統計物理在化學主方程式、艾根模型及表面機械研磨處理的應用

Abstract

In macro, the method and concept of statistical physics can be a powerful tool in analytical and numerical computation and applied to many fields such as chemical reaction, bio-evolution, and material science. In our work, the methods of statistical physics: chemical master equation (CME), Hamilton-Jacobi equation (HJE), and canonical ensemble are used to calculate various physical quantities. Our work is organized in three topics: the CME with the Gaussian and compound Poisson noise, bio-evolution of Eigen model, and energy conversion in the surface mechanical attrition treatment (SMAT) experiment. In the CME part, the chemical reaction among DNA, mRNA, and protein can be regarded as a stochastic process. We consider the CME with compound Poisson and Gaussian noises and obtain the exact solution of steady state probability density function (PDF) verified by the algorithm of forward finite difference in large-scale time. Without Gaussian white noise, the solution of CME (set diffusion coefficient ϵ = 0) can be returned to that of CME derived by Long Cai, et al. In the bio-evolution part, we use the method of expansion in O(1/N) to obtain the HJE for probability distribution in Hamming class which is applied to calculate the correction of O(1/N) accuracy for the steady-state probability distribution in Hamming class and mean fitness in Eigen model. The steady-state distributions of O(1/N) correction are well-consistent with the Runge-Kutta simulation with relative errors less than 1 %, while the mean fitness of O( 1/N) is the same one derived by Michael Deem, et al. in quantum field theory. In the SMAT part, we consider the collisions among the 304-steel balls, motor top, and chamber bottom, where the chamber or motor can be treated as a hot reservoir. Since we assume that all the collisions among them are elastic except the ball-sample collisions, the balls with negligible potential among them can be regarded as the canonical ensemble. By this concept, we construct the link for energy conversion among the motor top, sample bottom, and balls, where the kinetic energy, heat energy, and internal energy are included in the energy conversion. We also introduce the one-dimensional heat equation with uniform-distributed heat source to obtain the temperature distribution of sample, and we use this temperature distribution of sample to connect the Zenner-Hollmann parameter and the heat energy and surface hardness of sample.