Hasegawa Hideo
Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Mar;77(3 Pt 1):031133. doi: 10.1103/PhysRevE.77.031133. Epub 2008 Mar 27.
The Tsallis entropy and Fisher information entropy (matrix) are very important quantities expressing information measures in nonextensive systems. Stationary and dynamical properties of the information entropies have been investigated in the N -unit coupled Langevin model subjected to additive and multiplicative white noise, which is one of typical nonextensive systems. We have made detailed, analytical and numerical study on the dependence of the stationary-state entropies on additive and multiplicative noise, external inputs, couplings, and number of constitutive elements (N) . By solving the Fokker-Planck equation (FPE) by both the proposed analytical scheme and the partial difference equation method, transient responses of the information entropies to an input signal and an external force have been investigated. We have calculated the information entropies also with the use of the probability distribution derived by the maximum-entropy method, whose result is compared to that obtained by the FPE. The Cramér-Rao inequality is shown to be expressed by the extended Fisher entropy, which is different from the generalized Fisher entropy obtained from the generalized Kullback-Leibler divergence in conformity with the Tsallis entropy. The effect of additive and multiplicative colored noise on information entropies is discussed also.
Tsallis熵和Fisher信息熵（矩阵）是表示非广延系统中信息度量的非常重要的量。在受加性和乘性白噪声作用的N单元耦合朗之万模型中研究了信息熵的稳态和动态性质，该模型是典型的非广延系统之一。我们对稳态熵对加性和乘性噪声、外部输入、耦合以及组成元素数量（N）的依赖性进行了详细的解析和数值研究。通过所提出的解析方案和偏微分方程方法求解福克-普朗克方程（FPE），研究了信息熵对输入信号和外力的瞬态响应。我们还利用最大熵方法导出的概率分布计算了信息熵，并将其结果与通过FPE获得的结果进行了比较。结果表明，克拉美-罗不等式由扩展的Fisher熵表示，它不同于从符合Tsallis熵的广义库尔贝克-莱布勒散度得到的广义Fisher熵。还讨论了加性和乘性有色噪声对信息熵的影响。
