流分解揭示马尔可夫过程的动力学结构
Flow Decomposition Reveals Dynamical Structure of Markov Process
摘要：
马尔可夫过程在文献的各个方面都有着广泛的应用，对于理解非平衡过程尤其重要。本文引入了一般马氏过程的一种分解。该分解将过程分解为3个独立的部分：平稳分布、对称详细平衡部分和反对称破坏详细平衡部分。这种完整的分解捕获了过程的稳态和动态，为构建或分析问题提供了一个优雅的视角。在此基础上，给出了相对熵的唯一定义，形式化地分离了详细平衡部分和破坏详细平衡部分的影响。我们发现，相对基尼熵生产的文件中引入的不受影响的非细致平衡部分的过程。这一性质在一般离散情况下不适用于其它熵定义。
英文摘要：
Markov process is widely applied in almost all aspects of literature, especially important for understanding non-equilibrium processes. We introduce a decomposition to general Markov process in this paper. This decomposition decomposes the process into 3 independent parts: stationary distribution, symmetric detailed-balance part and anti-symmetric breaking detailed-balance part. This complete decomposition captures the steady state as well as the dynamics of the process, providing an elegant perspective for construction or analyzing problems. In light of the decomposition, a unique definition of relative entropy is found to formally separate the effect of detailed-balance part and breaking detailed-balance part. We find that the relative Gini entropy production introduced in the paper is not affected by the non-detailed balance part of the process. This property do not holds for other entropy definition in general discrete case.
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