姆潘巴效应中的最优制备与可达状态约束
Optimal preparation and reachable-state constraints in the Mpemba effect
AI总结：
研究聚焦均匀加热颗粒流体，通过最优控制问题研究姆潘巴效应，将制备阶段能量注入调整以生成特定初始条件，应用庞特里亚金极大值原理确定最优协议及过量峰度极值，发现随机恒温器对可达状态有限制，建立了非平衡系统中多方面的联系
AI中文摘要：
姆潘巴效应指初始较热的系统比冷系统更快弛豫到共同终态，常通过先验选择非平稳初始条件在动力学框架内分析。本文通过明确纳入制备状态的老化协议重新审视此观点。聚焦均匀加热的颗粒流体，将制备阶段表述为最优控制问题，通过能量注入调整生成使后续弛豫率最大或最小的初始条件。在一阶索宁近似下，该优化归结为使速度分布函数的过量峰度极值化，其控制固定温度下的冷却速率。应用庞特里亚金极大值原理表明最优制备协议总是单脉冲协议并确定过量峰度的相应极值。最重要的是，发现随机恒温器施加了非平凡的可达状态约束，可及非高斯性受均匀冷却状态限制，从而限制弛豫率不对称性和最大可及姆潘巴效应。这些结果表明动力学姆潘巴效应强度与潜在非平衡态可达性不可分割。更一般地，本文在非平衡系统中建立了反常弛豫、最优控制和状态可达性之间的联系。
英文摘要：
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.
