论动态信息理论

邢修三

doi:10.7498/aps.63.230201

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摘要

本文综述了作者的研究成果. 近十年, 作者将现有静态统计信息理论拓展至动态过程, 建立了以表述动态信息演化规律的动态信息演化方程为核心的动态统计信息理论. 基于服从随机性规律的动力学系统(如随机动力学系统和非平衡态统计物理系统)与遵守确定性规律的动力学系统(如电动力学系统)的态变量概率密度演化方程都可看成是其信息符号演化方程, 推导出了动态信息(熵)演化方程. 它们表明: 对于服从随机性规律的动力学系统, 动态信息密度随时间的变化率是由其在系统内部的态变量空间和传递过程的坐标空间的漂移、扩散和耗损三者引起的, 而动态信息熵密度随时间的变化率则是由其在系统内部的态变量空间和传递过程的坐标空间的漂移、扩散和产生三者引起的. 对于遵守确定性规律的动力学系统, 动态信息(熵)演化方程与前者的相比, 除动态信息(熵)密度在系统内部的态变量空间仅有漂移外, 其余皆相同. 信息和熵已与系统的状态和变化规律结合在一起, 信息扩散和信息耗损同时存在. 当空间噪声可略去时, 将会出现信息波. 若仅研究系统内部的信息变化, 动态信息演化方程就约化为与表述上述动力学系统变化规律的动力学方程相对应的信息方程, 它既可看成是表述动力学系统动态信息的演化规律, 亦可看成是动力学系统的变化规律都可由信息方程表述. 进而给出了漂移和扩散信息流公式、信息耗散率公式和信息熵产生率公式及动力学系统退化和进化的统一信息表述公式. 得到了反映信息在传递过程中耗散特性的动态互信息公式和动态信道容量公式, 它们在信道长度和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和静态信道容量公式. 所有这些新的理论公式和结果都是从动态信息演化方程统一推导出的.

关键词:

作者及机构信息

邢修三

1.
北京理工大学物理系, 北京 100081

参考文献

[1]	Cover T M, Thomas J A 1991 Elements of Information Theory (New York: John Wiley and Sons)
[2]	Zhu X L 2000 Fundamentals of Applied Information Theory (Beijing: Tsinghua University Press) (in Chinese) [朱雪龙 2000 应用信息论基础 (北京:清华大学出版社)]
[3]	Zhong Y X 1996 Principles of Information Science (Beijing: Beijing University of Posts and Telecommunications Press) (in Chinese) [钟义信 1996 信息科学原理 (北京:北京邮电大学出版社)]
[4]	Wiener N 1948 Cybernetics (Cambridge: MIT Press)
[5]	Hofkirchner W 1996 The Quest for a Unified Theory of Information (Amsterdam: Gordom and Breach Publishers)
[6]	Xing X S 2004 Trans. Beijing Inst. Technol. 24 1 (in Chinese) [邢修三 2004 北京理工大学学报 24 1]
[7]	Xing X S 2004 Acta Phys. Sin. 53 2852 (in Chinese) [邢修三 2004 物理学报 53 2852]
[8]	Xing X S 2006 Sci. China G 49 1
[9]	Xing X S 2010 Sci. China: Phys. Mech. Astron. 53 607
[10]	Xing X S 2001 Sci. China A 44 1331
[11]	Yang Z R 2007 Quantum Statistical Physics (Beijing: Higher Education Press) (in Chinese) [杨展如 2007 量子统计物理 (北京: 高等教育出版社)]
[12]	Xing X S 1996 Sci. China A 39 1193
[13]	Xing X S 1998 Int. J. Mod. Phys. B 12 2005
[14]	Xing X S 2010 Sci. China: Phys. Mech. Astron. 53 2194
[15]	Xing X S 2003 Acta Phys. Sin. 52 2969 (in Chinese) [邢修三 2003 物理学报 52 2969]
[16]	Frieden B R 1998 Physics from Fisher Information (Cambridge: Cambridge University Press)
[17]	Haken H 1983 Synergetics (Berlin: Springer Verlag)

施引文献

[1]	Cover T M, Thomas J A 1991 Elements of Information Theory (New York: John Wiley and Sons)
[2]	Zhu X L 2000 Fundamentals of Applied Information Theory (Beijing: Tsinghua University Press) (in Chinese) [朱雪龙 2000 应用信息论基础 (北京:清华大学出版社)]
[3]	Zhong Y X 1996 Principles of Information Science (Beijing: Beijing University of Posts and Telecommunications Press) (in Chinese) [钟义信 1996 信息科学原理 (北京:北京邮电大学出版社)]
[4]	Wiener N 1948 Cybernetics (Cambridge: MIT Press)
[5]	Hofkirchner W 1996 The Quest for a Unified Theory of Information (Amsterdam: Gordom and Breach Publishers)
[6]	Xing X S 2004 Trans. Beijing Inst. Technol. 24 1 (in Chinese) [邢修三 2004 北京理工大学学报 24 1]
[7]	Xing X S 2004 Acta Phys. Sin. 53 2852 (in Chinese) [邢修三 2004 物理学报 53 2852]
[8]	Xing X S 2006 Sci. China G 49 1
[9]	Xing X S 2010 Sci. China: Phys. Mech. Astron. 53 607
[10]	Xing X S 2001 Sci. China A 44 1331
[11]	Yang Z R 2007 Quantum Statistical Physics (Beijing: Higher Education Press) (in Chinese) [杨展如 2007 量子统计物理 (北京: 高等教育出版社)]
[12]	Xing X S 1996 Sci. China A 39 1193
[13]	Xing X S 1998 Int. J. Mod. Phys. B 12 2005
[14]	Xing X S 2010 Sci. China: Phys. Mech. Astron. 53 2194
[15]	Xing X S 2003 Acta Phys. Sin. 52 2969 (in Chinese) [邢修三 2003 物理学报 52 2969]
[16]	Frieden B R 1998 Physics from Fisher Information (Cambridge: Cambridge University Press)
[17]	Haken H 1983 Synergetics (Berlin: Springer Verlag)

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论动态信息理论

1. 北京理工大学物理系, 北京 100081

收稿日期: 2014-06-05

修回日期: 2014-07-28

刊出日期: 2014-12-05

摘要: 本文综述了作者的研究成果. 近十年, 作者将现有静态统计信息理论拓展至动态过程, 建立了以表述动态信息演化规律的动态信息演化方程为核心的动态统计信息理论. 基于服从随机性规律的动力学系统(如随机动力学系统和非平衡态统计物理系统)与遵守确定性规律的动力学系统(如电动力学系统)的态变量概率密度演化方程都可看成是其信息符号演化方程, 推导出了动态信息(熵)演化方程. 它们表明: 对于服从随机性规律的动力学系统, 动态信息密度随时间的变化率是由其在系统内部的态变量空间和传递过程的坐标空间的漂移、扩散和耗损三者引起的, 而动态信息熵密度随时间的变化率则是由其在系统内部的态变量空间和传递过程的坐标空间的漂移、扩散和产生三者引起的. 对于遵守确定性规律的动力学系统, 动态信息(熵)演化方程与前者的相比, 除动态信息(熵)密度在系统内部的态变量空间仅有漂移外, 其余皆相同. 信息和熵已与系统的状态和变化规律结合在一起, 信息扩散和信息耗损同时存在. 当空间噪声可略去时, 将会出现信息波. 若仅研究系统内部的信息变化, 动态信息演化方程就约化为与表述上述动力学系统变化规律的动力学方程相对应的信息方程, 它既可看成是表述动力学系统动态信息的演化规律, 亦可看成是动力学系统的变化规律都可由信息方程表述. 进而给出了漂移和扩散信息流公式、信息耗散率公式和信息熵产生率公式及动力学系统退化和进化的统一信息表述公式. 得到了反映信息在传递过程中耗散特性的动态互信息公式和动态信道容量公式, 它们在信道长度和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和静态信道容量公式. 所有这些新的理论公式和结果都是从动态信息演化方程统一推导出的.

关键词:

On dynamic information theory

1.
Department of Physics, Beijing Institute of Technology, Beijing 100081, China

Received Date: 05 June 2014

Accepted Date: 28 July 2014

Published Online: 05 December 2014

Abstract: In this paper, the author presents an overview on his own research works. In recent ten years, we extended the present static statistical information theory to dynamic processes and established a dynamic statistical information theory whose core is the dynamic information evolution equation describing the evolution law of dynamic information. Starting from the idea that the state variable probability density evolution equations of the stochastic dynamic system, the classical and quantum nonequilibrium statistical physical systems obeying stochastic law and the electrodynamic system obeying decterministic law can be regarded as their information symbol evolution equations and the definitions of dynamic information and dynamic entropy, we derived the evolution equations of dynamic information and dynamic entropy that express the evolution laws of dynamic information. These show that for the dynamic systems obeying a stochastic law, the time rate of change of dynamic information densities originates from their drift, diffusion and dissipation in state variable space inside the systems and coordinate space in the transmission processes, and that the time rate of change of dynamic entropy densities is caused by their drift, diffusion and production in state variable space inside the systems and coordinate space in the transmission processes. For the dynamic systems obeying the deterministic law, the evolution equations of dynamic information and dynamic entropy are the same mathematical type as the former except that dynamic information (entropy) density only has drift in state variable space inside the systems. Information and entropy have been connected with the state and change law of the system. Information diffusion and information dissipation occur at the same time. When the space noise can be neglected, information wave will appear. If we only consider the information change inside the systems, the dynamic information evolution equations reduce to information equations corresponding to the dynamic equations which express evolution laws for the above dynamic systems. This reveals that the evolution laws of the respective dynamic systems can be expressed by information equations in a unified fashion. Furthermore, we have presented the formulas for drift and diffusion information flow, information dissipation rate, and entropy production rate and a unified information expression for degradation and self-organizing evolution. Obtained the dynamic mutual information and dynamic channel capacity reflecting the dynamic dissipative character in transmission process, in when in the limiting case the ratio of channel length to signal transmission rate approaches zero, reduces itself to the present static mutual information and static channel capacity. All these new theoritical formulas and results are derived from the dynamic information evolution equation.

Keywords:

dynamic information evolution equation /
information wave /
information flow /
dynamic channel capacity

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