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.

结果表明，描述非规范哈密顿体系中扩散过程的Fokker-Planck方程表现出一种中电结构，即代数括号形式，生成了与能量守恒和熵增长的热力学原理一致的方程。首先，为表征运动的随机运动方程派生了一个微观的中电支架，该运动方程描述了构成统计集合的元素的随机游动。这种托架完全由泊松算子确定，该泊松算子生成孤立（无扰动）粒子的哈密顿动力学。然后，从微观的中电支架诱发与集合的分布函数的演化相关联的宏观的中电支架。同样，宏观的卡西米尔不变式是从微观动力学继承的。该理论被用于构造无限维哈密顿系统的Fokker-Planck方程，即Charney-Hasegawa-Mima方程。最后，根据与卡西米尔不变量所形成的辛叶上的欧几里得度量张量相对应的“经典度量张量”，确定了中晶托槽的对称（耗散）部分的规范形式。