Farago Oded, Grønbech-Jensen Niels
Department of Mechanical and Aerospace Engineering, University of California, Davis, California 95616, USA and Department of Biomedical Engineering, Ben Gurion University of the Negev, Be'er Sheva, 84105 Israel and Ilse Katz Institute for Nanoscale Science and Technology, Ben Gurion University of the Negev, Be'er Sheva, 84105 Israel.
Department of Mechanical and Aerospace Engineering, University of California, Davis, California 95616, USA and Department of Chemical Engineering and Materials Science, University of California, Davis, California 95616, USA and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Jan;89(1):013301. doi: 10.1103/PhysRevE.89.013301. Epub 2014 Jan 6.
The diffusive dynamics of a particle in a medium with space-dependent friction coefficient is studied within the framework of the inertial Langevin equation. In this description, the ambiguous interpretation of the stochastic integral, known as the Itô-Stratonovich dilemma, is avoided since all interpretations converge to the same solution in the limit of small time steps. We use a newly developed method for Langevin simulations to measure the probability distribution of a particle diffusing in a flat potential. Our results reveal that both the Itô and Stratonovich interpretations converge very slowly to the uniform equilibrium distribution for vanishing time step sizes. Three other conventions exhibit significantly improved accuracy: (i) the "isothermal" (Hänggi) convention, (ii) the Stratonovich convention corrected by a drift term, and (iii) a newly proposed convention employing two different effective friction coefficients representing two different averages of the friction function during the time step. We argue that the most physically accurate dynamical description is provided by the third convention, in which the particle experiences a drift originating from the dissipation instead of the fluctuation term. This feature is directly related to the fact that the drift is a result of an inertial effect that cannot be well understood in the Brownian, overdamped limit of the Langevin equation.
在惯性朗之万方程的框架内，研究了粒子在具有空间依赖摩擦系数的介质中的扩散动力学。在此描述中，避免了随机积分的模糊解释，即所谓的伊藤-斯特拉托诺维奇困境，因为在小时间步长的极限情况下，所有解释都收敛到相同的解。我们使用一种新开发的朗之万模拟方法来测量粒子在平坦势场中扩散的概率分布。我们的结果表明，对于消失的时间步长，伊藤和斯特拉托诺维奇解释都非常缓慢地收敛到均匀平衡分布。其他三种约定表现出显著提高的精度：（i）“等温”（亨吉）约定，（ii）通过漂移项修正的斯特拉托诺维奇约定，以及（iii）一种新提出的约定，采用两个不同的有效摩擦系数，分别表示时间步长内摩擦函数的两个不同平均值。我们认为，最符合物理实际的动力学描述由第三种约定提供，在该约定中，粒子经历的漂移源于耗散而非涨落项。这一特征直接与以下事实相关：漂移是一种惯性效应的结果，在朗之万方程的布朗、过阻尼极限中无法很好地理解。
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