  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 a representation for the
generators of such processes that acts directly in 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.
