非约束场Fokker-Planck方程的非正规化准平衡解
Non-normalizable quasi-equilibrium solution of the Fokker-Planck equation for nonconfining fields
摘要：
研究了渐近平坦势阱中粒子的过阻尼朗之万运动。当势阱比温度深时，像均方位移这样的物理观测量在很长的时间间隔内基本上是与时间无关的，即停滞期。然而，标准玻尔兹曼-吉布斯（BG）分布是不可归一化的，因为通常的配分函数是发散的。对于这种制度，我们以前已经表明，BG统计的正则化允许在非归一化准平衡态的动力学和物理观测值的预测。在这项工作中，本征函数扩展的相关的Fokker-Planck方程的自由边界条件下的时间依赖的解决方案的基础上，我们得到了BG形式的近似的时间无关的解决方案，有效的时间是长的，但仍然短的指数大逃逸时间相比。逃逸粒子遵循一般的自由粒子统计，其中解是一个误差函数，由于最初的斗争，以克服潜在的。通过Fokker-Planck方程的本征函数解，我们证明了正则化BG统计的有效性，以及它如何完美地描述了准定态的非标准化状态下的时间无关状态。
英文摘要：
We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep compared to 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 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, valid for times which are long, but still short compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is a an error function, 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.
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