过阻尼RLC电路中的制备控制弛豫：通向谱姆彭巴效应的教学路径
Preparation-controlled relaxation in an overdamped RLC circuit: A pedagogical route to the spectral Mpemba effect
- Universidade Federal de Goiás（戈亚斯联邦大学）
- Hainan Bielefeld University of Applied Sciences（海南比勒费尔德应用科学大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过过阻尼RLC电路展示谱姆彭巴效应，证明初始能量并非决定弛豫路径的唯一因素，并推广至随机、开放及量子系统，揭示制备模态与观测模式对弛豫反转的关键作用。
AI中文摘要：
一个储存更多能量的系统能否比储存较少能量的系统冷却得更快？过阻尼电阻-电感-电容（RLC）电路提供了一个经典的、实验上可及的谱姆彭巴效应类比：两种状态的能量都单调减少，然而初始能量较高的状态可以下穿至能量较低的状态之下。其原因在于，初始能量本身并不决定弛豫路径；初始电荷和电流决定了慢衰减模式和快衰减模式被激发的强度。我们将这一机制从确定性描述发展到随机描述和开放系统描述。对于经典电路，我们推导了模态制备条件和交叉判据，从几何上解释了为何等能量状态可以不同地弛豫，并通过从示波器迹线重建电荷、电流和能量在实验上确认了这一效应。当电路由外部注入的白噪声驱动时，这种反转仍然存在。平衡的Ornstein-Uhlenbeck处理将系综均值携带的信息与协方差中存储的信息区分开来：中心化的吉布斯制备保持平均能量排序，而位移或各向异性的高斯态可以编码不同的快慢扇区。然后，Fokker-Planck生成元桥接至马尔可夫量子动力学。对于热阻尼量子谐振子，平均裸能量过剩以单一指数因子衰减，禁止有限平均占据数下的能量序反转。低于阈值的参量振荡器恢复了快慢正交扇区，允许由位移和协方差控制的交叉。Liouvillian表述表明，观察到的弛豫共同取决于生成元谱、制备所激发的模式以及可观测量可见的模式。
英文摘要：
Can a system with more stored energy cool faster than one with less? An overdamped resistor--inductor--capacitor (RLC) circuit offers a classical, experimentally accessible spectral Mpemba analogue: both states lose energy monotonically, yet the initially higher-energy state can cross below the lower-energy one. The reason is that initial energy alone does not determine the relaxation path; initial charge and current set how strongly the slow and fast decay modes are excited. We develop this mechanism from deterministic to stochastic and open-system descriptions. For the classical circuit, we derive the modal preparation conditions and crossing criterion, explain geometrically why equal-energy states can relax differently, and confirm the effect experimentally by reconstructing charge, current, and energy from oscilloscope traces. The inversion survives when the circuit is driven by externally injected white noise. An equilibrium Ornstein--Uhlenbeck treatment separates information carried by the ensemble mean from that stored in its covariance: centered Gibbs preparations preserve mean-energy ordering, while displaced or anisotropic Gaussian states can encode distinct slow and fast sectors. A Fokker--Planck generator then bridges to Markovian quantum dynamics. For a thermally damped quantum harmonic oscillator, the mean bare-energy excess decays with a single exponential factor, forbidding energy-order inversion for finite mean occupation. A below-threshold parametric oscillator restores slow and fast quadrature sectors, allowing crossings controlled by displacement and covariance. The Liouvillian formulation shows that observed relaxation depends jointly on the generator spectrum, the modes populated by preparation, and the modes visible to the observable.
