流体中的分子随机行走与Bogolyubov生成函数方程的不变性群
Molecular random walks in a fluid and an invariance group of the Bogolyubov generating functional equation
摘要：
本文用Bogolyubov发展方程的BBGKY方程族分析了经典流体中分子随机游动的统计问题。引入了一组合适的等价关联函数，使得它们都是可积的，在统计平衡中消失，否则积累了关于"分子布朗粒子“（测试分子）与流体中其他分子碰撞历史的统计信息。导出了这种关联函数的生成泛函的精确演化方程。然后，它表明，时间依赖的解决方案，这个方程，以及一个适当定义的生成功能的静态磁平衡的相关性，具有不变性相对于一组确定的变换的功能的独立变量，如果密度的流体（每单位体积的分子数）被视为一个独立的变量。这种不变性导致了分子布朗粒子的关联函数与轨道概率分布之间存在无穷多个精确关系。即使是最简单的这些关系表明路径概率分布的配置文件的显着限制，即使没有文字解决BBGKY层次结构。
英文摘要：
The problem of statistics of molecular random walks in a classical fluid is analyzed by means of the BBGKY hierarchy of equations reformulated in terms of the Bogolyubov evolution equation for generating functional of many-particle distribution functions. A proper equivalent set of correlation functions is introduced so that all they are integrable, vanish in statistical equilibrium, otherwise accumulate statistical information about history of collisions of a ``molecular Brownian particle'' (test molecule) with other molecules of the fluid. An exact evolution equation for generating functional of such correlation functions is derived. Then it is shown that time-dependent solution to this equation, as well as a properly defined generating functional of static thermodynamically equilibrium correlations, possesses invariance with respect to a definite group of transformations of independent variables of the functional, if density of the fluid (number of molecules per unit volume) is treated as one of the independent variables. Such invariance results in infinitely many exact relations between the correlation functions and probability distribution of path of the molecular Brownian particle. Even simplest of these relations suggest significant restrictions on a profile of the path probability distribution, even without literal solving the BBGKY hierarchy.
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