In this paper, continuous-time master equations with finite states employed in nonequilibrium statistical mechanics are formulated in the language of discrete geometry. In this formulation, chains in algebraic topology play roles, and master equations are described on graphs that consist of vertexes representing states and of directed edges representing transition matrices. It is then shown that master equations under the detailed balance conditions are equivalent to discrete diffusion equations, where the Laplacians are defined as self-adjoint operators with respect to introduced inner products. An isospectral property of these Laplacians is shown for non-zero eigenvalues, and its applications are given. The convergence to the equilibrium state is shown by analyzing this class of diffusion equations. In addition, a systematic way to derive closed dynamical systems for expectation values is given. For the case that the detailed balance conditions are not imposed, master equations are expressed as a form of a continuity equation.

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来自主方程的扩散方程——离散几何方法

在本文中，非平衡统计力学中使用的具有有限状态的连续时间主方程是用离散几何的语言表述的。在这个公式中，代数拓扑中的链发挥作用，主方程在由表示状态的顶点和表示转移矩阵的有向边组成的图上描述。然后证明在详细平衡条件下的主方程等效于离散扩散方程，其中拉普拉斯算子被定义为关于引入内积的自伴随算子。这些拉普拉斯算子的等谱性质显示为非零特征值，并给出了它的应用。通过分析这类扩散方程可以看出收敛于平衡状态。此外，给出了推导出期望值的封闭动力系统的系统方法。对于不施加详细平衡条件的情况，主方程表示为连续方程的一种形式。