Optimal preparation and reachable-state constraints in the Mpemba effect
The Mpemba effect, whereby an initially hotter system relaxes faster than a colder one towards a common final state, is often analysed within the kinetic framework by assuming non-stationary initial conditions that are selected a priori. Here, we revisit this viewpoint by explicitly incorporating the aging protocol used to prepare those states. Focusing on uniformly heated granular fluids, we formulate the preparation stage as an optimal-control problem in which the energy injection is tuned to generate the initial conditions that maximise or minimise the subsequent relaxation rate. Within the first Sonine approximation, this optimisation reduces to extremising the excess kurtosis of the velocity distribution function, the quantity controlling the cooling rate at fixed temperature. Applying Pontryagin's maximum principle, we show that the optimal preparation protocol is always a one-bang protocol and determine the corresponding extremal values of the excess kurtosis. Most importantly, we find that the stochastic thermostat imposes non-trivial reachable-state constraints: the accessible non-Gaussianities are bounded by those of the homogeneous cooling state, thereby limiting the relaxation-rate asymmetry and the maximum attainable Mpemba effect. These results demonstrate that the strength of the kinetic Mpemba effect cannot be disentangled from the accessibility of the underlying non-equilibrium states. More generally, our work establishes a connection between anomalous relaxation, optimal control, and state accessibility in non-equilibrium systems.