论动态统计信息理论

On Dynamical Statistical Information Theory

将现有的静态统计信息理论拓展至动态过程,建立以动力学系统的信息和信息熵的演化规律为主题及其应用的动态统计信息理论.从态变量演化方程出发,推导出表述信息熵和信息的演化规律的动态信息熵密度和动态信息密度的非线性演化方程.这两个方程表明:信息熵密度随时间的变化率是由其在坐标空间和态变量空间的漂移、扩散和产生三者引起的;而信息密度随时间的变化率则是由其在坐标空间和态变量空间的漂移、扩散和减损三者引起的.给出了漂移信息流和扩散信息流的表达式、信息减损率和信息熵产生率的简明公式.证明了动力学系统内的信息减损(或增加)率等于它的信息熵产生(或减少)率,信息扩散与信息减损同时发生.得到了反映传递过程动态特性的动态互信息公式,它在信道长度与信号传递速度之比趋于零的极限情况下变为现有的静态互信息公式.这些结果都是从信息和信息熵演化方程统一得出的,未增加新的假设.作为这些理论公式的应用,研究和计算了布朗运动的漂移扩散传递、热缺陷产生动力学及分子马达3个动力学课题的信息熵和信息的变化式,给出了高斯信道的动态互信息公式.

Static statistical information theory is extended to dynamic processes and a dynamical statistical information theory is built up, whose subject is the evolution law of the information entropy and information of dynamical systems. Starting from the state variable evolution equation nonlinear evolution equations of dynamic information entropy density and dynamic information density are derived, that describes respectively the evolution law of information entropy and information. These two equations show that the time rate of change of information entropy density originates together from the drift, diffusion and production in coodinate space and state variable space; and that the time rate of change of information density is caused by the drift, diffusion and dissipation in coodinate space and state variable space. Expressions of drift information flow and diffusion information flow, and concise formulas of information dissipation rate and information entropy production rate are given. It is proved that the rate of the information dissipation (or increase) is equal to the rate of information entropy production (or decrease) in the dynamic system, and that information diffusion and information dissipation happen at the same time. Dynamic mutual information reflecting the dynamic character in the transmission process is presented, which in the limiting case when the proportion of channel length to signal transmission rate approaches zero reduces itself to the present static mutual information. All the above results are derived in a unified fasion from evolution equations of information and information entropy without the addition of any extra assumptions. As exampless of application of the above theoretical formulation, information and information entropy as well as their time rates for three dynamic topics are investigated, viz.: the drift-diffusion transmission of Brownian motion, the kinetics of production of thermal defects, and molecular motors; and the dynamic mutual information of the Gaussian channel are presented.