optimal control of levitated nanoparticles through finite-stiffness confinement
• Optimal synchronisation to a limit cycle
• Active charge and discharge of a capacitor: scaling solution and energy optimization
• Kinetic glass transition in granular gases and non-linear molecular fluids
• Control protocols for harmonically confined run-and-tumble particles
• Relaxation-speed crossover in anharmonic potentials