\section{Introduction: directly solving for fluctuation moments in
stochastic Chemical Reaction Networks}

When solving for the steady states of stochastic processes,
approximation methods are needed in all but the simplest cases.  The
choice of representation for the process and its generator can dictate
the efficiency or comprehensibility of approximation methods that are
available in that representation.

Two representations for the generators of stochastic processes that
are now more than a century old~\cite{vanKampen:Stoch_Proc:07} are the
transition matrix of the master equation, acting directly on
probability distributions, and the generator of the Liouville
equation, acting on generating functions that are the Laplace
transforms of those distributions.

A third representation can be constructed for the generator acting
directly on the hierarchy of moments. For a class of discrete-state
stochastic processes, we have obtained an explicit form for such a
generator acting on the moment
hierarchy~\cite{Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17}.
These are the models of stochastic Chemical Reaction Networks (CRNs),
which have been extensively
studied~\cite{Feinberg:def_01:87,Anderson:product_dist:10,Gunawardena:CRN_for_bio:03,Baez:QTRN_eq:14,Polettini:open_CNs_I:14,Polettini:diss_def:15}
and are of both mathematical and practical interest.  The moments of a
probability distribution for a CRN arrange into a lattice with
coordinates $k$, where $k$ is a vector whose components determine the
moment of interest.

For CRNs with only finitely many reactions, the generator in the
moment lattice takes the form of a discrete Laplacian, and is solved
analogously to a heat equation.  Because the dynamical equation for
moments is linear, the steady-state condition can be reduced to a
collection of constraints, one per lattice site.  Time-invariance of
the moment at index $k$ implies a constraint on the \emph{ratios} of
neighboring moments to the moment at $k$.

If a CRN has only one dynamical degree of freedom, a discrete
Laplacian acting on its moments reduces to a recursion relation on the
moment ratios.  We showed
in~\cite{Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17} that such
recursions can be solved by a method of matched asymptotic expansions,
in which low-order modes are solved recursively as one would expect,
upward from the mean or variance, but high-order modes are solved
(counterintuitively) by anchoring the moment ratios at large $k$,
where they converge asymptotically to values set by parameters in the
generator.  For multivariate distributions, however, the corresponding
method of solution encounters the difficulty faced in the solution of
other Laplacian equations, which is the specification of boundary
conditions.  For the CRN moment hierarchies, the difficult boundaries
are those on which one or more of the moments is \emph{not} of
asymptotically large order, since there is no approximation scheme
which gives an estimate of such moments.


An alternative to direct solution of Laplacian equations, which
addresses the problem of boundary conditions both computationally and
conceptually, is the WKB method, which computes the leading
exponential dependence of probability distributions in large-system
limits.  The key quantities in the WKB approximation are eikonals:
locally least-improbable trajectories for improbable excursions, along
which balance of probability currents determines the large-deviation
behavior of the
distribution.\footnote{See~\cite{Touchette:large_dev:09} for the
explanation that large-deviations scaling is the separation of scale
from structure in probability distributions for aggregate statistics.
Trajectories of eikonals are properties of the large-deviation rate
functions, while the probabilities associated with them are
exponential in the scale factors.}  Eikonal solutions for stationary
distributions, boundary values, and first-passage statistics of
diffusion solutions in basins of attraction were extensively developed
by Freidlin and Wentzell~\cite{Freidlin:RPDS:98}.  Related
formulations in terms of 2-field functional integrals, for stochastic
processes on discrete state spaces including those of CRNs, were
derived by Doi and
Peliti~\cite{Doi:SecQuant:76,Doi:RDQFT:76,Peliti:PIBD:85,Peliti:AAZero:86}.


The stability properties of WKB methods are dual to those of the
moment recursions
in~\cite{Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17}, in that
they propagate probability
distributions outward into attenuating tails from the regions near the
mean where probability is most concentrated.  Those central regions
are the main scale factors in the recursion approach, where matching
conditions must be imposed between low and high moment orders.
It is by grounding asymptotic regions in the centers of mass of probability
distributions that WKB methods provide approximate solution methods
for boundary-value problems~\cite{Freidlin:RPDS:98}.


In this paper we derive the eikonal method to directly approximate the
ratios of adjacent moments in the moment hierarchy.  Our starting
point will be the generator
from~\cite{Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17}.  The
steady-state condition will reduce to a set of stationary-path
equations that are the same as those for eikonals in Doi-Peliti
theory.  The new feature resulting from the moment representation is a
mapping from the order $k$ of a moment to a specific position along
the particular escape trajectory that governs the ratio of that moment
to its local neighbors.

The analysis leads to three results: 1) The generator in the moment
representation reduces in the large-system limit to the Liouville
operator from the generating-function representation.  2) Boundary
solutions for the WKB approximation are anchored on a discrete set of
interior fixed points in the steady state, as they are in
Freidlin-Wentzell theory.  3) The reduction of the generator from the
moment representation to the Liouville operator clarifies the roles of
number fields relative to coherent-state fields in Doi-Peliti theory,
as these represent both activities and relative migration rates in
steady states. 

The presentation is organized as follows: Sec.~\ref{sec:three_reps}
reviews three representations of the generator for a population
process such as a CRN: the transition matrix of the master equation,
the Liouville operator, and the generator in the moment
representation.  Sec.~\ref{sec:eikonal_solns} derives the eikonal
approximation for the moment hierarchies of CRNs.  The large-number
limit of the steady-state recursions of the moment hierarchy is
derived, recovering the Liouville operator. It is then shown how the
orders of the moments map to positions on the eikonals of the
Liouville represenation.  Sec.~\ref{sec:examples} illustrates the
eikonal construction on three example CRNs, beginning with a model
that is exactly solvable to demonstrate convergence and show how
finite-size effects are handled, and culminating in a model where
eikonal methods are needed, but where they also break down in some
regions due to trajectory crossing.


\section{Representations of the generator for discrete-state
stochastic processes}
\label{sec:three_reps}

To begin, we specify and introduce notation for the class of
stochastic processes to be considered, and review the three
representations of the generator.  We will consider continuous-time
Markov processes on discrete state spaces that are lattices of
non-negative integers in some dimension $P$.  Examples include a
variety of population processes~\cite{Smith:evo_games:15}, among which
are stochastic CRNs described in more detail below.

States are indexed by vectors ${\rm n} \equiv \left[ {{\rm n}}_p \right]$, in
which $p \in 1 , \ldots , P$ are called \textit{species} and ${{\rm n}}_p$
is a non-negative integer giving the count (particles, individuals,
etc.) of species $p$ in the population.  Probability distributions
over states ${\rm n}$ are denoted ${\rho}_{{\rm n}}$.  A time-dependent
probability evolves under a master equation of the form
\begin{equation}
  \frac{d}{d\tau}
  {\rho}_{{\rm n}} = 
  \sum_{{{\rm n}}^{\prime}}
  {\rm T}_{{\rm n} {{\rm n}}^{\prime}}
  {\rho}_{{{\rm n}}^{\prime}} . 
\label{eq:master_eq_popproc}
\end{equation}
${\rm T}_{{\rm n} {{\rm n}}^{\prime}}$ is a left-stochastic
matrix\footnote{Left-stochastic means that $\sum_{{\rm n}} {\rm T}_{{\rm n}
{{\rm n}}^{\prime}} = 0, \; \forall {{\rm n}}^{\prime}$, so that
Eq.~(\ref{eq:master_eq_popproc}) conserves probability.} called the
\textit{transition matrix}, and is one representation of the
generator.  


\subsection{Moment-generating function and Liouville operator}

The Laplace transform of a distribution ${\rho}_{{\rm n}}$ is called its
moment-generating function (MGF).  While the analytic structure of the
Laplace transform is often a source of elegant solution methods and
the reason to use the generating
function~\cite{Flajolet:anal_comb:09}, many problems require only its
formal representation as a power series, and the ensuing linear
algebra.  For these, an abstract operator algebra such as that of
Doi~\cite{Doi:SecQuant:76,Doi:RDQFT:76} is more economical, especially
for evaluating quadratures of Eq.~(\ref{eq:master_eq_popproc}).  The
Doi algebra is reviewed in numerous
places~\cite{Mattis:RDQFT:98,Kamenev:DP:02,Smith:LDP_SEA:11,Baez:QTRN:13,Baez:QTSM:17},
and we provide only a summary here.

\textit{Lowering operators} $a_p$ that remove particles from a
population state, and \textit{raising operators} $a^{\dagger}_q$ that
introduce particles, are defined by their commutation relation
\begin{equation}
  \left[
    a_p , a^{\dagger}_q
  \right] = 
  {\delta}_{pq} , 
\label{eq:ladd_comm}
\end{equation}
where $p , q \in 1 , \ldots , P$ and ${\delta}_{pq}$ is the Kronecker
$\delta$ symbol.

\textit{Number states}, corresponding to point distributions at each
lattice index ${\rm n}$, are defined by the action of the lowering and
raising operators on them, as 
\begin{align}
  a_p \left| {\rm n} \right)
& = 
  {{\rm n}}_p \left| {\rm n} - 1_p \right)
& 
  a^{\dagger}_p \left| {\rm n} \right)
& = 
  \left| {\rm n} + 1_p \right) . 
\label{eq:numb_ladd_def}
\end{align}
It follows that all number states can be built up from a ground state
$\left| 0 \right)$ by multiplication with suitable powers of the
$\left\{ a^{\dagger}_p \right\}$.  

The MGF of a distribution ${\rho}_{{\rm n}}$ is the sum of number states
weighted by the coefficients of ${\rho}_{{\rm n}}$, denoted 
\begin{equation}
  \left| \phi \right) = 
  \sum_{{\rm n}}
  {\rho}_{{\rm n}}
  \left| {\rm n} \right) . 
\label{eq:Doi_state_n_basis}
\end{equation}
Eq.~(\ref{eq:master_eq_popproc}) implies a corresponding evolution
equation for the MGF, known as the Liouville equation, 
\begin{equation}
  \frac{d}{d\tau}
  \left| \phi \right) = 
  - \mathcal{L} \! \left( a^{\dagger} , a \right)
  \left| \phi \right) 
\label{eq:Liouville_eq}
\end{equation}
in which the \textit{Liouville operator} $\mathcal{L} \! \left(
a^{\dagger} , a \right)$ is derived from the matrix of values ${\rm
T}_{{\rm n} {{\rm n}}^{\prime}}$. 

Computing the trace of a probability distribution is represented in
the operator algebra in terms of an inner product with a dual state
$\left( 0 \right|$ (which has the interpretation of a projection
operator).  Under an inner product called the Glauber norm, 
\begin{equation}
  \left( 0 \right| 
  e^{\sum_p a_p}
  \left| {\rm n} \right) = 
  1 \; ; \; \forall {\rm n} ,
\label{eq:Glauber_norm}
\end{equation}
in which all number states are normalized, the generating functions
become vectors in a Hilbert space, completing the definition of the
linear algebra.  For some applications, including the Peliti
functional integral introduced below, it remains useful to express the
generating function as a function of a vector $z \equiv \left[ z_p
\right]$ of complex numbers.  The analytic representation is extracted
from the formal power series by the inner product
\begin{equation}
  \left( 0 \right|
  e^{\sum_p z_p a_p}
  \left| \phi \right) = 
  \phi \! \left( z \right) . 
\label{eq:Doi_to_anal}
\end{equation}

\subsubsection{Stochastic CRNs; the form of their Liouville operator} 

The subset of population processes that we will consider in detail are
the stochastic CRNs.  Each elementary event is an instance of some
\textit{reaction}, which removes one set of particles from the
population and places another set of particles into it.  The number of
particles of each type that are removed or added give the
stoichiometry of the reaction, and the rate of the reaction equals the
probability for an instance to occur per unit time.  Particles are
sampled proportionally and without replacement, leading to mass-action
kinetics at large numbers.

In a decomposition of CRNs due to Horn and
Jackson~\cite{Horn:mass_action:72} and
Feinberg~\cite{Feinberg:def_01:87}, the sets of particles either
removed or {\textcolor{red} created} are called \textit{complexes}.
We denote the complexes that appear in a CRN with subscripts $i$, $j$;
each reaction corresponds to a unique ordered pair $\left( ji \right)$
that removes complex $i$ and adds complex $j$.

The reaction network is specified by an adjacency matrix
${\mathbb{A}}_{\kappa}$ on the set of complexes, with the form
\begin{equation}
  {\mathbb{A}}_{\kappa} \equiv 
  \sum_{\left( i , j \right)}
  \left( w_j - w_i \right)
  {\kappa}_{ji}
  w_i^T . 
\label{eq:CRN_compl_adj}
\end{equation}
$w_i$, a column vector, is an indicator function with 1 in the $i$th
position and zero elsewhere; $w_i^T$ is its transpose.
${\kappa}_{ji}$ is the {\textcolor{red} reaction} rate constant for
reaction $\left( ji \right)$, with respect to units $d\tau$ for time.
${\mathbb{A}}_{\kappa}$ is left-stochastic on complexes, as ${\rm
T}_{{\rm n} {{\rm n}}^{\prime}}$ is on states.

The matrix ${\mathbb{A}}_{\kappa}$ acts to the right on a column
vector of activities for complexes.  These activities are extracted
from a state $\left| \phi \right)$ by a vector ${\psi}_Y \equiv \left[
{\psi}_Y^i \right]$ in which the components are the operators 
\begin{equation}
  {\psi}_Y^i \! \left( a \right) \equiv 
  \prod_p 
  a_p^{y^i_p}
\label{eq:Psi_Yi_def}
\end{equation}
$Y \equiv \left[ y^i_p \right]$ is the matrix of stoichiometric
components, giving the counts of species $p$ in each complex $i$.  

The {\textcolor{blue} {rate}} equation for the CRN defined by
${\mathbb{A}}_{\kappa}$ and $Y$ can be written in the form 
\begin{equation}
  \frac{dn}{d\tau} = 
  Y 
  {\mathbb{A}}_{\kappa}
  {\psi}_Y \! \left( n \right)   
\label{eq:CRN_firstmom_EOM}
\end{equation}
Here $n$ is a vector that corresponds 
in the mean-field approximation
to the vector of expectations $\left< {\rm n} \right> \equiv \sum_{{\rm n}}
{\rm n} {\rho}_{{\rm n}}$.  

As we show
in~\cite{Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17}, the
Liouville operator for this class of CRNs takes the compact form 
\begin{equation}
  - \mathcal{L} \! \left( a^{\dagger} , a \right) = 
  {\psi}^T_Y \! \left( a^{\dagger} \right)
  {\mathbb{A}}_{\kappa} \, 
  {\psi}_Y \! \left( a \right)
\label{eq:L_of_as}
\end{equation}
The exact equation for the time evolution of all moments in the
distribution ${\rho}_{{\rm n}}$, follows from this form and from the
commutation relations~(\ref{eq:ladd_comm}).

\subsubsection{The Doi-Peliti functional integral and its action
functional}

Peliti~\cite{Peliti:PIBD:85,Peliti:AAZero:86} introduced a
path-integral representation for the quadrature of the Liouville
equation~(\ref{eq:Liouville_eq}) based on an expansion in
\textit{coherent states}.  For $\xi \equiv \left[ {\xi}_p \right]$ a
vector of complex coefficients, the coherent state 
\begin{equation}
  \left| \xi \right) \equiv 
  e^{
    \left( a^{\dagger} - 1 \right) \xi
  } 
  \left| 0 \right) 
\label{eq:CS_right_def}
\end{equation}
is an eigenstate of the lowering operators, with eigenvalues
\begin{equation}
  a_p \left| \xi \right) = 
  {\xi}_p \left| \xi \right) . 
\label{eq:CS_ladd_eigstate}
\end{equation}
In Eq.~(\ref{eq:CS_right_def}) we introduce a convention that
$a^{\dagger}$ is a row vector, $\xi$ is a column vector, and
$a^{\dagger} \xi$ is the vector inner product of the two.

Through constructions that are now
standard~\cite{Smith:LDP_SEA:11,Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17},
the time-dependent solution to Eq.~(\ref{eq:Liouville_eq}), converted
to complex arguments by Eq.~(\ref{eq:Doi_to_anal}), may be written as
the 2-field functional integral
\begin{equation}
  {\phi}_T \! \left( z \right) = 
  \int_0^T 
  \mathcal{D} {\xi}^{\dagger}
  \mathcal{D} {\xi}
  e^{
    \left( z - {{\xi}^{\dagger}}_T \right) {\xi}_T 
  }
  e^{- S}
  {\phi}_0 \! 
  \left( 
    {\xi}_0^{\dagger}
  \right) . 
\label{eq:DP_2FFI_CS}
\end{equation}
${\xi}^{\dagger}$ and $\xi$ are now histories indexed by time $\tau \in
\left[ 0 , T \right]$.  ${\phi}_0$ is the generating function for the
initial distribution at $\tau = 0$, and ${\phi}_T$ is its image evolved
to $\tau = T$.

All influence from the CRN dynamics in the
integral~(\ref{eq:DP_2FFI_CS}) comes through the action functional
\begin{equation}
  S = 
  \int_0^T d\tau 
  \left\{ 
    - \left( d_{\tau} {\xi}^{\dagger} \right)
    \xi + 
    \mathcal{L} \! \left( {\xi}^{\dagger} , \xi \right)
  \right\} .
\label{eq:S_from_L_CS}
\end{equation}
$\xi$ appears as a coordinate variable, ${\xi}^{\dagger}$ is its
conjugate momentum, and the place of the Hamiltonian is filled by the
Liouville function $\mathcal{L}$ in which $\xi$ takes the place of
operator $a$ and ${\xi}^{\dagger}$ takes the place of $a^{\dagger}$.
For the generator class~(\ref{eq:L_of_as}) of CRNs, we have 
\begin{equation}
  - \mathcal{L} \! \left( {\xi}^{\dagger} , \xi \right) = 
  {\psi}^T_Y \! \left( {\xi}^{\dagger} \right)
  {\mathbb{A}}_{\kappa} \, 
  {\psi}_Y \! \left( \xi \right) . 
\label{eq:L_of_xis}
\end{equation}


\subsubsection{Canonical transformations: coherent-state and
number-potential coordinates}

In terms of the coherent states used to construct the functional
integral~(\ref{eq:DP_2FFI_CS}), the components of the number index
${\rm n}$ correspond to a bilinear form $n_p = {\xi}^{\dagger}_p {\xi}_p$.
The number field may be made an elementary degree of freedom as part
of a canonical transformation~\cite{Goldstein:ClassMech:01} of the
coordinates of the Hamiltonian dynamical system, 
\begin{align}
  {\xi}_p^{\dagger}
& \equiv 
  e^{{\eta}_p}
& 
  {\xi}_p
& \equiv 
  e^{-{\eta}_p}
  n_p .
\label{eq:xis_to_n_eta}
\end{align}
$\eta$, the conjugate momentum to the number coordinate, admits a
variety of interpretations.  In the construction of the equilibrium
free energy of a chemical mixture, it has the dimensions of a chemical
potential~\cite{Smith:LDP_SEA:11}, so we will refer to $\left( \eta ,
n \right)$ as number-potential coordinates.  Both $\xi$ and $n$ have
dimensions of species numbers, and one of the subtleties of Doi-Peliti
theory is understanding the difference of their meaning in WKB-type
approximations.  The eikonal approximation for moment ratios is
interesting because it establishes a relation between the two in the
same observable.

The Doi-Peliti integral in number-potential coordinates becomes
\begin{equation}
  {\phi}_T \! \left( z \right) = 
  \int_0^T 
  \mathcal{D} \eta
  \mathcal{D} n
  e^{
    \left( z e^{-{\eta}_T} - 1 \right) n_T 
  }
  e^{- S}
  {\phi}_0 \! 
  \left( 
    e^{{\eta}_0}
  \right) , 
\label{eq:DP_2FFI_neta}
\end{equation}
in which the action is 
\begin{equation}
  S = 
  \int_0^T d\tau 
  \left\{ 
    - \left( d_{\tau} \eta \right)
    n + 
    \mathcal{L} \! \left( \eta , n \right)
  \right\} . 
\label{eq:S_from_L_neta}
\end{equation}
Preservation of the form of the kinetic term as $- \left( d_{\tau}
{\xi}^{\dagger} \right) \xi = - \left( d_{\tau} \eta \right) n$
ensures that Eq.~(\ref{eq:DP_2FFI_neta}) is canonical.  The Liouville
function $\mathcal{L} \! \left( \eta , n \right)$ is the same as
$\mathcal{L} \! \left( {\xi}^{\dagger} , \xi \right)$ in
Eq.~(\ref{eq:S_from_L_CS}) evaluated in the transformed variables.


\subsection{Moment hierarchy and its generator}

For CRNs with mass action rates, the macroscopic expression of
proportional sampling without replacement at the microscale, the
moments that have a simple behavior are neither ordinary moments nor
cumulants, but rather the expectations of falling
factorials~\cite{Baez:QTRN:13}, which we term \textit{factorial
moments}.  A compact notation for the falling factorial of number
$n_p$ at order $k_p$ is
\begin{align}
  {{\rm n}}_p^{\underline{k_p}} 
& \equiv 
  \frac{
    {{\rm n}}_p !
  }{
    \left( {{\rm n}}_p - k_p \right) ! 
  }
& ; \; k_p \le {{\rm n}}_p 
\nonumber \\ 
& \equiv 
 0 
& ; \; k_p > {{\rm n}}_p .
\label{eq:factorial_moment_not}
\end{align}

In a CRN, the activity products corresponding to the complexes $i$ in
a state ${\rm n}$ make up a column vector that we denote ${\Psi}_Y \equiv
\left[ {\Psi}_Y^i \right]$ with
\begin{equation}
  {\Psi}_Y^i \! \left( {\rm n} \right) \equiv 
  \prod_p 
  {{\rm n}}_p^{\underline{y_p^i}} 
\label{eq:Psi_Y_def}
\end{equation}
In the Liouville equation~(\ref{eq:Liouville_eq}), the
complex-lowering operators ${\psi}_Y^i$ from Eq.~(\ref{eq:Psi_Yi_def})
act respectively on number states and coherent states as
\begin{align}
  {\psi}_Y^i \! \left( a \right) 
  \left| {\rm n} \right) 
& = 
  {\Psi}_Y^i \! \left( {\rm n} \right)
  \left| {\rm n} - y^i \right) 
\nonumber \\ 
  {\psi}_Y^i \! \left( a \right) 
  \left| \xi \right) 
& = 
  {\psi}_Y^i \! \left( \xi \right) 
  \left| \xi \right) . 
\label{eq:Psi_psi_on_n_xi}
\end{align}
Finally, in parallel to the designation $\phi \! \left( z \right)$
from Eq.~(\ref{eq:Doi_to_anal}) for the MGF, we denote the lattice of
factorial moments $\Phi \equiv \left[ {\Phi}_k \right]$ with
\begin{equation}
  {\Phi}_k \equiv 
  \left<
    \prod_p
    {{\rm n}}_p^{\underline{k_p}}
  \right> . 
\label{eq:Phi_def}
\end{equation}

We showed in~\cite{Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17}
that the Liouville equation~(\ref{eq:Liouville_eq}), used to evolve
the probability distribution defining the average in
Eq.~(\ref{eq:Phi_def}), implies the equation of motion for the moment
hierarchy
\begin{align}
\lefteqn{
  \frac{d}{d\tau} {\Phi}_k = 
  \dot{\prod_p}
  \left[ 
    \sum_{j_p = 0}^{k_p}
    \left(
      \begin{array}{c}
        k_p \\ j_p 
      \end{array}
    \right)
    Y_p^{\underline{j_p}} \,  
    e^{
      {\left( k - j \right)}_p \, \partial / \partial Y_p
    }
  \right]
  {\mathbb{A}}_{\kappa}  
  \left< 
    {\Psi}_{Y} \! \left( {\rm n} \right) 
  \right>
} & 
\nonumber \\ 
& = 
  \sum_{j_1 = 0}^{k_1}
  \left(
    \begin{array}{c}
      k_1 \\ j_1 
    \end{array}
  \right) \ldots 
  \sum_{j_P = 0}^{k_P}
  \left(
    \begin{array}{c}
      k_P \\ j_P 
    \end{array}
  \right)  
  \left[ 
    \dot{\prod_p}
    Y_p^{\underline{j_p}} \,  
  \right]
  {\mathbb{A}}_{\kappa}  
  \left< 
    {\Psi}_{Y + \left( k - j \right)} \! \left( {\rm n} \right) 
  \right> 
\nonumber \\ 
& \equiv 
  \sum_{k^{\prime}}
  {\Lambda}_{k k^{\prime}} 
  {\Phi}_{k^{\prime}} . 
\label{eq:Glauber_moment_fact_prod}
\end{align}
The notation ${\dot{\prod}}_p$ stands for a product of terms over the
species index $p$ that is performed within each component of a row
vector over complexes $i$.  $Y_p \equiv \left[ y_p^i \right]$ is the
row vector of stoichiometric coefficients for species $p$ in each
complex, and both factorials $Y_p^{\underline{j_p}}$ and integer shift
operators $e^{ {\left( k - j \right)}_p \, \partial / \partial Y_p }$
are evaluated component-wise in indices $i$.  The resulting row vector
is contracted on the left with ${\mathbb{A}}_{\kappa}$.  The matrix
$\Lambda \equiv \left[ {\Lambda}_{k k^{\prime}} \right]$ is the
representation of the generator acting on the moment hierarchy.


\section{The eikonal approximation for the moment hierarchy}
\label{sec:eikonal_solns}

Eikonal approximations for the probability distribution
${\rho}_{{\rm n}}$, in steady state, are based on a leading-exponential
expansion $ {\rho}_{{\rm n}} \sim e^{- \Xi \left( {\rm n} \right)}$ expressing
the large-deviation scaling. In Section \ref{subsec:LD scaling}, we
demonstrate how this approximation for the steady-state probability
distribution results in a corresponding approximation for the
factorial moments. We show first however that this approximation for
the factorial moments, in a large-number limit in steady state, may be
arrived at directly from the moment hierarchy as well.

The factorial moments Eq.~(\ref{eq:Glauber_moment_fact_prod}), under
the steady state condition, satisfy a hierarchy of relations $0 =
\sum_{k^{\prime}} {\Lambda}_{k k^{\prime}} {\Phi}_{k^{\prime}}$ By
dividing through by ${\Phi}_k$, each of these may be written as a
constraint on ratios of neighboring moments ${\Phi}_{k^{\prime}}$ to
${\Phi}_k$.  In one dimension (for CRN's involving only one species)
these become recursion relations, which may be solved by a method of
matched asymptotic
expansions~\cite{Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17}
downward from $k \rightarrow \infty$ and upward from $k = 0$.

It will not be surprising that the steady-state condition is
equivalent to a condition $\mathcal{L} = 0$ in the Liouville
representation, a relation suggested by
Eq.~(\ref{eq:Liouville_eq}). We show in the following that the
steady-state condition for moments reduces in the large-system limit
to a particular form of the condition $\mathcal{L} = 0$ equivalent to
the coherent-state expression in the action~(\ref{eq:S_from_L_CS}).
However, as is well known in the literature on \textit{momentum-space
WKB methods}~\cite{Dykman:chem_paths:94,Assaf:mom_WKB:17}, the
$\mathcal{L} = 0$ submanifold contains ``momentum'' values (our
potential field $\eta$) consistent with derivatives of many different
distributions.  To isolate the stationary distribution requires
additional information from boundary conditions; which is the problem
that makes direct solution of higher-dimensional moment equations
difficult.  What is gained in passing to the continuum limit is a
non-local relation of the momentum values at each point through
stationary trajectories under variation of the
action~(\ref{eq:S_from_L_CS}) anchored in the
\emph{classical fixed points} of the rate equation, a result developed
extensively for the solution of boundary value and first-passage
problems by Freidlin and Wentzell~\cite{Freidlin:RPDS:98}, and since
generalized to many classes of stochastic
processes~\cite{Dykman:chem_paths:94,Assaf:mom_WKB:17}.

\subsection{Large-number limit of the generator in the moment
representation} 

The counterpart to a continuum limit in the moment hierarchy comes
from the observation that, if both particle numbers and the orders of
moments are large, the ratios of factorial moments at adjacent $k$
indices should change slowly with $k$.  If that is the case, then
factorial moments offset by finite lattice shifts will be related
through one ratio of moments in each direction $p$, raised to powers.
For the offset term appearing in
Eq.~(\ref{eq:Glauber_moment_fact_prod}), that limiting behavior
becomes
\begin{align}
  \left< 
    {\Psi}_{Y + \left( k - j \right)}^i \! \left( {\rm n} \right) 
  \right> 
& \underset{n \gg 1}{\rightarrow}
  {
    \left. 
      \left(
        \prod_p
        R_p^{y^i_p - j_p}
      \right)
    \right|
  }_k
  \left< 
    \prod_p 
    {{\rm n}}_p^{\underline{k_p}}
  \right> 
\nonumber \\ 
& = 
  {
    \left. 
      {\psi}_{Y-j}^i \! \left( R \right) 
    \right|
  }_k
  {\Phi}_k .
\label{eq:jacquard_walk}
\end{align}
Here ${\left.  R_p \right| }_k$ denotes $\left< {{\rm n}}^{\underline{k +
1_p}} \right> / \left< {{\rm n}}^{\underline{k}} \right>$: the ratio of
factorial moments incremented by the $p$th component of $k$, in a
neighborhood of the index $k$ of the reference moment ${\Phi}_k$.  $R
\equiv \left[ R_p \right]$ is the vector with these components at any
$k$.  The product of powers of $R_p$ in Eq.~(\ref{eq:jacquard_walk})
will be independent of the walk on the lattice between $k$ and $k + Y
- j$, and at large $k$ corrections from the exact matrices
${\Lambda}_{k k^{\prime}}$ will scale as $\mathcal{O} \! \left( 1 / k
\right)$ or smaller.

Dividing Eq.~(\ref{eq:Glauber_moment_fact_prod}) through by ${\Phi}_k$
and then taking the steady-state condition, produces the sum
\begin{align}
  0 
& = 
  \dot{\prod_p}
  \left[ 
    \sum_{j_p = 0}^{k_p}
    \left(
      \begin{array}{c}
        k_p \\ j_p 
      \end{array}
    \right)
    Y_p^{\underline{j_p}} \,  
    e^{
      - j_p \, \partial / \partial Y_p
    }
  \right]
  {\mathbb{A}}_{\kappa}
  {
    \left. 
      {\psi}_Y \! \left( R \right) 
    \right|
  }_k
\nonumber \\ 
& = 
  \sum_{j_1 = 0}^{k_1}
  \left(
    \begin{array}{c}
      k_1 \\ j_1 
    \end{array}
  \right) \ldots 
  \sum_{j_P = 0}^{k_P}
  \left(
    \begin{array}{c}
      k_P \\ j_P 
    \end{array}
  \right)  
  \left[ 
    \dot{\prod_p}
    Y_p^{\underline{j_p}} \,  
  \right]
  {\mathbb{A}}_{\kappa}  
  {
    \left. 
      {\psi}_{Y-j} \! \left( R \right) 
    \right|
  }_k
\nonumber \\ 
& = 
  \sum_{j_1 = 0}^{k_1}
  k_1^{\underline{j_1}}
  \ldots 
  \sum_{j_P = 0}^{k_P}
  k_P^{\underline{j_P}}
  \left[ 
    \dot{\prod_p}
    \left(
      \begin{array}{c}
        Y_p \\ j_p 
      \end{array}
    \right)
  \right] 
  e^{
    - j \cdot \partial / \partial Y
  } 
  {\mathbb{A}}_{\kappa}
  {
    \left. 
      {\psi}_Y \! \left( R \right) 
    \right|
  }_k
\nonumber \\ 
& = 
  \dot{\prod_p}
  \left[ 
    \sum_{j_p = 0}^{k_p}
    \left(
      \begin{array}{c}
        Y_p \\ j_p 
      \end{array}
    \right)
    k_p^{\underline{j_p}} 
    e^{
      - j_p \, \partial / \partial Y_p
    }
  \right]
  {\mathbb{A}}_{\kappa}
  {
    \left. 
      {\psi}_Y \! \left( R \right) 
    \right|
  }_k
\label{eq:Glauber_moment_SS_ctm}
\end{align}
The coefficient in the row vector on the third line of
Eq.~(\ref{eq:Glauber_moment_SS_ctm}) at index $i$ is the product of
binomial coefficients 
\begin{displaymath}
  \prod_p
  \left(
    \begin{array}{c}
      y^i_p \\ j_p 
    \end{array}
  \right)
\end{displaymath}
for $j_p \le y^i_p$ and zero otherwise.  If $k_p$ are large while
$j_p$ remain bounded by fixed $y^i_p$, up to corrections of order
$1/k_p$ we may approximate $k_p^{\underline{j_p}} \approx k_p^{j_p}$.
Then the sum on $j_p$ reduces back to
\begin{equation}
  0 = 
  {\psi}^T_Y \! 
  \left( 
    1 + k e^{- \partial / \partial Y}
  \right) 
  {\mathbb{A}}_{\kappa}
  {
    \left. 
      {\psi}_Y \! \left( R \right) 
    \right|
  }_k , 
\label{eq:Lamb_sum_ctm_lim}
\end{equation}
recognizable as the CRN Liouville operator from
Eq.~(\ref{eq:L_of_as}).  (As elsewhere, the argument $1 + k e^{-
\partial / \partial Y}$ is to be understood as a vector with
components $1 + k_p e^{- \partial / \partial Y_p}$ taking the place of
$a^{\dagger}_p$.)

The operator $e^{- \partial / \partial Y_p}$, acting on functions
${\psi}^i_Y \! \left( R \right)$ has the effect of reducing any such
product by one factor of $R_p$.  Thus we can rewrite the
expression~(\ref{eq:Lamb_sum_ctm_lim}) as
\begin{equation}
  0 = 
  {\psi}^T_Y \! 
  \left( 
    1 + k / R 
  \right) 
  {\mathbb{A}}_{\kappa}
  {
    \left. 
      {\psi}_Y \! \left( R \right) 
    \right|
  }_k .
\label{eq:L0_kR_args}
\end{equation}
Vanishing of $\Lambda \Phi$ in Eq.~(\ref{eq:jacquard_walk}) is thus
the same in the large-number limit as vanishing of the Liouville
operator in the steady-state condition~(\ref{eq:Liouville_eq}) for the
generating function, with numbers $R$ replacing the operator $a$ and
numbers $1 + k / R$ replacing the operator $a^{\dagger}$.


\subsection{Connection to the standard eikonal approximation}

\subsubsection{Preliminaries: stationary-path approximations in
Doi-Peliti theory, and the source of boundary conditions for the
steady state}

The Peliti construction~\cite{Peliti:PIBD:85,Peliti:AAZero:86}
likewise replaces $a^{\dagger}$ and $a$ in $\mathcal{L}$ with the
coherent-state parameters that are their eigenvalues. Stationary
trajectories arise in the saddle-point approximation of the functional
integral~(\ref{eq:DP_2FFI_CS}), where they are solutions to the
vanishing of the variational derivative of $S$. These stationary paths
are those which provide the eikonal approximation to the probability
density\footnote{For a discussion of when this property does or does
not apply to stationary paths used to calculate generating functions
and functionals, see~\cite{Smith:LDP_SEA:11}.  It will always apply to
the solutions computed here.  For much earlier work on a condition
equivalent to $\mathcal{L} = 0$ as a property of stationary
distributions, see~\cite{Eyink:action:96}.} and $\mathcal{L}
\! \left( {\xi}^{\dagger} , \xi \right) = 0$ is a property of these
paths.

The equations of motion from the
variation of $S$ in Eq.~(\ref{eq:S_from_L_CS}) are
\begin{align}
  \frac{d \xi}{d\tau}
& = 
  - \frac{\partial \mathcal{L}}{\partial {\xi}^{\dagger}}
& 
  \frac{d {\xi}^{\dagger}}{d\tau}
& = 
  \frac{\partial \mathcal{L}}{\partial \xi} . 
\label{eq:EOM_CS}
\end{align}
For the action~(\ref{eq:S_from_L_neta}) in number-potential fields,
the equivalent forms are:
\begin{align}
  \frac{d n}{d\tau}
& = 
  - \frac{\partial \mathcal{L}}{\partial \eta}
& 
  \frac{d\eta}{d\tau}
& = 
  \frac{\partial \mathcal{L}}{\partial n} . 
\label{eq:EOM_neta}
\end{align}

Solutions with ${\xi}^{\dagger} \equiv 1$ or $\eta \equiv 0$ reproduce
the mean-field rate equations~(\ref{eq:CRN_firstmom_EOM}) for $n$, which
then coincides with $\xi$. 

Potential solutions for $R$ joint with $1 + k / R$ come from points
along trajectories with ${\xi}^{\dagger} > 1$, describing improbable
excursions, for which $n \neq \xi$.  If the dimension of the subspace
of dynamical degrees of freedom in a CRN is some number $s \le P$, the
space of tangent vectors to stationary trajectories through a point
has dimension $2s$, and $\mathcal{L} = 0$ specifies a $\left( 2s-1
\right)$-dimensional manifold of possible solutions.

The saddle-point approximation (denoted $\sim$) to the value of the
generating function~(\ref{eq:DP_2FFI_neta}) is the integral over an
eikonal
\begin{align}
  {\phi}_T \! \left( z \right) 
& \sim
  {
    \left. 
      \left(
        \prod_p
        z_p^{{{\rm n}}_p}
      \right)
      {\rho}_{{\rm n}}
    \right|
  }_{{\rm n} = {\bar{n}}_T}
  \sim 
  e^{
    \int_0^T d\tau
    \left( d_{\tau} \bar{\eta} \right)
    \bar{n} 
  } , 
\label{eq:MGF_LD_approx}
\end{align}
where the trajectory $\left( {\bar{\eta}}_{\tau}, {\bar{n}}_{\tau}
\right)$ satisfies equations~(\ref{eq:EOM_neta}) and has terminal
conditions for $\bar{\eta}$ derived from the variation of $n_T$, and
initial conditions for $\bar{n}$ derived from variation of ${\eta}_0$
including the contribution to the initial-value function ${\phi}_0 \!
\left( e^{{\eta}_0} \right)$.  

For the steady-state distribution, $T \rightarrow \infty$ and any
trajectory from the initial conditions passes arbitrarily close to one
of the attracting fixed points and remains there arbitrarily long
before escaping to the final values $\left( {\bar{\eta}}_T,
{\bar{n}}_T \right)$.  Therefore the relevant eikonals are the escape
trajectories from the attracting fixed points of the rate equation
~(\ref{eq:CRN_firstmom_EOM}), with initial values ${\eta}_0 = 0$ from
the converging solutions, and final values ${\bar{\eta}}_T$ determined
by $z$.

The distribution is then obtained from the Legendre transform of the
cumulant-generating function $\log {\phi}_T \! \left( z \right)$.
Choosing a value $z$ for which ${\bar{n}}_T = {\rm n}$, and subtracting
${\bar{n}}_T \log z$, subtracts a total derivative of $\bar{\eta} \,
\bar{n}$ from the integral in Eq.~(\ref{eq:MGF_LD_approx}), leaving
\begin{align}
  {\rho}_{{\rm n}} 
& \sim 
  e^{
    - \int^{{\rm n}}
    \bar{\eta} \, d\bar{n}
  } 
  \equiv 
  e^{
    - \Xi \left( {\rm n} \right) 
  } .
\label{eq:eff_pot_introd}
\end{align}
In Eq.~(\ref{eq:eff_pot_introd}) we introduce the \textit{effective
potential} $\Xi \left( {\rm n} \right)$ defined as the eikonal integral
\begin{equation}
  \Xi \! \left( {\rm n} \right) \equiv 
  \int^T d\tau \, \bar{\eta} \, d_{\tau} \bar{n} = 
  \int^{{\rm n}} \bar{\eta} \, d\bar{n} .
\label{eq:eff_pot_def}
\end{equation}


\subsubsection{Large-deviations scaling and saddle-point
approximation of factorial moments}
\label{subsec:LD scaling}

The leading exponential dependence captured by the eikonal integral
gives the system's \textit{large-deviations}
scaling~\cite{Ellis:ELDSM:85} if, when we descale number ${\rm n}$ by some
characteristic scale $n_C$\footnote{In the examples below, we will use
one of the fixed points of the first-moment
condition~(\ref{eq:CRN_firstmom_EOM}) as the scale factor, but that
choice is not essential.} as
\begin{equation}
  \frac{{\rm n}}{n_C} \equiv x , 
\label{eq:descaled_number}
\end{equation}
and let $n_C$ become large, stationary properties such as trajectories
$\left( \bar{\eta} , \bar{n} / n_C \right)$ become asymptotically
independent of $n_C$.  In that case the effective potential separates
into scale and structure components~\cite{Touchette:large_dev:09} as 
\begin{align}
  \Xi \! \left( {\rm n} \right) 
& = 
  n_C \int^{{\rm n}} \bar{\eta} \, d\left( \bar{n} / n_C \right) \equiv 
  n_C \int^x \bar{\eta} \, d\bar{x}
  \equiv 
  n_C \hat{\Xi} \! \left( x \right) . 
\label{eq:descaled_eff_pot}
\end{align}
$\hat{\Xi} \! \left( x \right)$ is called the \textit{rate function}
of the large-deviations approximation for ${\rho}_{{\rm n}}$.

From the large-deviations approximation to ${\rho}_{{\rm n}}$, the
factorial moment at order $k$ has an implied approximation 
\begin{align}
  \left< {{\rm n}}^{\underline{k}} \right> 
& \sim 
  \int dx n_C 
  e^{- n_C \hat{\Xi} \! \left( x \right)}
  \frac{
    \left( n_C x \right) ! 
  }{
    \left( n_C x - n_C y \right) !
  }
\nonumber \\
& \approx 
  n_C^k
  \int dx n_C 
  e^{- n_C \hat{\Xi} \left( x \right)}
  e^{
    n_C 
    \left\{
      x \left[ \log x - 1 \right] - 
      \left( x-y \right) 
      \left[
        \log \left( x-y \right) - 1
      \right]
    \right\}
  } , 
\label{eq:fact_mom_int_form}
\end{align}
where we have defined $y \equiv k / n_C$ in parallel with
Eq.~(\ref{eq:descaled_number}).  $\sim$ in the first line of
Eq.~(\ref{eq:fact_mom_int_form}) denotes the saddle-point
approximation to the effective potential, while $\approx$ in the
second line indicates the Stirling approximation for factorials.

The scale factor $n_C$ in Eq.~(\ref{eq:fact_mom_int_form}) separates
from a rate function including the contribution from the falling
factorials, which we therefore denote
\begin{align}
  \mathbf{\hat{\Xi}} \! \left( x ; y \right) 
& = 
  \hat{\Xi} \! \left( x \right) + 
  \left( x-y \right) 
  \left[
    \log \left( x-y \right) - 1
  \right] - 
  x \left[ \log x - 1 \right] 
\label{eq:0123_xy_eff_Pot}
\end{align}
A second saddle-point approximation in the integral yields an
expression for the descaled factorial moment as a finite sum 
\begin{equation}
  {\hat{\Phi}}_k \equiv 
  \frac{
    \left< {{\rm n}}^{\underline{k}} \right> 
  }{
    n_C^k
  } \sim
  \sum_{\bar{x} \left( y \right)}
  e^{ - n_C
    \mathbf{\hat{\Xi}} \left( \bar{x} ; y \right) 
  } , 
\label{eq:fact_mom_SPA_form}
\end{equation}
where the saddle points $\bar{x} \! \left( y \right)$ are any
solutions to the equation
\begin{equation}
  \log 
  \left( 
    \frac{
      {\bar{x}}_p
    }{
      {\bar{x}}_p - y_p 
    } 
  \right) = 
  {
    \left. 
      \frac{d\hat{\Xi}}{dx_p}
    \right|
  }_{\bar{x}} = 
  {\eta}_p \! \left( \bar{x} \right) . 
\label{eq:fact_mom_saddle_pt}
\end{equation}

The ratio of two such solutions at values of $k$ separated by $1_p$
produces an expression for descaled moment ratios from
Eq.~(\ref{eq:jacquard_walk}) as
\begin{align}
\lefteqn{
  {\left. {\hat{R}}_p \right|}_y
  \equiv 
  \frac{
    {\left. R_p \right|}_k
  }{
    n_C
  } = 
  \frac{
    \left< {{\rm n}}^{\underline{k+1_p}} \right> / 
    \left< {{\rm n}}^{\underline{k}} \right> 
  }{
    n_C 
  }
} & 
\nonumber \\ 
& \underset{n_C \rightarrow \infty}{\rightarrow}
  \exp
  \left\{
    \frac{d}{dy_p}
    \mathbf{\hat{\Xi}} \! \left( \bar{x} ; y \right) 
  \right\}
\nonumber \\ 
& = 
  \exp
  \left\{
    \log \left( {\bar{x}}_p - y_p \right) + 
    \frac{ 
      {d\bar{x}}_p 
    }{ 
      dy_p
    } 
    {
      \left[
        \log 
        \left(
          \frac{x_p}{x_p-y_p}
        \right) - 
        \frac{d\hat{\Xi}}{dx_p}
      \right]
    }_{x = \bar{x}}
  \right\}
\nonumber \\ 
& = 
  {\bar{x}}_p - y_p .
\label{eq:hat_mom_rats}
\end{align}

From the saddle-point condition~(\ref{eq:fact_mom_saddle_pt}) and the
expression~(\ref{eq:hat_mom_rats}) for $\hat{R}$ it follows that 
\begin{align}
  y_p 
& = 
  {\bar{x}}_p 
  \left( 
    1 - e^{-{\eta}_p}
  \right)
\nonumber \\
  \frac{{\hat{R}}_p}{{\bar{x}}_p} 
& = 
  \frac{{\bar{x}}_p - y_p}{{\bar{x}}_p} = 
  e^{-{\eta}_p} . 
\label{eq:main_SP_relns}
\end{align}
Together with the definition of field
variables~(\ref{eq:xis_to_n_eta}) we confirm that ${\hat{R}}_p$
corresponds to the descaled coherent-state field ${\hat{\xi}}_p \equiv
{\xi}_p / n_C$.  Some algebra then gives 
\begin{align}
  1 + \frac{k_p}{R_p} 
& = 
  \frac{{\hat{R}}_p + y_p}{{\hat{R}}_p} = 
  \frac{{\bar{x}}_p}{{\hat{R}}_p} = 
  e^{{\eta}_p}
\label{eq:duplicate_main_SP_relns}
\end{align}
confirming that $1 + k / R$ corresponds to ${\xi}^{\dagger}$,
consistent with Eq. (\ref{eq:L0_kR_args}). Eq. (\ref{eq:hat_mom_rats})
and Eq. (\ref{eq:fact_mom_saddle_pt}) hence directly give us a
non-trivial approximation for the factorial moment ratios, consistent
with the eikonal approximation for the probability distribution.


\subsubsection{Direct derivation from the definitions of factorial
moments}

The mappings~(\ref{eq:main_SP_relns},\ref{eq:duplicate_main_SP_relns})
can be obtained directly from the definitions of factorial moments.  
The second line in Eq.~(\ref{eq:main_SP_relns}) is equivalent to
\begin{equation}
  \bar{\xi} = 
  \bar{n} - k . 
\label{eq:xi_n_k}
\end{equation}
The relations~(\ref{eq:Psi_psi_on_n_xi}) between actions of the
lowering operator give 
\begin{equation}
  {\bar{\xi}}_p = 
  \frac{
    \left< {{\rm n}}^{\underline{k + 1_p}} \right>
  }{
    \left< {{\rm n}}^{\underline{k}} \right>
  } . 
\label{eq:xi_as_rat}
\end{equation}
Then, from the definition~(\ref{eq:factorial_moment_not}),
\begin{equation}
  {{\rm n}}^{\underline{k + 1_p}} = 
  {{\rm n}}^{\underline{k}}
  \left( n_p - k_p \right) , 
\label{eq:n_und_k_rats}
\end{equation}
Eq.~(\ref{eq:xi_n_k}) is nothing more than the large-number-limiting
relation 
\begin{equation}
  \frac{
    \left< {{\rm n}}^{\underline{k}} {\rm n} \right>
  }{
    \left< {{\rm n}}^{\underline{k}} \right>
  } = 
  \left< {\rm n} \right> . 
\label{eq:large_n_exp}
\end{equation}  
In the background of a factorial moment, taking the expectation of an
additional factor of $n$ results in just the mean-field value to
$\mathcal{O} \! \left( n^0 \right)$.

The identification of Eq.~(\ref{eq:L0_kR_args})  as $\mathcal{L} \! 
\left( {\xi}^{\dagger} , \xi \right)$ along stationary trajectories is
thus confirmed, along with a map from $y = k / n_C$ to the position
$\bar{x}$ on the trajectory that controls the corresponding moment
ratio $\hat{R}$.  The mapping depends on convexity of the effective
potential $\Xi$.  If $\Xi$ is only locally convex, as occurs in
systems with multiple metastable fixed points, then $\hat{R}$ is
assigned to a stable branch by a rule equivalent to a coexistence
condition.  We provide examples in Sec.~\ref{sec:examples}.


\section{Examples}
\label{sec:examples}

A brief treatment will be given of three CRN examples, with complexity
added at each stage to illustrate the uses, variants, and limitations
of eikonal solutions.  The first example is a 1-species autocatalytic
system, illustrating the handling of bistability.  Methods are
introduced to characterize large-$n_C$ convergence of the eikonal
approximation and to estimate finite-size corrections.  The second
example is a symmetric 2-species cross-catalytic system that
reproduces the fixed-point structure of the 1-dimensional model along
its axis of symmetry, and introduces a boundary-value problem that is
difficult to solve without eikonal methods.  The third example is a
2-species CRN known as the Selkov model, which may be given a
fixed-point structure similar to the previous cases, but lacks axes of
symmetry.

Closed-form solution of the 1-dimensional CRN emphasizes why the
Hamilton-Jacobi dual coordinates $n$ and $\eta$ are also the index and
log-derivative of the stationary distribution, with $e^{\eta}$
representing a ratio of transport rates for probability current upward
and downward in $n$.  In higher dimensions where closed-form solutions
are no longer available, the stationarity condition under the
action~(\ref{eq:S_from_L_CS}) ensures that the planes normal to $\eta$
remain isoclines of the steady-state probability, so that the
interpretation of current balance remains as in the 1-dimensional
case.

In the symmetric cross-catalytic model,
values of the effective potential at the fixed points may be computed
directly from the recursion relations, without solving for eikonals.
In the final model without symmetry, eikonal methods are required to
estimate the relative residence times at fixed points, but they may
also exhibit caustics and other regions where the eikonal
approximation breaks down.


\subsection{Exactly solvable 1-variable autocatalytic CRN: multiple
fixed points and finite-size scaling}

Fig.~\ref{fig:g_0123_unconn} shows a 1-species autocatalytic CRN,
which for suitable parameters possesses two metastable states,
expressed as two stable fixed points of the mean-field
equations~(\ref{eq:CRN_firstmom_EOM}).  The bipartite graphical
notation in the figure represents the decomposition of CRNs due to
Horn and Jackson~\cite{Horn:mass_action:72} and
Feinberg~\cite{Feinberg:def_01:87}, and is elaborated in detail
in~\cite{Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17}.

\begin{figure}[ht]
  \begin{center} 
  \includegraphics[scale=0.6]{g_0123_unconn.pdf}
  \caption{
  %
  A network with one species and four complexes, which supports two
  metastable states at some values of the rate constants.  Filled
  circles are species; open circles are complexes.  Dashed lines
  between complexes indicate (directed) reactions; solid lines from
  species to complexes reflect the stoichiometric coefficients of the
  CRN.
 
    \label{fig:g_0123_unconn} 
  }
  \end{center}
\end{figure}

The recursion relations for this model were considered in detail
in~\cite{Smith:CRN_moments:17}, where they were solved by methods of
matched asymptotic expansions and validated against Gillespie
simulations.  Here we use the model to illustrate the relation between
the moment and Liouville representations and to consider finite-size
deviations from the eikonal approximation.  Additional detail is given
in App.~\ref{sec:autocat_numerics}.

The reaction schema corresponding to Fig.~\ref{fig:g_0123_unconn} is 
\begin{align}
  \varnothing
& \xrightleftharpoons[k_2]{\epsilon}
  {\rm A} & 
  2 {\rm A} 
& \xrightleftharpoons[{\bar{k}}_1]{k_1}
  3 {\rm A} .
\label{eq:cubic_1spec_scheme}
\end{align}
where $A$ is the species name, and $\epsilon$, $k_2$, $k_1$, and
${\bar{k}}_i$ are reaction rate constants.  $\varnothing$ indicates
the \textit{null species}, which may stand for buffered external
species for which fluctuations are not considered, or depending on the
context, on \textit{de novo} creation or annihilation of particles.
The mean-field equation of motion~(\ref{eq:CRN_firstmom_EOM}) for the
number $n$ of $A$ particles in the system is
\begin{equation}
  \frac{dn}{d\tau} = 
  \epsilon - k_2 n + k_1 {n}^2 -{\bar{k}}_1 {n}^3 .
\label{eq:0123_y1A_rate_eqn}
\end{equation}

The exact recursion relation for $R_k$ following from
Eq. (\ref{eq:Glauber_moment_fact_prod}) was shown
in~\cite{Smith:CRN_moments:17} to be
\begin{widetext}
\begin{equation}
  R_k =  
  \frac{
      \epsilon + 
      \left( k-1 \right) \left( k-2 \right) k_1 
  }{
    k_2 - k_1 R_{k+1} +
    {\bar{k}}_1 R_{k+2} R_{k+1} - 
    \left( k-1 \right) 
    \left( 
      2k_1 - 2 {\bar{k}}_1 R_{k+1} 
    \right) + {\bar{k}}_1 
    \left( k-1 \right) \left( k-2 \right)
  } . 
\label{eq:rec_0123}  
\end{equation}
\end{widetext}
Descaled variables to extract the large-deviations system behavior
will be denoted
\begin{align}
  \epsilon 
& = 
  \varepsilon n_C 
& 
  k_2 
& = 
  {\kappa}_2
\nonumber \\
  k_1 
& = 
  {\kappa}_1 / n_C
& 
  {\bar{k}}_1 
& = 
  {\bar{\kappa}}_1 / n_C^2 
\nonumber \\ 
  n 
& \equiv 
  x n_C 
&
  R 
& \equiv 
  \hat{R} n_C 
\nonumber \\
  k 
& \equiv 
  y n_C
& 
  \mathcal{L} 
& = 
  \hat{\mathcal{L}} n_C .
\label{eq:0123_parm_descaling}
\end{align}
In terms of these Eq.~(\ref{eq:rec_0123}) has the large-$n_C$ and
large-$k$ limit
\begin{equation}
  \hat{R} = 
  \frac{
    \varepsilon + y^2 {\kappa}_1 
  }{
    {\kappa}_2 - 
    {\kappa}_1 \hat{R} + 
    {\bar{\kappa}}_1 {\hat{R}}^2 - 
    2 y 
    \left( 
      {\kappa}_1 - {\bar{\kappa}}_1 \hat{R}
    \right) + 
    {\bar{\kappa}}_1 y^2 
  } . 
\label{eq:0123_R_recur_from_CRNLP}
\end{equation}
Non-negative real solutions to 
\begin{equation}
  y = 
  \sqrt{
    \frac{
      {\kappa}_2 \hat{R} - \varepsilon
    }{
      {\kappa}_1 - {\bar{\kappa}}_1 \hat{R}
    }
  } - 
  \hat{R}
\label{eq:0123_two_y_sols}
\end{equation}
give the two branches of $\hat{R} \! \left( y \right)$ satisfying
Eq.~(\ref{eq:0123_R_recur_from_CRNLP}).

The Liouville function in number-potential fields, derived following
standard methods
(see~\cite{Smith:LDP_SEA:11,Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17}),
is
\begin{align}
  \hat{\mathcal{L}} 
& = 
  \left( 1 - e^{\eta} \right)
  \left( 
    \varepsilon + {\kappa}_1 x^2
  \right) + 
  \left( 1 - e^{- \eta} \right)
  \left( 
    {\kappa}_2 + {\bar{\kappa}}_1 x^2
  \right) x . 
\label{eq:0123_L_form}
\end{align}
It is immediate to check that under the
assignments~(\ref{eq:main_SP_relns}),
Eq.~(\ref{eq:0123_R_recur_from_CRNLP}) re-arranges to
\begin{align}
  0 
& = 
  -\varepsilon + 
  {\kappa}_2 \hat{R} - 
  {
    \left( y + \hat{R} \right)
  }^2
  \left( 
    {\kappa}_1 - {\bar{\kappa}}_1 \hat{R}
  \right) = 
  \frac{\hat{R}}{y}
  \hat{\mathcal{L}} . 
\label{eq:onespec_yR_sol}
\end{align}

For a 1-dimensional system, the condition $\mathcal{L} = 0$ is
sufficient to solve jointly for $x$ and $\eta$, giving a relation
\begin{equation}
  \hat{R} = 
  \frac{
    \varepsilon + {\kappa}_1 x^2
  }{
    {\kappa}_2 + {\bar{\kappa}}_1 x^2
  } 
\label{eq:twospec_diag_ray_sol}
\end{equation}
that is equivalent under the mapping~(\ref{eq:main_SP_relns}) to
Eq.~(\ref{eq:0123_two_y_sols}).  It follows also that escape
trajectories are the time reverses of the solutions to the
first-moment equations~(\ref{eq:CRN_firstmom_EOM}).  

These properties of the continuum limit express the more basic
property that in 1-dimensional systems at steady states, upgoing and
downgoing probability flows must balance through each position.  If
only steps $\delta {\rm n}$ of size 1 are allowed, as is the case for this
CRN, the process is called a \textit{birth-death} process, and
probability flux balance renders its steady-state probability
distribution solvable in closed
form~\cite{vanKampen:Stoch_Proc:07}.\footnote{Balance of upgoing and
downgoing probability currents between any two adjacent indices
$\left( {\rm n} , {\rm n} + 1 \right)$ is written in terms of the descaled
rate constants as
\begin{equation}
  \frac{
    {\rho}_{{\rm n} + 1} 
  }{
    {\rho}_{{\rm n}}
  } = 
  \frac{n_C}{\left( {\rm n} + 1 \right)}
  \left( 
  \frac{
    \varepsilon + {\kappa}_1
    {\rm n} \left( {\rm n} - 1 \right) / n_C^2
  }{
    {\kappa}_2 + {\bar{\kappa}}_1
    {\rm n} \left( {\rm n} - 1 \right) / n_C^2
  }
  \right) 
\label{eq:birth_death_exact}
\end{equation}
Eq.~(\ref{eq:birth_death_exact}) is the exact evaluation of $\hat{R} /
x = e^{-\eta}$ and goes at large $n_C$ to $\hat{R}$ in
Eq.~(\ref{eq:twospec_diag_ray_sol}).  $\log
\left( {\rho}_{{\rm n}} \right)$ is then the (exact) discrete sum
approximated by the continuum integral~(\ref{eq:eff_pot_introd}).
\label{fn:1D_birth_death}}


\subsubsection*{Finite-size scaling of the transition regions using
the eikonal approximation to the effective potential}

Ordinary position-space WKB methods can be used 
at finite $n_C$ to estimate low-order
moments.  Of particular interest for multistable systems is the order
at which fluctuations transition between control by different fixed
points.  To obtain an estimate, we make a simplifying approximation of
the integral~(\ref{eq:fact_mom_int_form}) as a sum of Poisson
distributions at the stable fixed points with weights from the
effective potential
\begin{equation}
  {\rho}_x \approx 
  \frac{
    e^{- \Xi \left( {\bar{x}}_1 \right)}
    \delta \! \left( x - {\bar{x}}_1 \right) + 
    e^{- \Xi \left( {\bar{x}}_3 \right)}
    \delta \! \left( x - {\bar{x}}_3 \right)
  }{
    e^{- \Xi \left( {\bar{x}}_1 \right)} + 
    e^{- \Xi \left( {\bar{x}}_3 \right)}
  } . 
\label{eq:0123_WKB_dist}
\end{equation}
The $k$th-order moment is approximated, without making use of the
curvature of the effective potential in neighborhoods of the fixed
points, as 
\begin{equation}
  \left<
    {{\rm n}}^{\underline{k}}
  \right> \approx 
  \frac{
    e^{- \Xi \left( {\bar{x}}_1 \right)}
    {\bar{x}}_1^k + 
    e^{- \Xi \left( {\bar{x}}_3 \right)}
    {\bar{x}}_3^k 
  }{
    e^{- \Xi \left( {\bar{x}}_1 \right)} + 
    e^{- \Xi \left( {\bar{x}}_3 \right)}
  }
\label{eq:0123_WKB_fact_moms}
\end{equation}

The dominant term in the ratio~(\ref{eq:0123_WKB_fact_moms}) shifts
from the lower to the higher fixed point where the weighted effective
potentials are equal, corresponding to an equal-area rule under the
curve for the ``force'' in the two-variable effective potential 
\begin{equation}
  \frac{\partial}{\partial \bar{x}}
  \mathbf{\Xi} \! \left( \bar{x} ; y \right) = 
  \eta \! \left( \bar{x} \right) - 
  \log 
  \left(
    \frac{\bar{x}}{\bar{x} - y}
  \right) , 
\label{eq:0123_xy_spinodal}
\end{equation}
as in the coexistence condition for systems with first-order phase
transitions.  The crossover point in this approximation is 
\begin{equation}
  y_0 = 
  \frac{
    \hat{\Xi} \! \left( {\bar{x}}_3 \right) - 
    \hat{\Xi} \! \left( {\bar{x}}_1 \right)
  }{
    \log {\bar{x}}_3 - 
    \log {\bar{x}}_1 
  } .
\label{eq:0123_y0_solve}
\end{equation}

Fig.~\ref{fig:scaled_moms_bistable_good_ks} compares the
approximation~(\ref{eq:0123_WKB_fact_moms}) to the exact
recursion~(\ref{eq:rec_0123}) at finite $n_C$, and characterizes the
convergence to the eikonal approximation at large $n_C$.  When the
small-$k$ asymptotic expansion is initialized with values
${\hat{R}}_0$ and ${\hat{R}}_1$ derived from
Eq.~(\ref{eq:0123_WKB_dist}), the exact recursion solutions remain
close to that approximation over the interval between ${\bar{x}}_1$
and ${\bar{x}}_3$.  The WKB solution has captured
much of the finite-size scaling even though the effective potential is
approximated with the large-deviations scaling limit.  When $y$ is
large enough that $\hat{R}$ exceeds the upper fixed-point value
${\bar{x}}_3$, the recursion solutions converge on the exact continuum
value~(\ref{eq:0123_two_y_sols}).

\begin{figure}[ht]
\begin{center} 
  \includegraphics[scale=0.4]{scaled_moms_bistable_good_ks_AI.pdf} 
  \caption{
 
  Finite-$n_C$ evaluations (solid) of the exact recursion
  relation~(\ref{eq:rec_0123}), initiated at ${\hat{R}}_0$ and
  ${\hat{R}}_1$ (circles) 
  using values from Eq.~(\ref{eq:0123_WKB_fact_moms}).  That
  approximation (red-dashed) saturates at ${\bar{x}}_3 /
  {\bar{x}}_1$.  The lower and upper branches of the solution to
  the quadratic equation~(\ref{eq:0123_R_recur_from_CRNLP}) are shown
  in black-dash (vertical connection is the upper limit where
  solutions exist on the lower branch).  Equal-area value $y_0$ of
  Eq.~(\ref{eq:0123_y0_solve}), shown connecting the upper and lower
  branches $R \! \left( y \right)$ is approached by the recursion
  solutions at large $n_C$.
 
  \label{fig:scaled_moms_bistable_good_ks} 
  } 
\end{center}
\end{figure}


\subsection{Two-dimensional cross-catalytic CRN, and eikonal
approximation at boundaries}

To demonstrate the use of eikonals in a model with similar qualitative
behavior that is not solvable in closed form,
Fig.~\ref{fig:two_species_bistable} shows a symmetric cross-catalytic
CRN with the same fixed point structure along its axis of symmetry as
the previous CRN.  Recursion relations in this model were solved and
compared to numerical simulations
in~\cite{Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17} only along
the axis of symmetry.

\begin{figure}[ht]
  \begin{center} 
  \includegraphics[scale=0.6]{two_species_bistable.pdf}
  \caption{
  %
  A network with two species labeled $A$ and $B$ in a cross-catalytic
  configuration.  The fixed points for both $A$ and $B$ are the same
  as those for $A$ in the CRN of Fig.~\ref{fig:g_0123_unconn}.
 
    \label{fig:two_species_bistable} 
  }
  \end{center}
\end{figure}

The reaction schema corresponding to
Fig.~\ref{fig:two_species_bistable} is 
\begin{align}
  \varnothing
& \xrightleftharpoons[k_2]{\epsilon}
  {\rm A} & 
  {\rm A} + {\rm B} 
& \xrightleftharpoons[{\bar{k}}_1]{k_1}
  2 {\rm A} + {\rm B} 
\nonumber \\ 
  \varnothing
& \xrightleftharpoons[k_2]{\epsilon}
  {\rm B} & 
  {\rm B} + {\rm A} 
& \xrightleftharpoons[{\bar{k}}_1]{k_1}
  2 {\rm B} + {\rm A} , 
\label{eq:cubic_2spec_scheme}
\end{align}
and its mean-field equation of motion~(\ref{eq:CRN_firstmom_EOM}) for
numbers $n_A$ of species $A$ and $n_B$ of species $B$ is
\begin{align}
  \frac{d n_A}{d\tau}
& = 
  \epsilon - 
  k_2 n_A + 
  n_B 
  \left( 
    k_1 n_A - 
    {\bar{k}}_1 n_A^2 
  \right) 
\nonumber \\
  \frac{d n_B}{d\tau}
& = 
  \epsilon - 
  k_2 n_B + 
  n_A 
  \left( 
    k_1 n_B - 
    {\bar{k}}_1 n_B^2 
  \right) . 
\label{eq:rate_ab_cross}
\end{align}
Examples of the mean-field flow solutions, and more detailed
properties of the escape eikonals, are provided in
App.~\ref{sec:crosscat_numerics}, and we summarize only novel features
here. 


\begin{figure}[ht]
\begin{center} 
  \includegraphics[scale=0.45]{two_spec_ctm_escape_sep_bkgnd_AI.pdf} 
  \caption{
 
  Skeleton of trajectories determining the topological domains of the
  CRN from Fig.~\ref{fig:two_species_bistable}.  Solutions to the
  mean-field equations~(\ref{eq:rate_ab_cross}) from
  Fig.~\ref{fig:two_spec_ctm_class_flows} in the appendix are shown in
  thin black.  A converging separatrix (blue heavy) passing through
  the saddle separates the basins of attraction of the two fixed
  points.  An escape separatrix (red heavy), the locus of points
  reached with equal probability from either fixed point, also passes
  through the saddle and is otherwise contained in the basin of
  attraction of the lower fixed point.
 
  \label{fig:two_spec_ctm_escape_sep_bkgnd} 
  } 
\end{center}
\end{figure}

The topology of the space of trajectories is delimited by the skeleton
shown in Fig.~\ref{fig:two_spec_ctm_escape_sep_bkgnd}.  An attracting
separatrix divides the basins of attraction of the two stable fixed
points.  For this CRN (see App.~\ref{sec:crosscat_numerics}), escape
eikonals never cross, so they form a foliation of the space of $n$
values.  They are partitioned across the axis of symmetry by escape
eikonals from the stable fixed points either to the saddle or to $x_A
= x_B = 0$ or $\infty$, along which the integral $\int \eta \, dx$
gives the relative depth of the minima of the effective potential
$\hat{\Xi} \! \left( x \right)$.  An escape separatrix, the locus of
points with equal probability to be reached upon escape from either
fixed point, emanates from the saddle, and is entirely contained
within the basin of attraction of the lower fixed point.


\subsubsection*{Eikonal solutions along the non-asymptotic boundary}

The $A \leftrightarrow B$ exchange symmetry of this model was used
in~\cite{Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17} to reduce
the recursion relations for moment ratios to a pair of self-consistent
solutions similar in form to those for the 1-species autocatalytic
CRN.  The generator in the moment
representation~(\ref{eq:Glauber_moment_fact_prod}) is a 2-dimensional
discrete Laplacian, given in Eq.~(\ref{eq:twospec_rull_gen}).  Its
solutions off the diagonal depend on boundary conditions at $k_A = 0$
or $k_B = 0$ where the nonzero moment order need not lie in any
asymptotic regime.  In the limit $y_B \rightarrow 0$, an estimate for
$y_A$ is given by
\begin{equation}
  y_A = 
  \frac{1}{{\hat{R}}_B}
  \left( 
    \frac{
      {\kappa}_2 {\hat{R}}_A - \varepsilon
    }{
      {\kappa}_1 - {\bar{\kappa}}_1 {\hat{R}}_A
    }
  \right) - 
  {\hat{R}}_A
\label{eq:twospec_yb_bdry_sol}
\end{equation}
In Eq.~(\ref{eq:twospec_yb_bdry_sol}) ${\hat{R}}_B \! \left( y_A , 0
\right)$ is not given, but for $y \sim 1$ where the boundary is not
very far from the diagonal, the approximation ${\hat{R}}_B \approx
{\hat{R}}_A$ permits a closed-form comparison with
Eq.~(\ref{eq:0123_two_y_sols}) which applies to both $y_A$ and $y_B$
along the diagonal.  An approximation of the effective potential by
time-reversing trajectories, developed in
App.~\ref{sec:crosscat_numerics} shows that the transfer of control
between the two fixed points occurs along a roughly linear contour
$y_A + y_B \approx 2 y_0$ from Eq.~(\ref{eq:0123_y0_solve}).

Fig.~\ref{fig:two_spec_ray_bdry_diag_comp_limits} illustrates the use
of eikonals that emerge from neighborhoods of the diagonal to
approximate the boundary conditions at $k_B = 0$.  It emphasizes that
the boundary values at any point are governed not by propagation along
the boundary, but by the values of moment ratios near the
diagonal. 
Thus, in a population process with $P$ fluctuating species,
although the boundary with undetermined conditions is a $\left( 2P - 1
\right)$-dimensional hypersurface, the behavior along the boundary is
controlled by neighborhoods of a discrete set of interior fixed points
and the parameters of $\mathcal{L}$.  

\begin{figure}[ht]
\begin{center} 
  \includegraphics[scale=0.45]{two_spec_ray_bdry_diag_saturated_zoom_AI} 
  \caption{
 
  ${\hat{R}}_A \! \left( y_A, y_B \right)$ (red-toned lines and
  symbols) and ${\hat{R}}_B \! \left( y_A, y_B \right)$ (blue-toned
  lines and symbols) for eikonals emanating from neighborhoods of the
  escape trajectory between the lower fixed point and the saddle into
  the region $y_B < y_A$.  Trajectories terminating in dots are
  truncated at the last point along the discrete numerical integral
  before the corresponding $y_B$ passes below zero.  Upper black
  dashed curve is the eikonal solution~(\ref{eq:0123_two_y_sols})
  along the diagonal, and lower green dashed curve is the
  estimation~(\ref{eq:twospec_yb_bdry_sol}).  The asymmetry
  ${\hat{R}}_B - {\hat{R}}_A \ge 0$ remains small over the regions
  shown and increases with increasing $x_A, x_B \lesssim
  {\bar{x}}_3$. 
 
  \label{fig:two_spec_ray_bdry_diag_comp_limits} 
  } 
\end{center}
\end{figure}


\subsection{Selkov model: asymmetric autocatalysis with breakdowns in
the simple eikonal approximation} 

To present a case in which even the coarsest WKB
approximation~(\ref{eq:0123_WKB_dist}) cannot readily be obtained
without continuum methods, Fig.~\ref{fig:g_selkov} shows a CRN known
as the Selkov model, which was treated with eikonal methods by Dykman
\textit{et al.}~\cite{Dykman:chem_paths:94}.  The Selkov model is an
asymmetric autocatalytic system for which we can choose fixed points
and residence times similar to those in the cross-catalytic CRN,
though the eikonals in the two models show qualitative differences.

\begin{figure}[ht]
  \begin{center} 
  \includegraphics[scale=0.6]{g_selkov.pdf}
  \caption{
  %
  The Selkov model: an asymmetric autocatalytic CRN with two species
  $A$ and $B$.  The model can yield fixed points with the same spacing
  and similar effective potential depths to the cross-catalytic
  network of Fig.~\ref{fig:two_species_bistable}, but on a line of
  negative rather than positive slope.
 
    \label{fig:g_selkov} 
  }
  \end{center}
\end{figure}

The reaction schema for Fig.~\ref{fig:g_selkov} is
\begin{align}
  \varnothing
& \xrightleftharpoons[{\bar{k}}_1]{k_1}
  {\rm A} & 
  {\rm A} + 
  2 {\rm B} 
& \xrightleftharpoons[{\bar{k}}_2]{k_2}
  3 {\rm B} & 
  {\rm B} 
& \xrightleftharpoons[{\bar{k}}_3]{k_3}
  \varnothing . 
\label{eq:Selkov_scheme}
\end{align}
The fixed points of the Selkov model lie along a line
\begin{align}
  n_{\rm A} + 
  n_{\rm B} = 
  \frac{
    2 \left( k_1 + {\bar{k}}_3 \right)
  }{
    \left( {\bar{k}}_1 + k_3 \right)
  } - 
  \frac{
    \left( {\bar{k}}_1 - k_3 \right) 
  }{
    \left( {\bar{k}}_1 + k_3 \right)
  }
  \left( 
    n_{\rm A} - 
    n_{\rm B}  
  \right) , 
\label{eq:Selkov_linpart_solve}
\end{align}
Fig.~\ref{fig:selkov_ctm_class_flows} shows the solutions to the
mean-field equations~(\ref{eq:CRN_firstmom_EOM}) for $\left(
{\bar{k}}_1 - k_3 \right) / \left( {\bar{k}}_1 + k_3 \right) = - 1/3$.  
\begin{figure}[ht]
\begin{center} 
  \includegraphics[scale=0.45]{selkov_ctm_class_flows_am1on3_AI.pdf} 
  \caption{
 
  Flowfields from the mean-field equations~(\ref{eq:CRN_firstmom_EOM})
  for the Selkov model.  Parameters were chosen to place fixed points
  at $\left( {\bar{x}}_A, {\bar{x}}_B \right) = \left\{ \left( 2, 80
  \right) , \left( 8, 68 \right) , \left( 18, 48 \right) \right\}$.
  Trajectories satisfy $d_{\tau} \left( {{\rm n}}_{\rm A} + {{\rm n}}_{\rm B}
  \right) = 0$ where they pass through the
  line~(\ref{eq:Selkov_linpart_solve}) connecting the fixed points.
 
  \label{fig:selkov_ctm_class_flows} 
  } 
\end{center}
\end{figure}
Fig.~\ref{fig:selkov_eiks_limrays} shows the skeleton eikonals for the
escape system of the Selkov model, reproducing features observed
in~\cite{Dykman:chem_paths:94}.\footnote{We do not try here to
identify the escape separatrix, which in~\cite{Dykman:chem_paths:94}
falls within the regions between the caustics and limiting rays, areas
where each state is reached by multiple trajectories.} Further detail
is provided in App.~\ref{sec:selkov_numerics}.

\begin{figure}[ht]
\begin{center} 
  \includegraphics[scale=0.45]{selkov_eiks_limrays_AI.pdf} 
  \caption{
 
  Skeleton for escape trajectories in the Selkov model.  Eikonals from
  stable fixed points to the saddle are (heavy black).  Trajectories
  originating counterclockwise from those double back and cross,
  creating a singular measure, in caustics (heavy red).  The innermost
  limit of these rebounding trajectories is the limiting ray (thin
  black).  Each point between the limiting ray and caustic is crossed
  by two trajectories, and each trajectory is crossed continuously
  before that point, requiring modification of the eikonal
  approximation.
 
  \label{fig:selkov_eiks_limrays} 
  } 
\end{center}
\end{figure}

In contrast to the cross-catalytic CRN~(\ref{eq:cubic_2spec_scheme}),
in which every point $n$ is traversed by a unique escape trajectory
and has a well-defined eikonal approximation to the escape
probability, the Selkov model at parameters shown possesses two
caustics emanating from the saddle point.  Caustics are regions where 
the momentum-space WKB approximation breaks down
as stationary trajectories from
finite measure converge on a subsurface of measure zero.  These
caustics are formed as escapes diverging counterclockwise from the
eikonal to the saddle turn back and cross.  The innermost returning
ray on each side of the saddle is the limiting ray, the boundary of
the domain of validity of the simple stationary-path approximation.

Finite-size effects may again be estimated by the coarse WKB
approximation~(\ref{eq:0123_WKB_dist}), but now the escape equations
of motion must be solved to estimate the relative depths of the
effective potential at the fixed point.
Fig.~\ref{fig:selkov_SaddleEiks_eff_pot} shows $\int \eta \, dn$ along
the escape eikonals from the two stable fixed points to the saddle in
Fig.~\ref{fig:selkov_eiks_limrays}.  These give values $\hat{\Xi}
\! \left( {\bar{x}}_1 \right)$ and $\hat{\Xi} \! \left( {\bar{x}}_3
\right)$ determining relative residence times and the first-passage
time as a function of $n_C$.

\begin{figure}[ht]
\begin{center} 
  \includegraphics[scale=0.35]{selkov_SaddleEiks_eff_pot_AI.pdf} 
  \caption{
 
  $\int \eta \, dn$ along the eikonals from the two stable fixed
  points to the saddle gives the well depths of the effective
  potential.  For these parameters $\hat{\Xi} \! \left( {\bar{x}}_1
  \right) = - 0.4462$ and $\hat{\Xi} \! \left( {\bar{x}}_3 \right) = -
  0.1606$.  The difference, $0.2856$, is $\sim 4 \times$ the
  difference for the 1-dimensional autocatalytic model shown in Fig.~7
  of~\cite{Smith:CRN_moments:17}, and about twice the difference of
  the cross-catalytic model shown in
  Fig.~\ref{fig:two_spec_pot_rev_flat}.
 
  \label{fig:selkov_SaddleEiks_eff_pot} 
  } 
\end{center}
\end{figure}

Using the convention for large-deviations scaling that $\bar{n} = n_C
\bar{x}$ with fixed points 
\begin{align}
  {\bar{x}}_1
& = 
  \left(
    80 , 2 
  \right) , 
& 
  {\bar{x}}_3
& = 
  \left(
    48 , 18
  \right) , 
\label{eq:selkov_scaled_FP}
\end{align}
control of both moment ratios ${\hat{R}}_A$ and ${\hat{R}}_B$ is
expected to switch between the fixed points along the line
\begin{align}
  \begin{array}{c}
    \left[ 
      \begin{array}{cc}
        y_A & y_B
      \end{array}
    \right] \\
    \phantom{\mbox{}}
  \end{array}
  \left[ 
    \begin{array}{c}
      \log \left( 48 / 80 \right) \\
      \log \left( 18 / 2 \right) 
    \end{array}
  \right] 
& = 
  \hat{\Xi} \! \left( {\bar{x}}_3 \right)  - 
  \hat{\Xi} \! \left( {\bar{x}}_1 \right) , 
\label{eq:selkov_twomin_cross_short}
\end{align}
similar to the linear front $y_A + y_B = 2 y_0$ seen for the
cross-catalytic CRN~(\ref{eq:cubic_2spec_scheme}).  Further details
concerning trajectories and also the phenomenology of the conjugate
variables $\eta$ is provided in App.~\ref{sec:selkov_numerics}.


\section{Concluding remarks}
\label{sec:conclusions}

We have considered the properties of a third representation of the
generator of a stochastic CRN complementing its transition matrix and
Liouville operator, expressed directly in the lattice of factorial
moments of a probability distribution over number states.  This
generator provides a potentially efficient connection between nearby
moments, but suffers the limitation that its boundary conditions are
not simple to handle for high-dimensional systems.

We have shown how a continuum approximation, assuming
slow variation of the ratios of adjacent factorial moments, combines
with information already familiar from momentum-space WKB methods, to
provide a leading exponential estimate for the values of moments
throughout the lattice.  The limit~(\ref{eq:L0_kR_args}), in which the
generator of the moment representation reduces to the Liouville
operator on a stationary $\mathcal{L} = 0$ solution, defines a map
from the orders $k$ of factorial moments to positions along
large-deviation trajectories that control the leading exponential
behavior of moment ratios in neighborhoods of $k$.

Whereas for direct solution of the moment recursion equations,
boundary conditions are difficult to specify where some components of
$k$ become small, in the momentum-space WKB approximation solutions
are propagated to the boundary along rays emanating from fixed points
in the interior, as in the Freidlin-Wentzell approach to computing
boundary values for solutions to diffusions on basins of attraction.
Position-space WKB methods can be used to approximate finite-size
corrections using the simplified scale factors and rate functions
computed from the large-deviations limit.

The moment representation also provides a natural interpretation for
the coherent-state fields $\left( {\xi}^{\dagger} , \xi \right)$ which
appear in Hamilton-Jacobi theory as a canonical transform of the
number-potential fields $\left( \eta , n \right)$ that are the natural
variables for momentum-space WKB methods.  Whereas $n$ is the index,
and $\eta$ the log-derivative, of the stationary distribution, $\xi$
takes a value equal to the ratio $R$ of adjacent factorial moments.
$e^{\eta} = {\xi}^{\dagger}$ is then the ratio of probability flow
rates normal to isoclines of the stationary probability.

The three representations of the generator of a stochastic CRN use the
three classes of random variables as bases: ${\sigma}_{{\rm n}}$ (the
discrete indicator function on states, for the distribution
${\rho}_{{\rm n}}$), ${{\rm n}}^{\underline{k}}$ (discrete powers, for moments
${\Phi}_k$), and $z^{{\rm n}}$ (continuous tilts, for the MGF $\phi \!
\left( z \right)$).  The well-known direct
interconversions~\cite{Doi:SecQuant:76,Doi:RDQFT:76,Peliti:PIBD:85,Peliti:AAZero:86}
and the new one derived
in~\cite{Krishnamurthy:CRN_moments:17,Smith:CRN_moments:17} facilitate
choosing bases with good convergence properties for different classes
of observables of interest.  The connection among the three through
the large-deviations limit derived here, in addition to providing
approximation methods, shows connections of meaning across the
representations that are sometimes not apparent in their more
indirect, detailed interconversions.


\subsection*{Acknowledgments}

DES was supported in part by NASA Astrobiology CAN-7 award NNA17BB05A
through the College of Science, Georgia Institute of Technology, and
by the Japanese Ministry of Education, Culture, Sports, Science, and
Technology (MEXT) through the World Premiere International program.
He also thanks the Physics Department at Stockholm University
for hospitality in May $2016-2019$, when some of
this work was carried out.
