Hamilton系统中Fokker-Planck方程的耗散括号及亚重流形的刻画
Dissipative Brackets for the Fokker-Planck Equation in Hamiltonian Systems and Characterization of Metriplectic Manifolds
摘要：
它表明，Fokker-Planck方程描述非正则哈密顿系统中的扩散过程表现出metriplectic结构，即代数括号形式主义，产生的能量守恒和熵增长的热力学原理一致的方程。首先，一个微观metriplectic括号推导出的随机运动方程，其特征在于构成统计系综的元素的随机行走。这样的括号是完全由泊松算子产生的孤立（未扰动）粒子的哈密顿动力学确定。然后，从微观矩阵括号中导出与系综分布函数演化相关的宏观矩阵括号。类似地，宏观Casimir不变量是从微观动力学继承的。应用该理论构造了无限维哈密顿系统的Fokker-Planck方程，即Charney-Hasegawa-Mima方程。最后，规范形式的对称（耗散）部分的metriplectic括号中确定的“规范度量张量”对应于欧几里得度量张量的辛叶的Casimir不变量。
英文摘要：
It is shown that the Fokker-Planck equation describing diffusion processes in noncanonical Hamiltonian systems exhibits a metriplectic structure, i.e. an algebraic bracket formalism that generates the equation in consistency with the thermodynamic principles of energy conservation and entropy growth. First, a microscopic metriplectic bracket is derived for the stochastic equations of motion that characterize the random walk of the elements constituting the statistical ensemble. Such bracket is fully determined by the Poisson operator generating the Hamiltonian dynamics of an isolated (unperturbed) particle. Then, the macroscopic metriplectic bracket associated with the evolution of the distribution function of the ensemble is induced from the microscopic metriplectic bracket. Similarly, the macroscopic Casimir invariants are inherited from microscopic dynamics. The theory is applied to construct the Fokker-Planck equation of an infinite dimensional Hamiltonian system, the Charney-Hasegawa-Mima equation. Finally, the canonical form of the symmetric (dissipative) part of the metriplectic bracket is identified in terms of a `canonical metric tensor' corresponding to an Euclidean metric tensor on the symplectic leaves foliated by the Casimir invariants.
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