Doi Masao, Zhou Jiajia, Di Yana, Xu Xianmin
Center of Soft Matter Physics and its Applications, Beihang University, 37 Xueyuan Road, Beijing 100191, China.
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.
Phys Rev E. 2019 Jun;99(6-1):063303. doi: 10.1103/PhysRevE.99.063303.
In 1931, Onsager proposed a variational principle which has become the base of many kinetic equations for nonequilibrium systems. We have been showing that this principle is useful in obtaining approximate solutions for the kinetic equations, but our previous method has a weakness that it can be justified, strictly speaking, only for small incremental time. Here we propose an improved method which does not have this drawback. The improved method utilizes the integral proposed by Onsager and Machlup in 1953, and can tell us which of the approximate solutions is the best solution without knowing the exact solution. The improved method has an advantage that it allows us to determine the steady state in nonequilibrium system by a variational calculus. We demonstrate this using three examples, (a) simple diffusion problem, (b) capillary problem in a tube with corners, and (c) free boundary problem in liquid coating, for which the kinetic equations are written in second or fourth-order partial differential equations.
1931年，昂萨格提出了一个变分原理，该原理已成为许多非平衡系统动力学方程的基础。我们一直表明，这个原理在获得动力学方程的近似解方面很有用，但我们之前的方法有一个弱点，严格来说，它仅在小的增量时间内才是合理的。在此我们提出一种改进方法，该方法没有这个缺点。改进后的方法利用了昂萨格和马赫卢普在1953年提出的积分，并且在不知道精确解的情况下能够告诉我们哪个近似解是最佳解。改进后的方法有一个优点，即它允许我们通过变分法确定非平衡系统中的稳态。我们用三个例子来证明这一点，（a）简单扩散问题，（b）有拐角的管中的毛细问题，以及（c）液体涂层中的自由边界问题，其动力学方程用二阶或四阶偏微分方程表示。
