Morrison Philip J, Updike Michael H
Department of Physics and Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712, USA.
Phys Rev E. 2024 Apr;109(4-2):045202. doi: 10.1103/PhysRevE.109.045202.
An inclusive framework for joined Hamiltonian and dissipative dynamical systems that are thermodynamically consistent, i.e., preserve energy and produce entropy, is given. The dissipative dynamics of the framework is based on the metriplectic 4-bracket, a quantity like the Poisson bracket defined on phase space functions, but unlike the Poisson bracket has four slots with symmetries and properties motivated by Riemannian curvature. Metriplectic 4-bracket dynamics is generated using two generators, the Hamiltonian and the entropy, with the entropy being a Casimir of the Hamiltonian part of the system. The formalism includes known previous binary bracket theories for dissipation or relaxation as special cases. Rich geometrical significance of the formalism and methods for constructing metriplectic 4-brackets are explored. Many examples of both finite and infinite dimensions are given.
给出了一个适用于热力学上一致的联合哈密顿和耗散动力系统的包容性框架，即该框架能保持能量并产生熵。该框架的耗散动力学基于度量辛4-括号，这是一个类似于在相空间函数上定义的泊松括号的量，但与泊松括号不同的是，它有四个槽，其对称性和性质由黎曼曲率激发。度量辛4-括号动力学是使用两个生成元生成的，即哈密顿量和熵，其中熵是系统哈密顿部分的一个卡西米尔量。该形式体系包括以前已知的用于耗散或弛豫的二元括号理论作为特殊情况。探讨了该形式体系丰富的几何意义以及构建度量辛4-括号的方法。给出了许多有限维和无限维的例子。
