Liang X San
Nanjing Institute of Meteorology, Nanjing 210044, China.
Chaos. 2018 Jul;28(7):075311. doi: 10.1063/1.5010253.
Recently, a rigorous formalism has been established for information flow and causality within dynamical systems with respect to Shannon entropy. In this study, we re-establish the formalism with respect to relative entropy, or Kullback-Leiber divergence, a well-accepted measure of predictability because of its appealing properties such as invariance upon nonlinear transformation and consistency with the second law of thermodynamics. Different from previous studies (which yield consistent results only for 2D systems), the resulting information flow, say T, is precisely the same as that with respect to Shannon entropy for systems of arbitrary dimensionality, except for a minus sign (reflecting the opposite notion of predictability vs. uncertainty). As before, T possesses a property called principle of nil causality, a fact that classical formalisms fail to verify in many situation. Besides, it proves to be invariant upon nonlinear transformation, indicating that the so-obtained information flow should be an intrinsic physical property. This formalism has been validated with the stochastic gradient system, a nonlinear system that admits an analytical equilibrium solution of the Boltzmann type.
最近，针对动力学系统中与香农熵相关的信息流和因果关系，已经建立了一种严谨的形式体系。在本研究中，我们针对相对熵（即库尔贝克 - 莱布勒散度）重新建立了该形式体系，相对熵是一种被广泛接受的可预测性度量，因为它具有诸如非线性变换下的不变性以及与热力学第二定律的一致性等吸引人的特性。与先前的研究（仅对二维系统得出一致结果）不同，所得的信息流（设为T）与任意维度系统相对于香农熵的信息流完全相同，只是相差一个负号（反映了可预测性与不确定性的相反概念）。和以前一样，T具有一种称为零因果性原理的性质，这是经典形式体系在许多情况下无法验证的事实。此外，它在非线性变换下被证明是不变的，这表明如此得到的信息流应该是一种内在的物理性质。这种形式体系已经在随机梯度系统中得到验证，随机梯度系统是一个非线性系统，它允许有玻尔兹曼类型的解析平衡解。
