We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep as compared to the temperature, physical observables, like the mean square displacement, are essentially time-independent over a long time interval, the stagnation epoch. However, the standard Boltzmann–Gibbs (BG) distribution is non-normalizable, given that the usual partition function is divergent. For this regime, we have previously shown that a regularization of BG statistics allows for the prediction of the values of dynamical and thermodynamical observables in the non-normalizable quasi-equilibrium state. In this work, based on the eigenfunction expansion of the time-dependent solution of the associated Fokker–Planck equation with free boundary conditions, we obtain an approximate time-independent solution of the BG form, being valid for times that are long, but still short as compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is an error function, which is shifted due to the initial struggle to overcome the potential well. With the eigenfunction solution of the Fokker–Planck equation in hand, we show the validity of the regularized BG statistics and how it perfectly describes the time-independent regime though the quasi-stationary state is non-normalizable.

中文翻译：

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非约束场的Fokker-Planck方程的非归一化拟平衡解

我们研究了渐近平坦的势阱中粒子的过度阻尼的Langevin运动。当势阱与温度相比更深时，物理观测值（如均方位移）在很长的时间间隔（停滞时期）中基本上与时间无关。但是，由于通常的分区函数是发散的，因此标准的Boltzmann–Gibbs（BG）分布不可归一化。对于此方案，我们之前已经证明，BG统计量的正则化可以预测不可归一化的准平衡状态下的动态和热力学可观测值。在这项工作中，基于带有自由边界条件的相关Fokker-Planck方程的时变解的本征函数展开，我们获得了BG形式的近似于时间的解，对于较长但与指数增长的逃逸时间相比仍然很短的时间有效。逃逸的粒子遵循一般的自由粒子统计，其中的解是一个误差函数，由于最初为克服势阱而作的努力而使该函数发生偏移。借助Fokker-Planck方程的本征函数解，我们证明了正则化BG统计量的有效性，以及尽管准平稳状态不可归一化，但它如何完​​美地描述了与时间无关的状态。由于最初为克服潜力而做出的努力，因此发生了变化。借助Fokker-Planck方程的本征函数解，我们证明了正则化BG统计量的有效性以及它如何完美地描述了与时间无关的状态，尽管准平稳状态是不可归一化的。由于最初为克服潜力而做出的努力，因此发生了变化。借助Fokker-Planck方程的本征函数解，我们证明了正则化BG统计量的有效性，以及尽管准平稳状态不可归一化，但它如何完​​美地描述了与时间无关的状态。