Trajectory-based methods are well-developed to approximate steady-state probability distributions for stochastic processes in large-system limits. The trajectories are solutions to equations of motion of Hamiltonian dynamical systems, and are known as eikonals. They also express the leading flow lines along which probability currents balance. The existing eikonal methods for discrete-state processes including chemical reaction networks are based on the Liouville operator that evolves generating functions of the underlying probability distribution. We have previously derived [1, 2] a representation for the generators of such processes that acts directly on the hierarchy of moments of the distribution, rather than on the distribution itself or on its generating function. We show here how in the large-system limit the steady-state condition for that generator reduces to a mapping from eikonals to the ratios of neighboring factorial moments, as a function of the order k of these moments. The construction shows that the boundary values for the moment hierarchy, and thus its whole solution, are anchored in the interior fixed points of the Hamiltonian system, a result familiar from Freidlin–Wenztell theory. The direct derivation of eikonals from the moment representation further illustrates the relation between coherent-state and number fields in Doi–Peliti theory, clarifying the role of canonical transformations in that theory.

基于轨迹的方法已得到很好的发展，以近似大系统限制中随机过程的稳态概率分布。轨迹是哈密顿动力系统运动方程的解，被称为 eikonals。它们还表达了概率流平衡的前导流线。包括化学反应网络在内的离散状态过程的现有 eikonal 方法基于 Liouville 算子，该算子演化出潜在概率分布的生成函数。我们之前已经推导出 [1, 2] 这种过程的生成器的表示，它直接作用于分布的矩层次，而不是作用于分布本身或其生成函数。k这些时刻。该构造表明矩层次的边界值及其整个解都锚定在哈密顿系统的内部不动点中，这是 Freidlin-Wenztell 理论中熟悉的结果。从矩表示中直接推导出 eikonals 进一步说明了 Doi-Peliti 理论中相干态和数场之间的关系，阐明了正则变换在该理论中的作用。