We study closed systems of particles that are subject to stochastic forces in addition to the conservative forces. The stochastic equations of motion are set up in such a way that the energy is strictly conserved at all times. To ensure this conservation law, the evolution equation for the probability density is derived using an appropriate interpretation of the stochastic equation of motion that is not the Itô nor the Stratonovic interpretation. The trajectories in phase space are restricted to the surface of constant energy. Despite this restriction, the entropy is shown to increase with time, expressing irreversible behavior and relaxation to equilibrium. This main result of the present approach contrasts with that given by the Liouville equation, which also describes closed systems, but does not show irreversibility.

中文翻译：

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恒定能量表面上相空间中的随机运动

我们研究除了保守力之外还受到随机力的封闭粒子系统。运动随机方程的建立方式使得能量在任何时候都严格守恒。为了确保这个守恒定律，概率密度的演化方程是使用对随机运动方程的适当解释得出的，该解释既不是伊藤也不是 Stratonovic 解释。相空间中的轨迹被限制在恒定能量的表面上。尽管有这种限制，但熵显示出随着时间的推移而增加，表现出不可逆的行为和放松到平衡。本方法的这一主要结果与刘维尔方程给出的结果形成对比，后者也描述了封闭系统，但没有显示出不可逆性。