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 are used, 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.
Long-term behavior of the master equation on a countable network and approximation methods of the (stationary) solutions via finite subsystems in the thermodynamic limit
HIFMB Oldenburg（奥尔登堡高等功能材料研究所）；Institute of Mathematics University of Rostock（罗斯托克大学数学研究所）
Intersection matrices associated to geometric-ordered bases of Feynman integrals
Johannes Gutenberg-Universität Mainz（约翰内斯·古腾堡美因茨大学）；Universität Bonn（波恩大学）；University of Zurich（苏黎世大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Xiamen University（厦门大学）
Discrete differential-geometric Poisson brackets: general theory and explicit constructions
University of Warsaw（华沙大学）；Università degli Studi di Milano（米兰大学）；INFN, Sez. di Milano（意大利国家核物理研究所米兰分部）；University of Basilicata（巴西利卡塔大学）；INFN, Sez. di Napoli（意大利国家核物理研究所那不勒斯分部）
