%From schuss@math.tau.ac.il Sun Nov 19 07:55:00 1995
%From: Zeev Schuss <schuss@math.tau.ac.il>
%Date: Sun, 19 Nov 1995 15:57:19 +0200
%To: bob@aix550.phys.rpslmc.edu
%Status: R

\documentstyle [12pt]{article}
\textwidth=6in
\textheight=8.0in
\oddsidemargin 0in
\topmargin -0.25in
\renewcommand{\baselinestretch}{2}
\begin{document}
\pagestyle{empty}
\centerline{\bf EQUILIBRIUM AND RECROSSINGS OF THE
 TRANSITION STATE:}
\centerline{\bf WHAT CAN BE LEARNED FROM DIFFUSION?}

\vspace*{0.5cm}
\centerline{by}

\vspace*{0.5cm}
\centerline {Z. Schuss\footnote{Department of Mathematics, Tel-Aviv
University, Ramat-Aviv, Tel-Aviv 69978, Israel}}

\vspace*{0.5cm}
\centerline{\bf Abstract}.

\vspace{0.5cm}
We examine the assumptions and conclusion of (generalized)
transition state theory (GTST) by considering the activation
process in the diffusion (Langevin) limit.
We find the asymptotic structure of the leading eigenfunctions and
eigenvalues of the Fokker-Planck operator with a bistable
potential, and hence the long time quasi equilibrium behavior
of the phase space probability density function (pdf).
Defining reactant and
product as small neighborhoods $\Omega_A$ and $\Omega_B$ of the stable
states $A$ and $B$, respectively, we examine all possible
recrossings of the transition state region (TSR) and find their
contribution to the mean first passage time (MFPT) $\tau_{AB}$
from $\Omega_A$ to $\Omega_B$. We show that the mean number of
recrossings of the TSR is 1, hence  $\tau_{AB}=2\tau_{AS}$,
where $\tau_{AS}$ is the MFPT from $\Omega_A$ to the
{\em stochastic separatrix} $S$, which we use as a generalized
transition state (GTS).
The activation rate, that is, the rate at which trajectories
arrive to $\Omega_B$ from $\Omega_A$, is then shown to be
given by $\kappa=1/2\tau_{AS}$, and in the limit of small noise is
independent of the choice of $\Omega_A$ and $\Omega_B$.  We
conclude that to obtain the correct rate in (G)TST (i) the
quasi equilibrium density (qepdf) rather than the equilibrium density
(epdf) has to be used, (ii) the qepdf contains a boundary
layer near the stochastic separatrix, but otherwise the
reactant qepdf$\approx$epdf, and (iii) all recrossings of the
(G)TS are accounted for if (G)TS=$S$, but not otherwise.
We also consider the case of a single metastable state.
%
\newpage
\pagestyle{plain}
\setcounter{page}{1}
\renewcommand{\theequation}{1.\arabic{equation}}
\setcounter{equation}{0}
\setcounter{section}{1}
\noindent
{\bf 1. Introduction.}\\[0.3cm]

Transition state theory (TST) of activation and its
generalizations (GTST) (see e.g., \cite{Pechukas}) are based
on the following assumptions (1) an equilibrium probability
distribution of phase space trajectories, (2) that a certain
configuration space surface, called the (generalized)
transition state (G)TS, has the property that
trajectories that cross the (G)TS from reactant to product
never recross it. Because of (1) half the trajectories on
the (G)TS cross in the product direction, and because of (2)
they end up as product. It follows therefore that (3)
the reaction rate $\kappa$ is the outgoing half of the equilibrium
phase space probability flux on the (G)TS, normalized by the
reactant population. The underlying postulate which justifies
the equilibrium assumption is that after a sufficiently long
time the quasi equilibrium density of the reactant is
sufficiently similar to the equilibrium density so that the
outgoing half of the quasi equilibrium flux on the (G)TS,
normalized by the quasi equilibrium reactant population, is
well approximated by that of the equilibrium flux, normalized
by the equilibrium reactant population.

Both theoretical \cite{Kramers} and experimental results
\cite{Fleming} show that $\kappa$ depends on the dissipation,
contrary to the TST prediction. The failure of TST to show
this dependence brought (1) and (2) into question.
It has been widely recognized that both (1) and (2) are
unrealistic, because activation is a non equilibrium process
and because recrossings do occur. It is believed therefore that
due to (2),
(G)TST overestimates the rate \cite{Wigner}. There have been
different attempts to modify the equilibrium density on the
(G)TS to account for the non equilibrium nature of activation
\cite{BHL}-\cite{PHG}. Another method for
circumventing the equilibrium assumption is Kramers' method of
stationary flux, in which a source is placed at the bottom of
the reactant well and an absorbing barrier is imposed
sufficiently far outside it. Then the steady state normalized
outgoing flux on the (G)TS is related to the rate.
Attempts to account
for recrossings of the (G)TS introduced a transmission factor
$k$  \cite{Eyring35}, and assumed a relation between $k$ and
the (mean) number of crossings $\langle n\rangle$
\cite{Fleming},\cite{Miller},\cite{Frauenfelder}.  Also the choice of
the (G)TS in (2) has been the subject of study and
different authors proposed different surfaces, e.g., a
surface through the saddle point which is perpendicular to the
equipotential surfaces \cite{Wigner}, the configuration of least
probability \cite{Evans}, and a surface of minimal flux
\cite{Keck}-\cite{Pollak1}.

It seems that no satisfactory theoretical explanation of the
failure of the equilibrium assumption has been offered, other
than the incorrect predictions it implies. The methods used to
modify the epdf in order to obtain the qepdf are mostly based
on {\em ad hoc} assumptions about the behavior of the
trajectories or of the pdf on the chosen (G)TS
\cite{BHL}-\cite{PHG}. Also the choice of the
(G)TS and the treatment of recrossings have not been
adequately discussed in the literature.

In order to clarify some of these problems we consider them in
the diffusion (Langevin, or Fokker-Planck) limit \cite{Kramers},
\cite{Christiansen}.
The underlying assumption in this study is that
the behavior of the phase space trajectories of the diffusion
process approximates well that of the trajectories of a particle
coupled to a bath of oscillators \cite{Kubo}-\cite{Zwanzig}.
Such a limit corresponds to the Fokker-Planck approximation of the
Liouville evolution equation for the probability density
function (pdf) of the phase space trajectories \cite{Chandrasekhar}.
The Fokker-Planck equation describes the evolution of the
the pdf $p({\bf q,p},t)$ of the phase space trajectories of
the particle, whereas the Liouville equation describes that
of the joint pdf $p({\bf q,p,x,y},t)$ of the particle and the
oscillators in multidimensional phase space. Thus,
\begin{eqnarray}
p({\bf q,p},t)=\lim_{N\rightarrow\infty}\int\!\int
p({\bf q,p,x,y},t)\,d{\bf x}\,d{\bf y},\label{PP}
\end{eqnarray}
where ${\bf x}=(x_1,\dots,x_N)$ and ${\bf y}=(y_1,\dots,y_N)$
are the phase space coordinates of the bath oscillators.
In this approximation the behavior of the random trajectories
governed by a (generalized) Langevin equation (GLE)
\cite{Pollak86}, \cite{PollakPRL}
approximates that of the ensemble of trajectories of
 he particle, if the initial states of the bath oscillators
are chosen at random \cite{Kac}. Therefore the study of the
above mentioned problems in the diffusion limit may provide
some insight into (G)TST and its variants.

We take advantage of the eigenfunction expansion of the
nonequilibrium pdf $p({\bf q,p},t|{\bf q}_0,{\bf p}_0)$
\begin{eqnarray}
p({\bf q,p},t|{\bf q}_0,{\bf p}_0)=\sum_{n=0}^{\infty}
 \phi_n({\bf q,p})
\psi_n({\bf q}_0,{\bf p}_0)e^{-\lambda_nt},\label{efn}
\end{eqnarray}
where $\phi_n$ and $\psi_n$ are the eigenfunctions of the
Fokker-Planck operator (FPO) $L$ and its adjoint $L^*$,
respectively, corresponding to the eigenvalue $\lambda_n$, and
$({\bf q}_0,{\bf p}_0)$ is a point in the reactant part of
phase space. We denote by $D_A$ and $\partial D_A$ the
domain of attraction of the equilibrium state at the bottom of
the reactant well and its boundary, respectively.

First we examine the equilibrium assumption (1) and its consequences.
Since $\lambda_0=0$ it follows from (\ref{efn}) that for times
\begin{eqnarray}
\frac{1}{\lambda_2}\ll t\ll\frac{1}{\lambda_1}\label{t}
\end{eqnarray}
the rate of change of the reactant population is $\lambda_1$,
whereas for times
\begin{eqnarray}
t\gg\frac{1}{\lambda_1}\label{tg}
\end{eqnarray}
it vanishes. It is shown in Section 4 that
\begin{eqnarray}
\phi_1({\bf q,p})\psi_1({\bf q}_0,{\bf p}_0)\gg\phi_0({\bf q,p})
\psi_0({\bf q}_0,{\bf p}_0)\ \ \ \mbox{for $({\bf q,p})\in D_A$},
\end{eqnarray}
and $\psi_1({\bf q}_0,{\bf p}_0),\ \psi_1({\bf q}_0,{\bf p}_0)$
are independent of $({\bf q}_0,{\bf p}_0$ in $D_A$.
Thus for times (\ref{t}) $\exp\{-\lambda_1t\}=O(1)$
so that the reactant pdf $p({\bf q,p},t|{\bf q}_0,{\bf p}_0)$
is given by
\begin{eqnarray}
p({\bf q,p},t|{\bf q}_0,{\bf p}_0)\sim e^{-\lambda_1t}\phi_1({\bf q,p})
\psi_1({\bf q}_0,{\bf p}_0).\label{qepdf1}
\end{eqnarray}
It follows that the qepdf (\ref{qepdf1}) of the reactant is
represented by $\phi_1$ and not by the equilibrium density
$\phi_0$. We conclude that the root cause of error in TST is
the equilibrium assumption (1). There is however a certain
degree of similarity between the equilibrium and the quasi
equilibrium reactant densities. We show in Section 5 that
\begin{eqnarray}
\phi_1({\bf q,p})\sim\phi_0({\bf q,p})\psi_1({\bf q,p}),
\label{phi011}
\end{eqnarray}
in $D_A$, where $\psi_1$ is a boundary layer function which
connects smoothly to a large constant $C_A$ in the interior
of $D_A$ and to $\frac{1}{2}C_A$ on $\partial D_A$. This
similarity explains the need for introducing corrections to
the equilibrium density \cite{BHL}-\cite{PHG}. From the
boundary layer structure (\ref{phi011}) of
$\phi_1$ we conclude that in order to obtain the qepdf
(\ref{qepdf1}) from the epdf a modification has to be
introduced on $\partial D_A$ and it must have the
boundary layer form $\psi_1$.

Next we examine the problem of recrossings of the (G)TS. We
show in Section 4 that in the limit of high barrier the {\em
stochastic separatrix} $S$, the locus of points from which
trajectories are equally likely to become reactant and
product, is asymptotically $\partial D_A$. From the definition
of $S$ we find that
average number of times $\langle n \rangle$,
that a trajectory crosses a neighborhood of $S$
before becoming a product is 2, so that $k=1$.
The calculation of the rate from
the normalized flux on the (G)TS is based on the identity
\begin{eqnarray}
\lambda_1=\frac{\int_{\partial D}{\bf J}(\phi_1)\cdot\nu\,ds}
{\int_{D}\phi_1\,d{\bf q}\,d{\bf p}},\label{LA1}
\end{eqnarray}
where ${\bf J}(\phi_1)$ is the flux density corresponding to
the quasi equilibrium density $\phi_1$, $D$ is the
reactant region in phase space, and $\nu$ is the unit outer
normal to $\partial D$. If $\partial D=S$ is chosen, then it
follows from (\ref{LA1}) and (\ref{phi011}) that the outgoing
half of the quasi equilibrium flux on $S$, normalized by the
reactant population, is the rate. We conclude that if the
(G)TS is chosen to be the stochastic separatrix, then the
relation (3) between the rate and the quasi equilibrium flux
holds, but not otherwise.

Our conclusions about (G)TST can be summarized as follows. To
obtain the correct rate in (G)TST (i) the qepdf the reactant is
obtained from the epdf by a modification of the epdf,
(ii) the modification has to have the form
of a boundary layer at the stochastic separatrix $S$, and (iii)
all recrossings are accounted for if (G)TS=$S$, but not
otherwise. The structure of $S$ in multidimensional phase
space is needed only near the saddle point. In the high
barrier limit it can be determined from the memory function in
the GLE \cite{DMS}. The determination of $S$ in other cases has
been discussed in \cite{Tutzing}, \cite{anis}.

We find the quasi stationary behavior of the solution of
the Fokker-Planck equation (FPE) by finding the asymptotic
structure of the leading eigenvalues and eigenfunctions in the
limit of high barrier. We consider both bistable and
metastable dynamics, and discuss Kramers' method of stationary
flux. We find the average number of times a trajectory crosses
a neighborhood of an equiprobable surface $S_p$ before
becoming a product ($S_{1/2}\equiv S$). Hence
we find the relation between the flux on $S$ and the rate. In
Section 2 we formulate the problem and discuss the concept of
escape. In Section 3 we discuss the relation between the MFPT,
the absorption rate, the escape rate, and the eigenvalues of
the FPO in bounded and unbounded domains. In Section 4 we find
$S$ and the asymptotic structure of the leading
eigenfunctions. In Section 5 we consider the problem of
recrossings and of their contribution to the escape rate.
Finally, in section 6 we summarize the results and the conclusions
of this paper.\\
%
\renewcommand{\theequation}{2.\arabic{equation}}
\setcounter{equation}{0}
\setcounter{section}{2}

\noindent
{\bf 2. Formulation.}\\

We consider the system of $n$ It\^o stochastic differential
equations
\begin{equation}
{\bf \dot x = b(x)} + \sqrt{2\epsilon}\,\sigma({\bf x)\, \dot
w},\label{SDE}
\end{equation}
where ${\bf b(x)}=(b^1({\bf x}),\dots,b^n({\bf x}))$ is a smooth
vector field in ${\cal R}^n$, ${\bf \dot w}$ is a vector of $k$
independent standard Gaussian white noises ($k\leq n$),  and
${\bf \sigma(x)}$ is an $n\times k$ noise matrix. The parameter
$\epsilon$ is a measure of the noise intensity and is assumed to
be small relative to other parameters of the problem such as the
size of ${\bf b(x)}$ and ${\bf \sigma(x)}$, the sizes of the
domains in which (\ref{SDE}) is considered, and so on.
The diffusion matrix  ${\bf a(y)}\equiv\{a^{ij}({\bf y})\}$ is
given in terms of the noise matrix $\{\sigma^{ij}({\bf y})\}$ as
\begin{equation}
{\bf a(y)=\sigma(y)\sigma^T(y)}.\label{a}
\end{equation}
The noiseless dynamics
\begin{equation}
{\bf \dot x = b(x)}\label{dynamics}
\end{equation}
is assumed to have a finite attractor $A$ with domain of
attraction $D_A$ whose boundary $\partial D_A$
is a repeller (see fig.1). We assume that all domains mentioned
below have boundaries with a piecewise continuous normal. When
(\ref{SDE}) is used to model an activated process by the
Langevin dynamics of a particle in a potential well, activation
occurs when a trajectory leaves a neighborhood $\Omega_A$ of $A$
and does not return for a long period of time. Thus the
activation rate $\kappa$ is the escape rate from $\Omega_A$.
To be meaningful such a definition should be independent of
the choice of $\Omega_A$, of the location of $\partial D$, and
of the noise strength $\epsilon$, provided $\epsilon$ is small.

As noted in the Introduction, several definitions have been
given for the GTS. Calculations of $\kappa$ from the rate at
which trajectories arrive at boundaries of various domains,
have been used. One such calculation of $\kappa$ is  based
on the rate at which trajectories are absorbed in the
boundary $\partial\Omega_A$, or equivalently, $\kappa=
1/\langle \tau(\Omega_A)\rangle$, where
$\langle\tau(\Omega_A)\rangle$ is the mean first passage time (MFPT)
to $\partial \Omega_A$ \cite{FV}, \cite{Ludwig}. If $\Omega_A$
is replaced by the domain
of attraction $D_A$ of $A$, then for small $\epsilon$,
as noted in \cite{79}-\cite{Mangel}
trajectories arriving at $\partial D_A$ are equally likely
to return to $\Omega_A$ prior to absorption in the boundary
$\partial D$ of a larger domain $D$ containing $D_A$, as they are to
be absorbed before they return to $\Omega_A$. Therefore the
calculation of $\kappa$ was modified in  \cite{Tutzing}, \cite{79},
\cite{Direct}, to $\kappa=1/2\langle \tau(D_A)\rangle$.
Another calculation identifies $\kappa$ as the principal
eigenvalue $\lambda_1$ of the Fokker-Planck equation in
the whole space \cite{Risken} or in a specified domain
$D$ \cite{Cartling}. Not all
such calculations are consistent with one another or with
physically measured quantities. Thus for example, the normalized
outgoing flux at the top of a potential barrier may not represent
the rate, since trajectories arriving at the top of a potential
barrier with outward pointing velocity may have a non-negligible
probability of returning to the well.

The definition of an escape rate $\kappa$
from $\Omega_A$ as the absorption rate $\kappa_{abs}(D)$
in $\partial D$ requires the calculation of the MFPT
$\langle\tau(D)\rangle$ from $\Omega_A$ to $\partial D$, or
equivalently, of the principal eigenvalue $\lambda_1(D)$ of
the Fokker-Planck operator with absorbing boundary conditions
on $\partial D$. The calculation of $\langle\tau(D)\rangle$
and of $\lambda_1(D)$ for multidimensional systems
in domains $D$ containing an attractor has a long history.
Several types of domains have been considered, which we now
classify. A domain  $D$ is classified as type I if
$D\subset D_A$ and the drift enters $D$. In this case $\partial
D$ is a noncharacteristic boundary. If $D=D_A$, we classify it
as type II. Here the system does not drift into $D_A$,
$\partial D$ is a characteristic boundary, and
is an invariant manifold for the drift. If $D\supset D_A$
and the drift in $D-\bar D_A$ is attracted by
$\partial D$, we classify $D$ as type III (see the exact
definition below).

The MFPT for domains of type I was calculated in \cite{FV},
\cite{Ludwig}, \cite{book} with increasingly more
accurate results. Escape rates from domains of
type II with drift derivable from a potential were calculated in
\cite{Brinkman}-\cite{Langer}
by a generalization of Kramers' theory to higher dimensions. For
general systems, not derivable from a potential,
 $\langle\tau(D)\rangle$ was calculated in \cite{PLL}-\cite{Day}.
For domains of type III neither $\langle\tau(D)\rangle$ nor
$\lambda_1(D)$ seem to have been calculated, other than for
the one dimensional Kramers problem \cite{Kramers}.\\

\noindent
{\bf Definition.} {\it A domain $D$ is of type {\rm III}
relative to {\rm (\ref{dynamics})} if\\
{\rm (i)} $D\supset \bar D_A$,\\
{\rm (ii)}
\begin{eqnarray}
\delta_1<{\rm dist}({\bf x},\partial D)<\delta_2 \ \ \mbox{{\rm for all }
${\bf x}\in\partial D_A$}, \label{d12}
\end{eqnarray}
and $\delta_1,\ \delta_2$ are positive constants, independent
of $\epsilon$, and\\
{\rm (iii)} trajectories of {\rm (\ref{dynamics})} which start
in $D-\bar D_A$ reach $\partial D$ in finite time.}\\

An often encountered example of a domain of type III is the
following. Let (\ref{dynamics}) be a bistable system with
a finite attractor $A$ and a second, more stable
attractor $B$ (possibly at infinity). If $\Omega_B$ is a
neighborhood of $B$, inside the domain of attraction of
$B$, then $D\equiv {\cal R}^n-\bar\Omega_B$ is a domain of type
III relative to (\ref{dynamics}). This
example corresponds, e.g., to a model of a chemical reaction.
The noisy dynamics (\ref{SDE}) models the motion of an atom
bound by a stable chemical bond, corresponding to the attractor
$A$. When the bond is broken due to the noise (e.g., to
molecular collisions), a new more stable bond $B$ is formed
\cite{Kramers}. Another example of a domain of type III relative
to (\ref{dynamics}) corresponds to the dynamics
(\ref{SDE}) with a single metastable state. In
this case $D$ contains $D_A$ and trajectories which start
in $D-\bar D_A$ are assumed to exit $D$ in finite
time and never return. This corresponds, e.g., to dissociation,
where the metastable state $A$ models the undissociated state
of a molecule \cite{Weiss}.

Because of the weak fluctuations, the trajectories of the
system tend to stay in a small neighborhood $\Omega_A$ of $A$,
with rare escapes to
$\Omega_B$ or to an absorbing boundary $\partial D$, from
which they do not return. In the case of bistability
$\Omega_B$ usually corresponds to a more stable state of the
system, so that the time spent in $\Omega_B$ is usually much
longer than in $\Omega_A$. Thus we may assume that trajectories
arriving in $\Omega_B$ are absorbed there and never return to
$A$.  Therefore the bistable case can also be treated as metastable
with absorption in $\partial D$.

The transition probability density function (tpdf) of
trajectories of (\ref{SDE}), $p({\bf y},t|{\bf x})\equiv
Pr\{{\bf x}(t)={\bf y}|{\bf x}(0)={\bf x}\}$,
satisfies the forward Kolmogorov (Fokker-Planck) equation
\cite{Gardiner}
\begin{equation}
\frac{\partial p}{\partial t}= L_\epsilon p\equiv-\nabla
{\bf \cdot J},\label{FPE}
\end{equation}
where the probability current density ${\bf J(x,y},t)$
is defined as
\begin{equation}
J^i({\bf x,y},t)\equiv-\epsilon\sum_{j=1}^{n}
\frac{\partial}{\partial y^j}\left[a^{ij}({\bf y})p({\bf y},t|
{\bf x})\right]+b^i({\bf y})p({\bf y},t|{\bf x}).\label{J}
\end{equation}
The initial condition for (\ref{FPE}) is given by
\begin{eqnarray}
p({\bf y},0|{\bf x})=\delta({\bf x-y}).\label{IC}
\end{eqnarray}
If trajectories of (\ref{SDE}) are absorbed on the boundary
$\partial D$ of a given domain $D$, the tpdf satisfies
the boundary condition
\begin{equation}
p({\bf y},t|{\bf x})=0\ \ {\rm for}\ \ {\bf x}\in{D},\
{\bf y}\in\partial{D}.\label{BC}
\end{equation}
It also satisfies the backward Kolmogorov equation
\begin{equation}
\frac{\partial p}{\partial t}= L_{\epsilon}^{*} p
\equiv\epsilon\sum_{i,j=1}^{n}a^{ij}({\bf x})
\frac{\partial^2p}
{\partial x^i\partial x^j}
+\sum_{i=1}^{n}b^i({\bf x})\frac{\partial p}{\partial x^i}
\ \ {\rm for}\ {\bf x}\in D,\label{PDE}
\end{equation}
and
\begin{equation}
p=0\ \ {\rm for}\ \ {\bf x}\in\partial D,\ {\bf y}\in D.\label{BC1}
\end{equation}
We assume that the tpdf $p({\bf y},t|{\bf x})$
can be represented by the eigenfunction expansion
\begin{equation}
p({\bf y},t|{\bf x}) = \sum_{n=1}^{\infty}\psi_{n}({\bf
x})\phi_n({\bf y})e^{-\lambda_n(D)t},\label{efexpansion}
\end{equation}
where
\begin{eqnarray}
& &L_\epsilon\phi_n({\bf y})= -\lambda_n(D) \phi_n({\bf y})\ \
{\rm for}\ \ {\bf y}\in{D}, \label{phin}\\
& &\phi_n({\bf y})=0\ \ {\rm for}\ \ {\bf y}\in\partial{D},
\label{phinbc}
\end{eqnarray}
and
\begin{eqnarray}
& &L_{\epsilon}^{*}\psi_n({\bf x})=-\lambda_{n}(D)
\psi_n({\bf x})\ \
{\rm for}\ \ {\bf x}\in{D}, \label{psin}\\
& &\psi_n({\bf x})=0\ \ {\rm for}\ \ {\bf x}\in\partial{D}
\label{psinbc}.
\end{eqnarray}
This is the case for example, if $D$ is a bounded domain,
however we do not restrict our considerations to this case.
The eigenvalues are ordered as $0<\lambda_1(D)\leq \Re{\rm
e}\,\lambda_2(D)\leq \Re{\rm e}\,\lambda_3(D)\leq\dots$,
with $\lambda_1(D)$ real \cite{ProtterWeinberger}.
 We assume that the eigenfunctions are normalized so that
\begin{equation}
\int_{D}\phi_1({\bf y})\,d{\bf y}=1\label{phi=1},
\end{equation}
and are biorthogonal
\begin{equation}
\int_{D}\phi_m({\bf x})\psi_{n}({\bf x})\,d{\bf
x}=\delta_{mn}.\label{phipsi}
\end{equation}
In Section 4 we calculate the probability $P({\bf x})$ of
return from a point ${\bf x}$ to $\Omega_A$, and in Section 6
we identify it asymptotically as the principal eigenfunction
$\psi_1({\bf x})$. In Section 5 we calculate the principal eigenvalue
$\lambda_1(D)=\kappa=1/\langle\tau(D)\rangle$ for domains of
type III, and prove
\begin{eqnarray}
\lim_{\epsilon\rightarrow 0}\frac{\lambda_1(D)}{\lambda_1(D_A)}
=\frac{1}{2}.\label{lim}
\end{eqnarray}
\\
%
\renewcommand{\theequation}{3.\arabic{equation}}
\setcounter{equation}{0}
\setcounter{section}{3}
\vspace{0.5cm}

\noindent
{\bf 3. The MFPT, the absorption rate, and the principal
eigenvalue}\\

Let $D$ be a bounded domain which contains $A$.
 Then, under some mild positivity assumptions about ${\bf a(x)}$,
the trajectories of the noisy system (\ref{SDE}) which start in $D$,
cross $\partial D$ in finite (random)
time $\tau$ with probability 1. Moreover, the MFPT
\begin{equation}
\langle\tau({\bf x})\rangle\equiv E(\tau|{\bf x}(0)={\bf x})
\end{equation}
from a point ${\bf x}$ in $D$ to $\partial D$ is finite and is
independent of the behavior of the process at the boundary.
These results may hold as well even if $D$ is unbounded.
Thus we may assume that $\partial D$  is an absorbing boundary,
so that any trajectory of (\ref{SDE}) that reaches  $\partial D$
is instantaneously terminated. The MFPT is given by \cite{book}
\begin{equation}
\langle\tau({\bf x})\rangle=\int_{0}^{\infty}\!\int_{
D}p({\bf y},t|{\bf x})\,d{\bf y}\,dt,\label{taudydt}
\end{equation}
or by
\begin{equation}
\langle\tau({\bf x})\rangle=\int_{D}p({\bf x,y})\,d{\bf
y},\label{taudy}
\end{equation}
where $p({\bf x,y})$ is defined by
\begin{equation}
p({\bf x,y})\equiv\int_{0}^{\infty}p({\bf y},t|{\bf x})\,dt.
\end{equation}
>From the Fokker-Planck equation (\ref{FPE}) and the initial
condition (\ref{IC}) we find that
\begin{equation}
L_\epsilon p({\bf x,y})=-\delta({\bf
x-y}).\label{Lpdelta}
\end{equation}
If $\delta({\bf x-y})$ is replaced by $f({\bf y})$ in the
initial condition (\ref{IC}), the pdf $p({\bf x,y})$
is independent of ${\bf x}$, that is, $p({\bf x,y})=p({\bf
y})$, and satisfies the equation
\begin{equation}
L_\epsilon p({\bf y})=-f({\bf y}).\label{Lpf}
\end{equation}
In particular, if we choose $f({\bf y})=\phi_1({\bf y})$, then
$p({\bf y})=\phi_1({\bf y})/\lambda_1(D)$ and
(\ref{taudy}) implies that
\begin{eqnarray}
\langle\tau({D})\rangle\equiv\int_{D}\phi_1({\bf x})\langle
\tau({\bf x})\rangle\,d{\bf x}=\frac{1}{\lambda_1(D)}.
\label{taulambda1}
\end{eqnarray}
The structure of the functions $p({\bf x,y})$ and $p({\bf y})$
has been discussed in \cite{Direct}. Equation (\ref{Lpdelta})
with absorbing boundary conditions on $\partial D$ represents
Kramers' method of stationary flux, in which a source is
placed inside $D$ and an absorbing barrier is imposed outside
$D_A$. Equation (\ref{taulambda1}) relates the solution of
(\ref{Lpdelta}) with the escape rate. The relation of the out
going flux on the (G)TS, as obtained from (\ref{Lpdelta}), and
the rate is discussed in Sections 4 and 6.\\

Next we discuss the relationships between the absorption rate
$\kappa_{abs}(D)$ in $\partial D$, the MFPT
$\langle\tau(D)\rangle$, and the principal eigenvalue
$\lambda_1(D)$.  The absorption rate constant $\kappa_{abs}(D)$
represents the stationary relative rate of
change of the population $N({\bf x},t)$ in ${D}$, given
the initial condition (\ref{IC}), where
\begin{equation}
N({\bf x},t)\equiv\int_{D}p({\bf y},t|{\bf x})\,d{\bf
y}.\label{Nxt}
\end{equation}
The Fokker-Planck equation (\ref{FPE}) implies that
\begin{eqnarray}
\frac{\partial N({\bf x},t)}{\partial t}&=&\int_{
D}\frac{\partial p({\bf y},t|{\bf x})}{\partial t}\,d{\bf y}=
-\int_{D}\nabla_{\bf y}\cdot{\bf J(x,y},t)\,d{\bf y}\nonumber\\
&=&-\int_{\partial {D}}{\bf J(x,y},t)\cdot\nu({\bf
y})\,dS_{\bf y}\equiv -F({\bf x},t).\label{Ndot}
\end{eqnarray}
Recalling that ${\bf J}$ is the probability current density,
we see that the rate of change of the population in ${D}$
is the negative of the total probability flux on
$\partial {D}$. It follows that
the relative rate of change is given by
\begin{equation}
\kappa_{abs}({\bf x},t)\equiv
-\frac{\partial N({\bf x},t)/\partial t}{N({\bf x},t)}=
\frac{F({\bf x},t)}{N({\bf x},t)}.\label{kFN}
\end{equation}
Using the eigenfunction expansion (\ref{efexpansion}) we
obtain
\begin{equation}
\kappa_{abs}({\bf x},t)=
\frac{\sum_n\lambda_n(D)\\psi_{n}({\bf x})\int\phi_n({\bf y})\,
d{\bf y}\,e^{-\lambda_n(D)t}}{\sum_n\psi_{n}({\bf x})
\int\phi_n({\bf y})\,d{\bf y}\,e^{-\lambda_n(D)t}}.\label{relative}
\end{equation}
Obviously the relative rate of change (\ref{relative}) depends
on both ${\bf x}$ and $t$, however its limit as
$t\rightarrow\infty$ is constant. Thus the concept of a
rate constant for (\ref{SDE}) is meaningful only in
the limit $t\rightarrow\infty$, since the relative rate of
change becomes constant only after a sufficiently long time.
Therefore the natural definition of the absorption rate constant
$\kappa_{abs}(D)$ is given as
\begin{equation}
\kappa_{abs}(D)\equiv\lim_{t\rightarrow\infty}\kappa_{abs}
({\bf x},t),\label{kabsorption}
\end{equation}
which, in view of (\ref{relative}) is
\begin{equation}
\kappa_{abs}(D)=\lambda_1(D).\label{lambda1}
\end{equation}

The corresponding notion of the MFPT must be defined in a
manner consistent with the definition (\ref{lambda1}) of
rate as a long time limit. After a long time $t_0$ has
elapsed, a large proportion of the trajectories have already been
absorbed. Thus we have to consider only those trajectories
that survived (were not absorbed) in $\partial D$ at time
$t_0$. The probability density of finding a trajectory at a
particular point ${\bf y}$, among the surviving trajectories
in ${D}$ at time $t_0$, is the conditional density
\begin{equation}
p_C({\bf y},t_0|{\bf x}) \equiv\frac{p({\bf y},t_0|{\bf x})}
{\int_{D}p({\bf y},t_0|{\bf x})\,d{\bf y}}.\label{pC}
\end{equation}
Using the eigenfunction expansion (\ref{efexpansion}) and the
normalization (\ref{phi=1}), we
obtain
\begin{equation}
p_C({\bf y},t_0|{\bf x})=\frac
{\sum_n\psi_{n}({\bf x})\phi_n({\bf y})e^{-\lambda_n(D)t_0}}
{\sum_n
\psi_{n}({\bf x})
\int_{D}\phi_n({\bf y})\,d{\bf y}e^{-\lambda_n(D)t_0}}\rightarrow
\phi_1({\bf y})\ \ {\rm as}\ \ t_0\rightarrow\infty.
\end{equation}
Thus the conditional density $p^C({\bf y},t)$
of trajectories at time $t+t_0$,
which were observed at time $t_0\gg 1$ in ${D}$,  is the
solution of the Fokker-Planck equation (\ref{FPE}) with the
absorbing boundary condition (\ref{BC}), and the initial
condition
\begin{equation}
p^C({\bf y},0)=\phi_1({\bf y}).\label{pc0}
\end{equation}
Thus, for times $1/\lambda_2\ll t\ll 1/\lambda_1$ the qepdf is
given by
\begin{equation}
p^C({\bf y},t)=\phi_1({\bf y})e^{-\lambda_1(D)t}.\label{p1}
\end{equation}
Therefore the MFPT for a trajectory,
observed at a point ${\bf y}$ at a sufficiently large time
$t_0$, must be calculated by (\ref{taudydt}), with
the conditional density $p^C({\bf y},t)$ of surviving trajectories
replacing the tpdf $p({\bf y},t|{\bf x})$.
Now the concept of first passage time, which is consistent with
the definition (\ref{kabsorption}) of $\kappa_{abs}(D)$, is the first
passage time for the process (\ref{SDE}) which starts with the
initial density (\ref{pc0}). The long time MFPT is therefore given
by
\begin{equation}
\langle\tau(D)\rangle=\int_{0}^{\infty}\!\int_{D}p^C({\bf
y},t)\,d{\bf y}\,dt=\frac{1}{\lambda_1(D)}.\label{taulambda2}
\end{equation}
We see that, according to this definition, both $\langle\tau(D)
\rangle$ and $\kappa_{abs}(D)$ are independent of initial conditions.
Although $p^C({\bf y},0)$ is not a density of the process ${\bf x}(t)$,
defined by (\ref{SDE}), it represents the shape of the long
time tpdf with the decay $e^{-\kappa_{abs}(D) t}$.

Now we consider the FPE (\ref{FPE}) in ${\cal R}^n$. We assume
that (\ref{dynamics}) is a bistable system with attractors $A$
and $B$, whose domains of attraction, $D_A$ and $D_B$, are
separated by $\partial D_A$. We denote by $\phi_i,\ \psi_i$,
and $\lambda_i$ the corresponding eigenfunctions and
eigenvalues. We have $\lambda_0=0$ and $\phi_0$ is the
equilibrium pdf. We write (\ref{phin}) as
\begin{eqnarray}
-\nabla\cdot{\bf J}(\phi_n)=-\lambda_n\phi_n,\label{Jn}
\end{eqnarray}
where ${\bf J}(\phi_n)$ is defined in (\ref{J}) with $p$ replaced by
$\phi_n({\bf y})$. Then for any domain $\Omega\subset{\cal R}^n$
\begin{eqnarray}
\lambda_n=\frac{F_n}{N_n},\label{ln}
\end{eqnarray}
where
\begin{eqnarray}
F_n\equiv\int_{\partial \Omega}{\bf J}(\phi_n)\cdot\nu\,ds\label{Fn}
\end{eqnarray}
and
\begin{eqnarray}
N_n\equiv\int_\Omega\phi_n\,d{\bf y}.\label{Nn}
\end{eqnarray}
In particular we choose $\Omega=D_A$.

Next we describe the structure of the MFPT and its dependence
on $D$. For domains of type I it was shown  and II
\cite{Ludwig}, \cite{book}, \cite{MS} that
\begin{equation}
\langle\tau(D)\rangle =O(\epsilon^{1/2}
e^{\hat\Psi(D)/\epsilon}),\label{Keps}
\end{equation}
where $\hat\Psi(D)$ is a positive constant which depends on
$D$ and on the
coefficients $a^{ij}({\bf x})$ and $b^i({\bf x})$ in $D$. Explicit
expressions for $\hat\Psi(D)$ were given in \cite{FV}, \cite{Ludwig},
\cite{book} for various domains in terms of
an eikonal function $\Psi({\bf x})$, which is a solution of
the Hamilton-Jacobi type equation
\begin{equation}
\sum_{i,j=1}^{n}a^{ij}({\bf x})\frac{\partial \Psi({\bf
x})}{\partial x^i}\frac{\partial \Psi({\bf x})}{\partial x^j} +
\sum_{i=1}^{n}b^i({\bf x})\frac{\partial \Psi({\bf x})}{\partial
x^i}=0.\label{HJE}
\end{equation}
The constant $\hat\Psi(D)$ is given by
\begin{equation}
\hat\Psi(D)=\min_{{\bf x}\in\partial D}\Psi({\bf x}).\label{Kpsi}
\end{equation}

The function $\Psi({\bf x})$ is defined by the assumption that
the principal eigenfunction $\phi_0$ has the WKB structure
in $D$
\begin{eqnarray}
\phi_0({\bf y})=K({\bf y},\epsilon)e^{-\Psi({\bf
y})/\epsilon},\label{WKB}
\end{eqnarray}
where $K({\bf y},\epsilon)$ is a regular function of
$\epsilon$. The structure of $\Psi({\bf x})$ for bistable systems
(\ref{dynamics}) is more complicated. If we assume that
(\ref{dynamics}) has attractors $A$ and $B$, with domains of
attraction $D_A$ and $D_B$, respectively, then
a global smooth solution to (\ref{HJE}) in ${\cal R}^n$ may not
exist \cite{GrahamTel}. However locally, in $D_A$ and
$D_B$, (\ref{WKB}) holds. In the case of detailed balance
\cite{Gardiner} (\ref{WKB}) holds in ${\cal R}^n$ with $K({\bf
y},\epsilon)=1$ and $\Psi({\bf y})$ is the energy, so that
$\phi_0({\bf y})$ is the Boltzmann equilibrium density.

The function $\Psi({\bf x})$ decreases on the trajectories of
(\ref{dynamics}) in $D_A$, so that $\hat\Psi(D_1)\geq \hat\Psi(D_2)$
if $D_1$ and $D_2$ are of type I and $D_1\supset D_2$,
hence by (\ref{Keps})
\begin{eqnarray}
\frac{\langle\tau(D_1)\rangle}{\langle\tau(D_2)\rangle}=
O(e^{[\hat\Psi(D_1)-\hat\Psi(D_2)]/\epsilon}).
\end{eqnarray}
If in addition
$\delta_1\leq\mbox{dist}({\bf x},\partial D_2)\leq\delta_2$
for all ${\bf x}\in\partial D_1$, where $\delta_1$ and $\delta_2$
are positive numbers independent of $\epsilon$, then
$\hat\Psi(D_1)>\hat\Psi(D_2)$, so that
\begin{eqnarray}
\lim_{\epsilon\rightarrow 0}
\frac{\langle\tau(D_2)\rangle}{\langle\tau(D_1)\rangle}
=0.\label{frac0}
\end{eqnarray}
This however, is not the case for domains of type III,
since for such domains the limit (\ref{lim}) implies that
$\langle\tau(D)\rangle$ is to leading order independent of $D$.

We now discuss the relation between absorption and escape.
We distinguish between two types of escape. Let $\Omega_A$
be a neighborhood of $A$ in $D_A$. One type of escape is the event
of reaching $\partial\Omega_A$ for the first time, regardless
of the subsequent behavior of the trajectory. This type of
escape describes several physical systems in which trajectories
are terminated at $\partial\Omega_A$ \cite{Ludwig}, \cite{Hagan}.
The rate of this type
of escape is simply the absorption rate in $\partial \Omega_A$.
It involves neither multistability nor separation of time scales.
A second type of escape involves multistability and
separation of time scales. According to (\ref{Keps}) trajectories spend
time $O(\exp(\hat\Psi(\Omega_A)/\epsilon)$ in $\Omega_A$ before
reaching $\partial\Omega_A$ (necessarily crossing it several times
in rapid succession). If a trajectory reaches a distance $O(1)$
beyond $\partial\Omega_A$ in ${\cal R}^n-\bar\Omega_A$, it
either returns to $\Omega_A$ or is absorbed in $\partial D$
without returning to $\Omega_A$. In the first instance, if the
time of return is short relative to
$\langle\tau(\Omega_A)\rangle$, the trajectory
is not considered to have escaped, whereas in the latter it
is. This distinction means that relatively short departures
do not represent transitions to a new physical state, e.g., going from
reactant to product in a chemical reaction, whereas long
departures do represent such transitions. The rate at which
long departures occur can be defined in terms of an absorption
rate in $\partial D$, where $D$ is a domain of type III.\\

\noindent
{\bf 4. The probability of returns, $\psi_1$, $\phi_1$, and
$\lambda_1$}\\
\renewcommand{\theequation}{4.\arabic{equation}}
\setcounter{equation}{0}
\setcounter{section}{4}

In this section we consider the probability $P({\bf x})$
of return from a point ${\bf x}$ to $\Omega_A$,
prior to absorption in $\partial D$, where $D$ is a domain of
type III, and its relation to the eigenfunctions $\psi_1$ and
$\phi_1$ in ${\cal R}^n$ and in domains of type II and type
III. First we summarize the results
of \cite{MangelLudwig} for drifts which are continuous across
$\partial D_A$ (e.g., for smooth potential barriers), and then
show that similar results hold for drifts whose normal component
suffers a discontinuity across $\partial D_A$ (e.g., for sharp
potential barriers).

We denote by $\tau(\partial\Omega_A)$ ($\tau(\partial
D))$ the first passage time to $\partial\Omega_A$ ($\partial
D$). The function
\begin{eqnarray}
P({\bf x})\equiv {\rm Pr}\{\tau(\partial\Omega_A)
<\tau(\partial D)|{\bf x}_\epsilon(0)={\bf x}\}\label{Px}
\end{eqnarray}
is the probability that a trajectory starting at ${\bf x}$
will reach $\partial\Omega_A$ before $\partial D$. The {\em
stochastic separatrix} $S$ \cite{Tutzing}, \cite{79}-\cite{Mangel},
\cite{RyterJSP}, \cite{RyterPhysica}
is defined as the locus of points ${\bf x}$ such that
\begin{eqnarray}
P({\bf x})=1/2.\label{P12}
\end{eqnarray}
The probability $P({\bf x})$ is the solution of
\begin{eqnarray}
L^*_\epsilon P({\bf x})=0\ \ {\rm in}\ D-\bar\Omega_A\label{LP}
\end{eqnarray}
with the boundary conditions
\begin{eqnarray}
P({\bf x})=1\ \ {\rm for}\ {\bf x}\in\partial\Omega_A\label{BComegaA}
\end{eqnarray}
and
\begin{eqnarray}
P({\bf x})=0\ \ {\rm for}\ {\bf x}\in \partial D.\label{BCD}
\end{eqnarray}
(see e.g. \cite{book}). In Section 6 we show that
asymptotically (\ref{LP})-(\ref{BCD}) also define the
principal eigenfunction $\psi_1({\bf x})$ of $L^*_\epsilon$.

The outer expansion of $P({\bf x})$  in $D$ is found by
assuming the regular expansion
\begin{equation}
P({\bf x})=P^0({\bf x})+o(1)\ \ {\rm as}\ \epsilon\rightarrow
0.\label{outer}
\end{equation}
In view of (\ref{Keps}) the leading term $P^0({\bf x})$
must satisfy  the reduced equation
\begin{equation}
\sum_{i=1}^{n}b^i({\bf x})\frac{\partial P^0({\bf x})}{\partial
x^i} = 0,\label{reduced}
\end{equation}
or equivalently,
\begin{equation}
\frac{dP^0({\bf x}(t))}{dt}=0,\label{dpdt}
\end{equation}
where ${\bf x}(t)$ is any trajectory of (\ref{dynamics}). It
follows that $P^0({\bf x})=const.$ on every trajectory.
By assumption, all trajectories outside $D_A$
reach $\partial D$ in finite time, so that the boundary
condition (\ref{BCD}) implies that
\begin{equation}
P^0({\bf x})=0\ \ {\rm for}\ {\bf x}\in D-\bar{D_A}.\label{p0}
\end{equation}
Similarly, all trajectories in $D_A$ reach $\partial \Omega_A$
in finite time, hence by (\ref{BComegaA})
\begin{equation}
P^0({\bf x})=1\ \ {\rm for}\ {\bf x}\in D_A.\label{p11}
\end{equation}

For continuous drifts the discontinuity in the outer solution
is bridged by the uniform expansion \cite{MangelLudwig},
\cite{Mangel}, given by
\begin{eqnarray}
P({\bf x})=\sqrt{\frac{1}{2\pi}}\int_{-\infty}^{\chi({\bf
x})/\sqrt{\epsilon}}e^{-z^2/2}\,dz.\label{Pint}
\end{eqnarray}
The function $\chi({\bf x})$ satisfies (see (\ref{LP}))
\begin{eqnarray}
\sum_{i=1}^{n}b^i({\bf x})\frac{\partial\chi({\bf x})}{\partial
x^i}=\sum_{i,j=1}^{n}a^{ij}({\bf
x})\left(\chi({\bf x})\frac{\partial\chi({\bf x})}{\partial
x^i}\frac{\partial\chi({\bf x})}{\partial x^j}- \epsilon\frac{
\partial^2\chi({\bf x})}{\partial x^i\partial x^j}\right).\label{chi}
\end{eqnarray}
The boundary conditions (\ref{BComegaA}) and (\ref{BCD}) imply that
\begin{eqnarray}
\chi({\bf x})>0\ \ \mbox{for ${\bf x}\in D_A-\bar\Omega_A$,
outside a boundary layer}\label{+}
\end{eqnarray}
 and
\begin{eqnarray}
\chi({\bf x})<0\ \ \mbox{for ${\bf x}\in D-\bar D_A$, outside
a boundary layer}.\label{-}
\end{eqnarray}
The boundary conditions (\ref{BComegaA}) and (\ref{BCD})
are satisfied by (\ref{Pint}) asymptotically. In view of (\ref{P12})
$S$ is defined by $\chi({\bf x})=0$. Now (\ref{+}) and
(\ref{-}) imply that $\lim_{\epsilon\rightarrow 0}{\cal
S}=\partial D_A$ (see the discussion of this argument in Section
6).  Expanding $\chi({\bf x})=\chi^0({\bf x})+o(1)$ as
$\epsilon\rightarrow 0$ we obtain that on a trajectory ${\bf
x}={\bf x}(t)$ of (\ref{dynamics}).
\begin{eqnarray}
\frac{d}{dt}\chi^0({\bf x})=\sum_{i=1}^{n}b^i({\bf x})
\frac{\partial\chi^0({\bf x})}{\partial
x^i}=\sum_{i,j=1}^{n}a^{ij}({\bf x})\chi^0({\bf x})
\frac{\partial\chi^0({\bf x})}{\partial
x^i}\frac{\partial\chi^0({\bf x})}{\partial x^j}.\label{chi0}
\end{eqnarray}
In view of (\ref{+}), (\ref{-}), and (\ref{chi0}),
$\chi^0({\bf x})$ increases on the trajectories of
(\ref{dynamics}) outside $D_A$ and decreases inside $D_A$. \\

Next we consider drifts ${\bf b(x)}$ whose normal component
$b_n({\bf x})$ suffers a discontinuity across $\partial D_A$.
That is, we assume that $\pm\lim_{\rho\rightarrow 0\pm}
b_n({\bf x})>0$. This is the case, e.g., if the potential
barrier is sharp (see fig.1). The solution of (\ref{LP}) is
still given by (\ref{Pint}), however the local behavior of
$\chi({\bf x})$ near $\partial D_A$ is not as given in
\cite{MangelLudwig}. The boundary layer to bridge the
discontinuity of the outer solution
across $\partial D_A$ can be found in this case
by introducing the local coordinates $(\rho,{\bf s})$ near
$\partial D_A$, where $\rho({\bf x})\equiv{\rm dist}({\bf
x},\partial D_A)$ and ${\bf s(x)}=(s^2,\dots,s^n)$ are local
coordinates in $\partial D_A$. Then we stretch $\rho$ by
setting

\begin{eqnarray}
\zeta\equiv\frac{\rho}{\epsilon},\label{zeta}
\end{eqnarray}
$Q(\zeta,{\bf s})\equiv P({\bf x})$, and expanding
\begin{eqnarray}
Q(\zeta,{\bf s})\sim Q^0(\zeta,{\bf s})+\epsilon Q^1(\zeta,{\bf s})+
\dots\ .\label{Q}
\end{eqnarray}
Now (\ref{LP}) is to leading order
\begin{eqnarray}
a({\bf s})\frac{\partial^2Q^0(\zeta,{\bf s})}{\partial
\zeta^2}+b_n^\pm({\bf s})\frac{\partial Q^0(\zeta,{\bf
s})}{\partial\zeta}=0\ \ \mbox{for $\pm\zeta>0$},\label{Qeq}
\end{eqnarray}
where
\begin{eqnarray}
a({\bf s})\equiv\lim_{\rho\rightarrow 0}\sum_{i,j=1}^{n}
a^{ij}({\bf
x})\frac{\partial\rho({\bf x})}{\partial x^i}\frac{\partial\rho({\bf
x})}{\partial x^j}\label{as}
\end{eqnarray}
and
\begin{eqnarray}
b_n^\pm({\bf s})\equiv\lim_{\rho\rightarrow 0\pm}{\bf
b(x)\cdot\nabla\rho(x)}.\label{bn}
\end{eqnarray}
Since $\partial D_A$ is a repeller we have $\pm b_n^\pm({
\bf s})>0$. The matching conditions are
\begin{eqnarray}
\lim_{\rho\rightarrow\infty}Q^0(\rho,{\bf s})=1,\ \
\lim_{\rho\rightarrow-\infty}Q^0(\rho,{\bf s})=0.\label{MC}
\end{eqnarray}
The solution is given by
\begin{eqnarray}
Q^0(\zeta,{\bf s})=1+\frac{b_n^-({\bf s})}{b_n^+({\bf s})-b_n^-({\bf
s})}\,\exp\{-\frac{b_n^+({\bf s})}{a({\bf s})}\zeta\}\ \
\ {\rm if}\ \ \zeta>0\label{zeta>0}
\end{eqnarray}
and
\begin{eqnarray}
Q^0(\zeta,{\bf s})=\frac{b_n^+({\bf s})}{b_n^+({\bf s})-b_n^-({\bf
s})}\,\exp\{-\frac{b_n^-({\bf s})}{a({\bf s})}\zeta\}\ \
\mbox{if $\zeta<0$}\label{zeta<0}.
\end{eqnarray}
The stochastic separatrix $S$, given to leading order
by $Q^0(\zeta,{\bf s})=1/2$, is found to be
\begin{eqnarray}
\rho({\bf s})=\frac{\epsilon a({\bf s})}{b_n^\pm({\bf s})}\log
\frac{\pm 2b_n^\pm({\bf s})}{b_n^+({\bf s})-b_n^-({\bf s})}\
\ \ {\rm if}\ \  \frac{\pm 2b_n^\pm({\bf s})}{b_n^+({\bf s})-
b_n^-({\bf s})}>1.
\end{eqnarray}
Thus $\lim_{\epsilon\rightarrow 0}S=\{\rho({\bf
s})=0\}=\partial D_A$ in this case as well. Therefore
(\ref{Pint}) still holds, but with the local behavior
(\ref{zeta>0}) and (\ref{zeta<0}) rather than that in the
continuous case (see \cite{MangelLudwig}).

Next we consider the eigenvalue problems (\ref{phin}) and
(\ref{psin}) in ${\cal R}^n$ and in domains of type II and
III. We we begin with a bistable system (\ref{dynamics}) in
${\cal R}^n$, with attractors $A$ and $B$.
The domains of attraction $D_A$ and $D_B$ are separated by
$\partial D_A$. We denote by $\Psi_{A(B)}({\bf y})$ the
solution of (\ref{HJE}) in $D_{A(B)}$ and set
$\Delta\Psi_{A(B)}\equiv\min_{\partial D_A}\Psi_{A(B)}({\bf
y})-\Psi_{A(B)}(A(B))$. In the case of bistable Langevin dynamics
$\Delta\Psi_{A(B)}$ is the barrier height of the $A(B)$ well.
If
\begin{eqnarray}
\Delta\Psi_A<\Delta\Psi_B,\label{D<D}
\end{eqnarray}
then
\begin{eqnarray}
\lambda_1=O(e^{-\Delta\Psi_A/\epsilon}).\label{lO}
\end{eqnarray}
It follows that (\ref{psin}) with $n=1$ is asymptotically
the same as (\ref{LP}).

To find the expansion of $\psi_1$ in ${\cal R}^n$ we begin
with the outer expansion
\begin{eqnarray}
\psi_1\sim \psi_1^0+\epsilon\psi_1^1+\dots
\end{eqnarray}
and find that
\begin{eqnarray}
\psi_1^0\sim C_{A(B)}\ \ \ \mbox{in $D_{A(B)}$},
\end{eqnarray}
where $C_A$ and $C_B$ are constants \cite{book}. The boundary layer
connecting the constants across $\partial D_A$ is similar to
$P({\bf x})$ in (\ref{Pint}),
\begin{eqnarray}
\psi_1({\bf y})\sim\frac{C_A-C_B}{\sqrt{2\pi}}
\int_{0}^{\chi({\bf y})/\sqrt{\epsilon}}e^{-s^2/2}\,ds
+\frac{C_A+C_B}{2}.\label{CAB}
\end{eqnarray}
The orthogonality condition (\ref{phipsi}) and the WKB
assumption (\ref{WKB}) give
\begin{eqnarray}
0&=&\int_{{\cal R}^n} \phi_0({\bf x})\psi_1({\bf x})\,d{\bf x}
=\int_{{\cal R}^n}\psi_1({\bf x})
K({\bf x},\epsilon)e^{-\Psi({\bf x})/\epsilon}\nonumber\\
&\sim&(2\pi\epsilon)^{n/2}\sum_{i=A,B}C_iK(i,\epsilon)
{\cal H}^{-1/2}(\Psi(i)) e^{-\Psi(i)/\epsilon}
\equiv aC_A+bC_B,\label{0=}
\end{eqnarray}
where ${\cal H}(\Psi(i))$ is the Hessian of $\Psi$ at $i$
\cite{book}. Assuming without loss of generality that
$\min_{\partial D_A}\Psi_A({\bf x})=
\min_{\partial D_A}\Psi_B({\bf x})$ (this is the case, e.g., if
detailed balance holds), (\ref{D<D}) implies that
\begin{eqnarray}
\frac{a}{b}=O(e^{-[\Psi(A)-\Psi(B)]/\epsilon})\ll 1.
\end{eqnarray}
Hence $C_B\ll C_A$ so that
\begin{eqnarray}
\psi_1({\bf x})\sim C_AP({\bf x}),\label{psiP}
\end{eqnarray}
(see (\ref{Pint})). In particular we have $\psi_1\sim C_A/2$ on
$\partial D_A$. We normalize $\psi_1$ below.

Next we determine $\phi_1$ in ${\cal R}^n$. We consider two
cases, $K=1$ in (\ref{WKB}) and $K\neq 1$. For $K\neq 1$
the structure of the principal eigenfunction $\phi_1({\bf y})$
is a little more complicated than that for $K=1$.
The substitution
\begin{eqnarray}
\phi_1({\bf y})=q({\bf y})e^{-\Psi({\bf y})/\epsilon}\label{WKB1}
\end{eqnarray}
transforms (\ref{phin}) into
\begin{eqnarray}
\tilde L_\epsilon \phi_1({\bf y})=O(\epsilon),\label{tildeL}
\end{eqnarray}
where $\tilde L_\epsilon$ is an operator of the form
(\ref{PDE}) with ${\bf b(x)}$ replaced by
\begin{eqnarray}
\tilde b_\epsilon^i({\bf y}) \equiv
- \left (b^i({\bf y}) + \sum_{j=1}^{n} 2a^{ij}({\bf y})
\frac{\partial \Psi({\bf y})}{\partial y^j}\right ) +
O(\epsilon).\label{btld}
\end{eqnarray}
It is easy to see from (\ref{HJE}) that in appropriate local
variables near the critical points of ${\bf b(x)}$,
the dynamics (\ref{dynamics}) and
\begin{equation}
{\bf \dot x =\tilde b(x)}\label{dynamicstilde}
\end{equation}
have the same equilibrium points with the same local
stability properties. It follows that $A$ is an attractor for both
(\ref{dynamics}) and (\ref{dynamicstilde}), and similarly
saddle points are preserved. The domains of
attraction $D_A$ and $\tilde D_A$ of (\ref{dynamics}) and
(\ref{dynamicstilde}), respectively, however may be different.
In the case $D_A=\tilde D_A$ the structure of $q({\bf y})$ is
the same as that of $\psi_1({\bf y})$. This happens, for
example, if (\ref{dynamics}) has no equilibrium points on
$\partial D_A$ (e.g., if $D_A$ is a limit cycle in two-dimensions,
see the analysis in \cite[Section 4.2]{Direct}), or if
detailed balance holds. If $D_A\neq\tilde D_A$, their
intersection still contains a neighborhood $\Omega_A$ of $A$.
Thus the outer expansion of $q({\bf y})$ is still $C_A$ in
$\Omega_A$. From (\ref{WKB1}) it follows that $\phi_1({\bf y})$
is sharply peaked at $A$, so that the averaging in
(\ref{taulambda1}) asymptotically gives
\begin{eqnarray}
\frac{1}{\lambda_1(D)}=\langle\tau(D)\rangle\sim
\langle\tau(A)\rangle.\label{tauDA}
\end{eqnarray}
Since $\langle\tau({\bf x})\rangle$ is asymptotically
independent of ${\bf x}$ in $D_A$ \cite{book}, the point $A$
in (\ref{tauDA}) can be replaced by any point ${\bf x}$ in $D_A$
outside a boundary layer near $\partial D_A$.

We assume $K=1$, then \cite{Direct}
\begin{eqnarray}
\phi_1({\bf x})=\phi_0({\bf x})\psi_1({\bf x}).\label{phi01}
\end{eqnarray}
 Using the normalization condition (\ref{phipsi})) with
$m=n=1$ we obtain from (\ref{0=}), (\ref{CAB}), and (\ref{phi01})
\begin{eqnarray}
aC_A^2+bC_B^2=1,\label{ab}
\end{eqnarray}
and from (\ref{0=}) and (\ref{ab})
\begin{eqnarray}
C_A=\sqrt{\frac{b}{a}}, \ \ \ C_B=\sqrt{\frac{a}{b}}.\label{Cab}
\end{eqnarray}

>From the structure of the eigenfunctions we can determine the
quasi stationary density in $D_A$. For times such that
\begin{eqnarray}
\frac{1}{\lambda_2}\ll t\ll\frac{1}{\lambda_1}\label{tl12}
\end{eqnarray}
the eigenfunction expansion (\ref{efn}) is given by
\begin{eqnarray}
p({\bf y},t|{\bf x})\sim\phi_0({\bf y})\psi_0({\bf
x})+e^{-\lambda_1t}\phi_1({\bf y})\psi_1({\bf x}),
\end{eqnarray}
where $\exp\{-\lambda_1t\}=O(1)$.
>From (\ref{Cab}) it follows that
$\phi_0({\bf y})\ll\phi_1({\bf y})$, hence in $D_A$
\begin{eqnarray}
p({\bf y},t|{\bf x})\sim e^{-\lambda_1t}\phi_1({\bf y})
\psi_1({\bf x}).\label{pet}
\end{eqnarray}
However for times $t\gg 1/\lambda_1$
\begin{eqnarray}
p({\bf y},t|{\bf x})\sim\phi_0({\bf y})\psi_0({\bf x}).
\end{eqnarray}
It follows that the rate of change of the population in $D_A$
satisfies
\begin{eqnarray}
-\frac{\dot N}{N}\sim\lambda_1\label{qepdf}
\end{eqnarray}
for times (\ref{tl12}), however
\begin{eqnarray}
-\frac{\dot N}{N}\sim 0\label{epdf}
\end{eqnarray}
for times $t\gg 1/\lambda_1$.

To calculate the flux $F_1$ on $\partial D_A$ we note that the
drift ${\bf b(x)}$ is tangent to $\partial D_A$, so that the
contribution of the second term on the right hand side of
(\ref{J}) to $F_1$ vanishes. It follows that
\begin{eqnarray}
\int_{\partial D_A}{\bf J}(\phi_1)\cdot\nu\,ds&=&\int_{\partial
D_A}\psi_1({\bf x}){\bf J}(\phi_0)\cdot\nu\,ds+\int_{\partial
D_A}\phi_0({\bf x}){\bf J}(\psi_1)\cdot\nu\,ds\nonumber\\
&=&\frac{C_A}{2}
\int_{\partial D_A}{\bf J}(\phi_0)\cdot\nu\,ds+C_A\int_{\partial
D_A}\phi_0({\bf x})\epsilon\sum_{i,j}a^{ij}\psi_{1,j}({\bf x})
\nu^i\,ds\nonumber\\
&\sim&C_A\sqrt{\frac{\epsilon}{2\pi}} \int_{\partial D_A}
\phi_0 \sum_{i,j}a^{ij}\chi_j({\bf x})\nu^i\,ds.
\end{eqnarray}
Hence, by (\ref{ln}),
\begin{eqnarray}
\lambda_1=\frac{F_1}{N_1}\sim\sqrt{\frac{\epsilon}{2\pi}}
\int_{\partial D_A}\phi_0 \sum_{i,j}a^{ij}\chi_j\nu^i\,ds{\bigg /}
\int_{D_A}\phi_0\,d{\bf x}\equiv\frac{\tilde F_1}{N_0}.\label{ll1}
\end{eqnarray}
Thus $\lambda_1$ is the total flux $\tilde F_1$ on $\partial D_A$,
normalized by the equilibrium reactant population.
Equations (\ref{ll1}) and (\ref{epdf}) indicate that the
(G)TST assumption that the equilibrium flux is a good
approximation to the quasi equilibrium flux is incorrect.

Next we consider the principal eigenvalue problem (\ref{phin}),
(\ref{phinbc}) in domains of type II, that is, we choose
$D=D_A$. If absorbing conditions are imposed on $\partial D_A$, then
$\lambda_0=0$ is no longer an eigenvalue, since no equilibrium
density exists. The principal eigenfunction $\psi_{1}^{D_A}$,
which satisfies the boundary condition $\psi_{1}^{D_A}=0$ on
$\partial D_A$, is given by
\begin{eqnarray}
\psi_{1}^{D_A}=\psi_1-\frac{C_A}{2},
\end{eqnarray}
because for ${\bf x}\in \partial D_A$ (\ref{Pint}) and (\ref{psiP})
imply that $\psi_1\sim 1/2$. Hence
\begin{eqnarray}
\phi_{1}^{D_A}\sim\phi_0(\psi_1-\frac{C_A}{2}),
\end{eqnarray}
and consequently
\begin{eqnarray}
\lambda_1(D_A)\sim\frac{F_1(D_A)}{N_1(D_A)}=\frac{\tilde F_1}
{\frac{1}{2}N_0}=2\lambda_1.\label{l1}
\end{eqnarray}
It follows that the absorption rate in $\partial D_A$ is twice
the rate, that is, (\ref{lim}).

Finally, if $D$ is a domain of type
III, then asymptotically $\psi_1^D\sim\psi_1$, and (\ref{l1})
holds with $\lambda_1$ replaced by $\lambda_1(D)$.
The result (\ref{l1}) can be stated as\\

\noindent
{\bf Theorem 1.} {\it If $D$ is a domain of type {\rm III}
relative to {\rm (\ref{dynamics})}, then}
\begin{equation}
\lim_{\epsilon\rightarrow
0}\frac{\lambda_1(D)}{\lambda_1(D_A)}=\frac{1}{2}.\label{12D}
\end{equation}
Thus to leading order in $\epsilon$,
$\lambda_1(D)$ is independent of $D$, as long as $D$ is of
type III. That is, changes in the shape of $D$ do
not affect $\lambda_1(D)$ to leading order.
In contrast, as $D$ shrinks to $D_A$, $\lambda_1(D)$ abruptly
doubles as $D$ nears $D_A$. As it shrinks
yet further inside $D_A$, the quotient $\lambda_1(D)/
\lambda_1(D_A)$ increases exponentially fast in $1/\epsilon$.

A generalization of Theorem 1 to the eigenvalue problem in
${\cal R}^n$ can be stated as\\

\noindent
{\bf Theorem 2.} {\it If {\rm (\ref{dynamics})} is a bistable
system and $D={\cal R}^n$, then {\rm (\ref{12D})} holds.}\\

Theorems 1 and 2 show that in the limit of high barrier the
definitions of rate as an absorption rate of trajectories
outside $D_A$ or as the rate of change of the reactant population,
or as twice the absorption rate on $S$, or as the stationary
flux on an absorbing boundary $\partial D$ with a source in
$D_A$, all lead the same result.\\

We illustrate the difference between our non equilibrium
considerations and the equilibrium considerations of (G)TST
with an example. In the one dimensional Kramers problem
\cite{Kramers} (\ref{SDE}) is given in dimensionless phase
space by
\begin{eqnarray}
\dot x&=&y\nonumber\\
\dot y&=&-\gamma y-V'(x)+\sqrt{2\gamma\epsilon}\,\dot w,
\end{eqnarray}
where $V(x)$ is a bistable potential, $\gamma$ is a
dimensionless friction coefficient, and $\epsilon$ is
dimensionless temperature (normalized by the barrier height).
We denote the minima of $V(x)$ by $x_A$ and $x_B$, its
local maximum by $x_C$, and set $\omega_{A(B)}^2\equiv
V''(x_{A(B)})$, and $\omega_C^2\equiv -V''(x_C)$. We assume
$V(x_A)<V(x_B)<V(x_C)$. The domain of attraction $D_A$ of the
attractor $A\equiv (x_A,0)$ in phase space is bounded by a separatrix
$\partial D_A$, which passes through the saddle point
$C\equiv (x_C,0)$. The direction of $\partial D_A$ at $C$ is
that of the line $y=-\lambda (x-x_C)$, where
$\lambda=[\gamma+\sqrt{\gamma^2+4\omega_C^2}]/2$. Obviously,
$\phi_0=\exp(-E/\epsilon)$, where $E=\frac{1}{2}y^2+V(x)$. We
have locally near $C$
\begin{eqnarray}
\chi(x,y)=\frac{\omega_C^2}{\sqrt{\gamma(\lambda-\gamma)}}x+
\sqrt{\frac{\lambda-\gamma}{\gamma}}y,
\end{eqnarray}
hence, according to (\ref{ln}),
\begin{eqnarray}
\lambda_1\sim\frac{\omega_A}{4\pi\omega_C}(\sqrt{\gamma^2+
4\omega_C^2}-\gamma)\,e^{-\Delta V/\epsilon},
\end{eqnarray}
where $\Delta V\equiv V(x_C)-V(x_A)$, which is Kramers' result
(see detailed calculations in \cite{Direct}, \cite{MST},
and \cite{MSB}).

In contrast, according to (G)TST, the rate is the normalized
flux on the line $x=x_C,\ y>0$,
\begin{eqnarray}
\kappa=\int_{0}^{\infty}y\phi_0\,dy{\bigg
/}\int_{-\infty}^{x_C}\!\int_{-\infty}^{\infty}\phi_0\,dx\,dy
\sim\frac{\omega_A}{2\pi}\,e^{-\Delta V/\epsilon},\label{TST}
\end{eqnarray}
which is the TST result \cite{Glasstone}. We note that if the
outgoing flux  on $C$ is replaced by half the flux,
calculated on any curve emanating from $C$ to
$\infty$, the result (\ref{TST}) is unchanged. Indeed, we
have
\begin{eqnarray}
{\bf J}(\phi_0)=(y\phi_0,-\gamma\epsilon\phi_{0,y}-[\gamma
y+V'(x)]\phi_0)=(y\phi_0,-V'(x)\phi_0),
\end{eqnarray}
so that
\begin{eqnarray}
F_0=\int_{C}^{\infty}{\bf J}(\phi_0)\cdot\nu\,ds=\int_{C}^{\infty}
\phi_0(y\,dy+V'(x)\,ds)=\int_{E_C}^{\infty}e^{-E/\epsilon}\,dE=
\epsilon\,e^{-\Delta V/\epsilon},
\end{eqnarray}
hence (\ref{TST}). We conclude that $\phi_1$ cannot be
replaced by $\phi_0$ in the calculation of $\kappa$,
regardless of the choice of the (G)TS.\\

\noindent
{\bf 5. Accounting for recrossings and the MFPT}\\
\renewcommand{\theequation}{5.\arabic{equation}}
\setcounter{equation}{0}
\setcounter{section}{5}

A trajectory which starts in $\Omega_A$ and having crossed
$\partial D_A$ returns to $\Omega_A$ prior to reaching
$\partial D$ is said to recross $\partial D_A$. In the process
of escaping from $\Omega_A$ to $\partial D$ a trajectory may
recross $\partial D_A$ a random number of times. In this
section we describe the escape process, calculate the mean
number of times $\langle n\rangle$ that a trajectory crosses
$\partial D_A$ before it reaches $\partial D$, and show that for
domains of type III the mean time $\langle\tau(D)\rangle$
to reach $\partial D$ (i.e., to escape) is asymptotically twice
the MFPT $\langle\tau(D_A)\rangle$. We begin with  a\\

\noindent
{\bf Lemma.} {\it Let $D$ be a domain of type {\rm III} relative
to {\rm (\ref{dynamics})}. Then the {\rm MFPT} from $\partial
D_A$ to $\partial D$,
conditioned on reaching $\partial D$ before $\partial
\Omega_A$, and the {\rm MFPT} from $\partial D_A$ to
$\partial\Omega_A$, conditioned on reaching $\partial
\Omega_A$ before $\partial D$, are at most $O(1/\sqrt{\epsilon})$ as
$\epsilon\rightarrow 0$}.\\

\noindent
{\bf Proof.} We prove only the second part of the lemma, since the first
part follows from similar arguments. We denote by ${\bf x}^*(t)$
the process ${\bf x}(t)$ in $D-\bar\Omega_A$,
conditioned on the event $\{\tau(\partial\Omega_A)<\tau(D)\}$,
that is, the trajectories of ${\bf x}^*(t)$ consist of those
trajectories of (\ref{SDE}) which start in $D-\bar\Omega_A$ and
reach $\partial\Omega_A$ before $\partial D_A$.

The process ${\bf x}^*(t)$ is a diffusion process
with noise matrix $\sigma({\bf x})$ and drift
\begin{eqnarray}
{b^*}^i{\bf (x)}\equiv b^i{\bf (x)}+2\epsilon\sum_{j=1}^{n}
a^{ij}({\bf x}) \frac{\partial \log P({\bf x})}{\partial x^j},
\ \ (i=1,\dots,n)\label{bi*}
\end{eqnarray}
(see \cite{Karlin}), where $P({\bf x})$ is defined in
(\ref{Px}) and is given by (\ref{Pint}).
We will show that the drift ${\bf b^*(x)}$ is attracted to
$\Omega_A$ everywhere in $D-\bar\Omega_A$ and will estimate
its component in the direction "toward" $\Omega_A$. To this end we
change variables in the operator $L_\epsilon^*$ in (\ref{PDE}).
We introduce the stretched variable
\begin{eqnarray}
\xi\equiv\frac{\chi({\bf x})}{\sqrt{\epsilon}}\label{xi}
\end{eqnarray}
to measure distance from $\partial D_A$. We assume that there
exists a global set of variables ${\bf s}\equiv (s^2,\dots,s^n)$
on the hypersurfaces $\xi=const.$ (see \cite{Day} for a more
detailed discussion of this change of variables).  In the variables
$(\xi,{\bf s})$ the operator $L_\epsilon^*$ takes the form
\begin{eqnarray}
L_\epsilon^* p=A({\bf x}) \frac{\partial^2 p}
{\partial\xi^2}+B({\bf x}) \frac{\partial p}{\partial\xi}+
\sum_{k=2}^{n}C^k({\bf x})\frac{\partial p}{\partial s^k}+
O(\sqrt{\epsilon}),\label{Lxi}
\end{eqnarray}
where
\begin{eqnarray}
A({\bf x})\equiv \sum_{i,j=1 }^{n}a^{ij}({\bf
x})\frac{\partial\chi({\bf x})}{\partial x^i}
\frac{\partial\chi({\bf x})}{\partial x^j}\geq 0,\label{A}
\end{eqnarray}
\begin{eqnarray}
\sqrt{\epsilon}\,B({\bf x})\equiv\epsilon \sum_{i,j=1}^{n}a^{ij}({\bf
x})\frac{\partial^2\chi({\bf x})}{\partial x^i
\partial x^j}+\sum_{i=1}^{n} b^i({\bf
x})\frac{\partial\chi({\bf x})}{\partial x^i},\label{B}
\end{eqnarray}
and
\begin{eqnarray}
C^k({\bf x})\equiv\sum_{i=1}^{n}b^i({\bf x})\frac{\partial
s^k}{\partial x^i}.\label{Ck}
\end{eqnarray}
The $O(\sqrt{\epsilon})$ term in (\ref{Lxi}) contains mixed
derivatives with respect to $\zeta$ and $s^k$ and derivatives
with respect to $s^k$. In view of (\ref{chi}) and (\ref{xi}),
\begin{eqnarray}
B({\bf x})=A({\bf x})\xi.
\end{eqnarray}
The function $P({\bf x})$ in (\ref{bi*}), given by
(\ref{Pint}), can be written in terms of the variables
$(\xi,{\bf s})$  as
\begin{eqnarray}
P({\xi,{\bf s}})=\sqrt{\frac{1}{2\pi}}\int_{-\infty}^{\xi}
e^{-z^2/2}\,dz.\label{Pxi}
\end{eqnarray}
The components of ${\bf b(x)}$  in the $\xi$ and $s^k$
directions are denoted by $(b_\xi,b_{s^2},\dots,b_{s^n})$,
with
\begin{eqnarray}
b_\xi(\xi,{\bf s})=A({\bf x})\xi.\label{bxi}
\end{eqnarray}
We define the domains $D_\xi\equiv\{\xi({\bf x)}>\xi\}$ and
note that if $\xi_1<\xi_2$, then $D_{\xi_1} \supset D_{\xi_2}$.
In particular $D_0=D_A$. In view of (\ref{bxi}), the drift
${\bf b(x)}$ points into $D_\xi$ for $\xi>0$, outside $D_\xi$
for $\xi<0$, and is tangent to $\partial D_0$.
 Equation
(\ref{bi*}) gives the $\xi$ component of ${\bf b^*(x)}$ as
\begin{eqnarray}
b_\xi^*=b_\xi+2 A({\bf x})
\frac{\partial P(\xi,{\bf s})}{\partial\xi}+
2\sum_{k=2}^{n}A^{\xi,k}({\bf x})
\frac{\partial P(\xi,{\bf s})}{\partial s^k},\label{bxi*}
\end{eqnarray}
where $A^{\xi,k}({\bf x})$ is the coefficient of $\partial^2
P(\xi,{\bf s})/\partial\xi\partial s^k$ in (\ref{Lxi}).
In view of (\ref{Pxi}) we have $\partial P(\xi,{\bf
s})/\partial s^k=0$, so that
\begin{eqnarray}
b_\xi^*(\xi,{\bf s})= A({\bf x})\left(\xi+2\frac{\partial\log
P(\xi,{\bf s})}{\partial \xi}\right)
= A({\bf x})\left( \xi+ \frac{2
e^{-\xi^2/2}}{\int_{-\infty}^{\xi}e^{-z^2/2}\,dz}
\right).\label{b*xi}
\end{eqnarray}
We denote by ${\bf n(x)}$ the inner unit normal to the surface
$\xi=const.$ Next we show that the drift ${\bf b^*(x)}$ in
$D-\bar\Omega_A$ is attracted to $\partial\Omega_A$.
 On $\partial D_A$ we have $\xi({\bf x})=0$ and $P(0,{\bf s})=
1/2$. It follows from (\ref{b*xi}) that
$b_\xi^*(0,{\bf s})=2\sqrt{2/\pi}\,A({\bf x})$,
or in the original variables
\begin{eqnarray}
{\bf b^*(x)\cdot n(x)}=2\sqrt{\frac{2\epsilon}{\pi}}
A({\bf x})>0.\label{sqrte}
\end{eqnarray}
We note that (\ref{sqrte}) holds in a boundary layer near
$\partial D_A$.
 Thus the drift ${\bf b^*(x)}$ on $\partial D_A$ points into $D_A$.
As $\xi\rightarrow\infty$, that is,
for ${\bf x}$ outside a small neighborhood of
$\partial D_A$ in $D_A$, the numerator in (\ref{b*xi})
decays exponentially, whereas the denominator converges to
$\sqrt{2\pi}$. It follows that the second term in (\ref{bi*})
decays exponentially so that the directions of ${\bf b^*(x)}$
and ${\bf b(x)}$ coincide. Thus, by assumption, the drift
${\bf b^*(x)}$ is attracted by $\partial\Omega_A$ in
$D_A-\bar\Omega_A$. Next we consider ${\bf b^*(x)}$ in $D-\bar
D_A$, outside a boundary layer, that is, we consider
$\xi\rightarrow-\infty$.  First we note that  in this limit
\cite{Feller}
\begin{eqnarray}
P(\xi,{\bf s})\sim-\sqrt{\frac{1}{2\pi}}\,\frac{
e^{-\xi^2/2}}{\xi}.\label{Feller}
\end{eqnarray}
>From (\ref{b*xi}) and (\ref{Feller}) we obtain
\begin{eqnarray}
b_\xi^*(\xi,{\bf s})\sim-A({\bf x})\xi=-b_\xi(\xi,{\bf
s}),\label{b*-b}
\end{eqnarray}
where the equality is a restatement of (\ref{xi}).
It follows that ${\bf b^*(x)}$ points into $D_\xi$ for all
$\xi<0$ and is therefore attracted to $\partial D_A$ in
$D-\bar D_A$. Thus ${\bf b^*(x)}$ is attracted to $\Omega_A$
everywhere in $D-\bar\Omega_A$. We see from (\ref{sqrte})
and (\ref{b*-b}) that the component $b_\xi^*$ is asymptotically
identical to the component $b_\xi$ inside $D_A$ and to $-b_\xi$
outside $D_A$, except for a boundary layer, where it is
positive and $O(\sqrt{\epsilon})$. That is, the conditional
process ${\bf x}^*(t)$ drifts toward $\Omega_A$ across the
surfaces $\partial D_\xi$ with normal speed which is
either $O(1)$ or $O({\sqrt\epsilon})$.
Therefore the MFPT from any point in $D-\bar \Omega_A$ to
$\partial \Omega_A$, conditioned on $\{\tau(\partial\Omega_A)
<\tau(D)\}$, is at most $O(1/\sqrt{\epsilon})$. This proves
the lemma.\\

\noindent
{\bf Theorem 3.} {\it If $D$ is a domain of type {\rm III}
relative to {\rm (\ref{dynamics})}, then}
\begin{eqnarray}
\lim_{\epsilon\rightarrow
0}\frac{\langle\tau(D)\rangle}{\langle\tau(D_A)\rangle}=2.\label{lim1}
\end{eqnarray}
{\bf Proof.} We partition the trajectories of (\ref{SDE}), all of which
eventually reach $\partial D_A$, into two sets,
\begin{eqnarray}
S_{1,1}\equiv&&\{\mbox{trajectories which having reached
$\partial D_A$ for the first time,}\nonumber\\
&&\mbox{reach $\partial
\Omega_A$ before $\partial D$}\}\label{S11}
\end{eqnarray}
and
\begin{eqnarray}
S_{1,2}\equiv&&\{\mbox{trajectories which having reached
$\partial D_A$ for the first time,}\nonumber\\
&&\mbox{reach $\partial D$ before
$\partial \Omega_A$}\}.\label{S12}
\end{eqnarray}
We denote by $\tau_1$ the first passage time to $\partial
D_A$, that is $\langle\tau_1\rangle=\langle\tau(D_A)\rangle$.
It follows from the lemma that
\begin{eqnarray}
E[\tau(D)|S_{1,2}]=\langle\tau(D_A)\rangle+
O(1/\sqrt{\epsilon})\ \ {\rm as}\ \epsilon\rightarrow 0.
\end{eqnarray}
Also, as  shown in Section 4,
\begin{eqnarray}
P(S_{1,1})\sim P(S_{1,2})\sim \frac{1}{2}.
\end{eqnarray}
Next we partition $S_{1,1}$ into two sets,
\begin{eqnarray}
S_{2,1}\equiv&&\{\mbox{trajectories in $S_{1,1}$ which having reached
$\partial\Omega_A$ after the time $\tau_1$, reach}\nonumber\\
&&\mbox{$\partial D_A$ for the first time after $\tau_1$, and then
reach $\partial\Omega_A$ before $\partial D$}\}\label{S21}
\end{eqnarray}
and
\begin{eqnarray}
S_{2,2}\equiv&&\{\mbox{trajectories in $S_{1,1}$ which having reached
$\partial\Omega_A$ after the time $\tau_1$, reach}\nonumber\\
&&\mbox{$\partial D_A$ for the first time after $\tau_1$, and then
reach $\partial D$ before $\partial\Omega_A$}\}.\label{S22}
\end{eqnarray}
Then, as above,
\begin{eqnarray}
P(S_{2,1})\sim P(S_{2,2})\sim \frac{1}{2}P(S_{1,1})\sim \frac{1}{4}.
\end{eqnarray}
We denote by $\tau_2$ the first time after $\tau_1$, that a
trajectory in $S_{1,1}$ returns to $\partial D_A$ after
reaching $\partial\Omega_A$. By the lemma, the MFPT
from $\partial D_A$ to $\partial\Omega_A$ for trajectories
in $S_{2,1}$, is at most $O(1/\sqrt{\epsilon})$. The MFPT from
$\partial\Omega_A$ to $\partial D_A$ is
$\langle\tau(D_A)\rangle$, since it is independent of the initial
point of the trajectories in $D_A$, outside an
$\epsilon$-neighborhood of $\partial D_A$ \cite{book}. Thus
by the lemma,
\begin{eqnarray}
E[\tau_2-\tau_1|S_{2,1}]=\langle\tau(D_A)\rangle+O(1/\sqrt{\epsilon}).
\end{eqnarray}
Again, from the lemma
\begin{eqnarray}
E[\tau(D)|S_{2,2}]=2[\langle\tau(D_A)\rangle+O(1/\sqrt{\epsilon})]
\ \ {\rm as}\ \epsilon\rightarrow 0.
\end{eqnarray}

We proceed in a similar manner and obtain sets $S_{n,1},\
S_{n,2}$, and times $\tau_n$, such that
\begin{eqnarray}
S_{n,1}\equiv&&\{\mbox{trajectories in $S_{n-1,1}$ which having reached
$\partial\Omega_A$ after the time $\tau_{n-1}$, reach}\nonumber\\
&&\mbox{$\partial D_A$ for the first time after $\tau_{n-1}$, and then
reach $\partial\Omega_A$ before $\partial D$}\}\label{Sn1}
\end{eqnarray}
and
\begin{eqnarray}
S_{n,2}\equiv&&\{\mbox{trajectories in $S_{n-1,1}$ which having reached
$\partial\Omega_A$ after the time $\tau_{n-1}$, reach}\nonumber\\
&&\mbox{$\partial D_A$ for the first time after $\tau_{n-1}$, and then
reach $\partial D$ before $\partial\Omega_A$}\}.\label{Sn2}
\end{eqnarray}
Then, as above,
\begin{eqnarray}
P(S_{n,1})\sim P(S_{n,2})\sim\frac{1}{2}P(S_{n-1,1})\sim\frac{1}{2^{n}}.
\end{eqnarray}
We denote by $\tau_n$ the first time after $\tau_{n-1}$ that a
trajectory in $S_{n-1,1}$ returns to $\partial D_A$ after
reaching $\partial\Omega_A$. Again, by the lemma,
\begin{eqnarray}
E[\tau_n-\tau_{n-1}|S_{n,1}]=\langle\tau(D_A)\rangle+
O(1/\sqrt{\epsilon}),
\end{eqnarray}
and
\begin{eqnarray}
E[\tau(D)|S_{n,2}]=n[\langle\tau(D_A)\rangle+
O(1/\sqrt{\epsilon})],\ \ {\rm as}\ \epsilon\rightarrow 0.
\end{eqnarray}
Since every trajectory reaches $\partial D$ in finite time,
having gone from $\partial D_A$ to $\partial\Omega_A$ and
back a finite number of times, the union of the sets $S_{n,2},\
\ (n=1,2,\dots)$
contains all trajectories. In addition, the sets $S_{n,2}$ are
mutually disjoint. It follows that
\begin{eqnarray}
\langle\tau(D)\rangle&=&\sum_{n=1}^{\infty}E[\tau_{n+1}-\tau_n|
S_{n,1}]P(S_{n,1})=\sum_{n=1}^{\infty}\frac{n}{2^{n}}
[\langle\tau(D_A)\rangle+O(1/\sqrt{\epsilon})]\nonumber\\
&=& 2[\langle\tau(D_A)\rangle+O(1/\sqrt{\epsilon})].\label{t2t}
\end{eqnarray}
Since \cite{book}
\begin{eqnarray}
\langle\tau(D_A)\rangle=O(e^{\hat\Psi(D_A)/\epsilon}),
\end{eqnarray}
the $O(1/\sqrt{\epsilon})$ term in (\ref{t2t}) is negligible relative to
$\langle\tau(D_A)\rangle$, so that
\begin{eqnarray}
\langle\tau(D)\rangle\sim 2\langle\tau(D_A)\rangle,
\end{eqnarray}
hence (\ref{lim}).

Next we discuss the notion of recrossings and of the
transmission coefficient $k$. If the transition
state region (TSR) is chosen as a neighborhood of $S$, whose width
is independent of $\epsilon$, then, as is evident from the
proof of Theorem 1, the probability that a trajectory leaves
the TSR in the direction of $A$ or $B$ is independent of
the choice of the TSR. We therefore choose the TSR to be the domain
$D-\Omega_A$.\\

\noindent
{\bf Definition.} {\it A trajectory ${\bf x}(t)$
of {\rm (\ref{SDE})} which starts in $\Omega_A$
is said to cross the {\rm TSR} $n$ times, if
${\bf x}(t)\in S_{n,2}$. If a trajectory crosses the
{\rm TSR} $n$ times, it is said to recross it $n-1$ times}.\\

Thus the number $n$ of crossings of the TSR is a random
variable. This definition is asymptotically independent of
$\Omega_A$ and $D$ (or $\Omega_B$).\\

\noindent
{\bf Corollary 1}. {\it If $D$ is a domain of type {\rm III}
relative to {\rm (\ref{dynamics})}, then the mean number of
times a trajectory recrosses the {\rm TSR} is }1.\\

\noindent
{\bf Proof.} From the definition and from the proof of Theorem
1 it is obvious that
\begin{eqnarray}
\langle n\rangle=\sum_{n=1}^{\infty}\frac{n}{2^n}=2,
\end{eqnarray}
so that the mean number of recrossings is 1.\\

If the definition of the stochastic separatrix is modified to
replace the probability 1/2 in (\ref{P12}) with a probability $p$,
independent of $\epsilon$, then according to (\ref{Pint}),
the resulting surface $S_p$ will be in an $\epsilon$-neighborhood
of $S$. Then (\ref{lim}) is replaced by
\begin{eqnarray}
\tau_{AB}=\frac{1}{p}\tau_{AS_p}.\label{TSp}
\end{eqnarray}
It can be easily shown that $\tau_{AS_p}/p$ is asymptotically
independent of $p$. Thus the convenient choice $S=S_{1/2}$ does
not limit the generality of the definition. As in
the proof of Corollary 1, it can be shown that
the mean number of crossings is $1/p$.

Next we comment on the definition of the transmission
coefficient $k$ and its use. As mentioned in the Introduction,
$k$ was introduced to account for the recrossing trajectories
which are neglected in (G)TST. If in the bottleneck case
\cite{Truhlar} the (G)TS$\neq S$, but passes through the saddle
point, the probability of a return from the (G)TS strongly depends
on the point where a trajectory hits the (G)TS. This is the
source of the difficulty in calculating $\langle n\rangle$ in
(G)TST, and of the overestimates of $\kappa$ in the (G)TST
results \cite{Fleming}, \cite{Frauenfelder}. If however the
(G)TS is chosen as $S$, this difficulty is averted, as shown above.\\

\noindent
{\bf 6. Conclusions and discussion}\\
\renewcommand{\theequation}{6.\arabic{equation}}
\setcounter{equation}{0}
\setcounter{section}{6}

Noise induced escapes from an attractor of a dynamical system
account for a variety of physical phenomena, such as chemical reactions
\cite{Kramers}, \cite{Glasstone}, \cite{Benson}, nucleation
\cite{Langer}, \cite{Haken} the behavior of Josephson
junction devices \cite{green}, \cite{SQUID}, loss of lock in code
tracking loops \cite{PLL}, \cite{Viterbi}, to name but a few.
The noise in such systems describes internal or
external fluctuations, brought about, e.g., by thermal vibrations,
molecular collisions, noisy resistors, and so on.
In many cases the strength of the fluctuations relative to
the non fluctuating forces in such systems is small so that
an escape is a rare event on the time scale of the dynamics.
The escape rate determines physically measurable quantities,
such as rates of molecular dissociation, isomerization, chemical
reactions, the resistance of Josephson junction devices, the
reliability of logic elements, the performance of tracking
circuits, the stability of elastic structures, and so on.

In this paper we consider activation as a non equilibrium process.
We define reactant and product as neighborhoods
$\Omega_A$ and $\Omega_B$ of the stable states of the
noiseless dynamics and show that in the limit of small
noise this definition is independent of the choice of these
neighborhoods. We show that the rate at which trajectories
arrive from $\Omega_A$ to $\Omega_B$
for the first time is half the rate at which trajectories
arrive from $\Omega_A$ to the stochastic separatrix $S$, and
that this rate is also independent of these neighborhoods.
We conclude that in order to obtain the correct rate in
(G)TST (i) the qepdf has to be used rather than the epdf, (ii)
the qepdf differs from the epdf in a boundary layer near $S$
by a boundary layer factor, and (iii) all recrossings of the
TSR are accounted for if (G)TS=$S$, but not otherwise.
Thus no transmission coefficient is needed if (G)TS=$S$.

The need for (i)-(iii) in (G)TST is based on the assumption
that the qepdf in multidimensional phase space is well
approximated by the qepdf of the diffusion process in the
sense of (\ref{PP}). We conclude that the modified (G)TST
employed in \cite{PHG} can be correct only if the chosen (G)TS
is shown to be $S$, and if the modification of the epdf
defined by the integral equation \cite[(3.10]{PHG} has the
appropriate boundary layer structure.

The identification of $S\sim\partial D_A$ was based on the
premise that $\epsilon$ was the only small parameter in the
problem. This assumption is satisfied for example in Kramers'
model when the dissipation coefficient is not too small. In
this case the above results and \cite{MST}, \cite{MSB}
confirm Kramers' result to be exact, not merely an upper
bound.  However in problems which involve more than one
small parameter, then $S\neq \partial D_A$ in general. Indeed, $S$ may
differ considerably from $\partial D_A$. This is the case,
e.g., in the extremely anisotropic overdamped motion of a two
dimensional Brownian particle in a double well potential
$V(x,y)$.  Here $x$ and $y$ are the coordinates along which the
motion is fast and slow, respectively. If
$\partial^2V/\partial x^2<0$ at the saddle point, then
$S$ may be completely unrelated to $\partial D_A$. This
leads to many interesting phenomena such as the appearance of
long non exponential transients, decay rates not determined by
the height of the potential barrier, etc. \cite{anis},
\cite{BZ}-\cite{JCP}. Also in the limit of small damping
$S\neq\partial D_A$ \cite{Tutzing}. In such cases the the factor 1/2
in the relation between the normalized flux on the GTS
and $\kappa$ should be used only if GTS=$S$.

In \cite{RyterPhysica} an attempt was made to prove that
$\lim_{\epsilon\rightarrow 0}S=\partial D_A$, for
continuous drifts ${\bf b(x)}$, by employing
the following argument. Using (\ref{Pint})-(\ref{chi0}),
in the limit $\epsilon\rightarrow 0$, $\chi^0({\bf x})$ satisfies
\begin{eqnarray}
\sum_{i=1}^{n}b^i({\bf x})\frac{\partial\chi^0({\bf x})}{\partial
x^i}=0 \ \ \mbox{on $S$ (on $\chi^0({\bf x})=0$)}.\label{calS}
\end{eqnarray}
Hence $\lim_{\epsilon\rightarrow 0}S$ is a characteristic
surface, tangent to the vector field ${\bf b(x)}$.
Since $\partial D_A$ is a characteristic surface, it is concluded
in \cite{RyterPhysica}
that $\lim_{\epsilon\rightarrow 0}S=\partial D_A$. However
$\partial D_A$ is not necessarily the only closed
characteristic surface in $D$, even if there is only one attractor
in $D$, and the above argument fails to choose $S$
among the various closed characteristic surfaces. Therefore
this argument is not complete.
For example, the two-dimensional dynamics (in
polar coordinates)
\begin{eqnarray}
\dot r&=&-r(1-r)^2(2-r)\nonumber\\
\dot \theta&=&f(r,\theta),\label{r}
\end{eqnarray}
where $f(r,\theta)$ is a positive function, has a single
attractor in the disk $r<2$, centered at $r=0$, and an
unstable limit cycle at $r=1$ (attracting from the outside
and repelling from the inside). The second attractor is at infinity.
Both circles $r=1$ and $r=2$ are closed
characteristic surfaces, and the argument (\ref{calS}) does
not identify which circle is $S$ in the limit
$\epsilon\rightarrow 0$. In fact, our approach shows that
in the limit $\epsilon\rightarrow 0$, $S$ is the
circle $r=2$, as follows from the argument of Section 4.
Indeed, the outer solution to (\ref{LP}) is constant inside the
disk $r<1$. It is also
constant in the annulus $1<r<2$, since all trajectories
in the annulus come arbitrarily close together near the attractor
$r=1$. It is easy to see that
the solution of the boundary layer equation near $r=1$ cannot
match to two different constants on the two sides of $r=1$. It
follows that the outer solution is a single constant
throughout the disk $r<2$, so that
$\lim_{\epsilon\rightarrow 0}S=\{r=2\}$.\\

\noindent
{\bf Acknowledgment.} The authors thank E. Pollak for
useful discussions of TST. 
\newpage
\centerline{\bf FIGURE CAPTION}

\vspace{0.5cm}
\noindent
{\bf Figure 1.} The dynamics (\ref{dynamics}) is attracted to
$A$ and to $\partial D$ and is repelled from $\partial D_A$.
The domains $\Omega_A,\ D_A,$ and $D$ are of types I, II, and
III, respectively.

%
\newpage
\begin{thebibliography}{99}
\bibitem{Pechukas} P. Pechukas, in {\em Dynamics of
Molecular Collisions B}, (W.H. Miller, Ed.), Plenum, N.Y. 1976.
\bibitem{Kramers} H.A. Kramers, {\em Physica} (Utrecht) {\bf 7},
284 (1940).
\bibitem{Fleming} G.R. Fleming and P.G. Wolynes, {\em Phys.
Today} {\bf 43}, 36 (1990).
\bibitem{Wigner} E. Wigner, {\em Trans. Faraday Soc.} {\bf
34}, 29 (1938).
\bibitem{BHL} M. B\"uttiker, Harris, and R. Landauer
\bibitem{MelMesh} Melnikov and Meshkov
\bibitem{PHG} E. Pollak, H. Grabert, and P. H\"anngi, {\em J.
Chem. Phys.} {\bf 91}, 4073 (1989).
\bibitem{Eyring35} H. Eyring, {\em J. Chem. Phys.} {\bf 3},
107, 1935.
\bibitem{Miller} W.H. Miller, {\em J. Chem. Phys.} {\bf 61},
1283 (1974).
\bibitem{Frauenfelder} H. Frauenfelder and P.G. Wolynes, {\em
Science} {\bf 229}, 337 (1985).
\bibitem{Evans} M.G. Evans, {\em Trans. Faraday Soc.} {\bf
34}, 49 (1938).
\bibitem{Keck} J.C. Keck, {\em Adv. Chem. Phys.} {\bf 13}, 85
(1967).
\bibitem{Pollak86} E. Pollak, {\em J. Chem. Phys.} {\bf 85},
865 (1986).
\bibitem{Pollak1} E. Pollak, {\em J. Chem. Phys.} {\bf 93},
1116 (1990).
\bibitem{Christiansen} J.A. Christiansen, {\em Z. Phys. Chem.}
B{\bf 33}, 145 (1936).
\bibitem{Kubo} R. Kubo, {\em J. Phys. Soc. Japan} {\bf 12},
570 (1957), {\em Rep. Prog. Phys.} {\bf 29}, 255 (1966).
\bibitem{Kac} G.W. Ford, M. Kac, and P. Mazur, {\em J. Math.
Phys.} {\bf 6}, 504 (1965).
\bibitem{Zwanzig} R. Zwanzig, {\em J. Stat. Phys.} {\bf 9}, 215
(1973).
\bibitem{Chandrasekhar}  S. Chandrasekhar, {\em Rev. Mod.
Phys.} {\bf 15}, 1 (1943).
\bibitem{PollakPRL} E. Pollak, S.C. Tucker, and B.J. Berne,
{\em Phys. Rev. Lett.} {\bf 65}, 1399 (1990).
\bibitem{DMS} M.M. Dygas, B.J. Matkowsky, and Z. Schuss, {\em
SIAM J. Appl. Math.} {\bf 46} (2), 265 (1986).
\bibitem{Tutzing} M.M. K\l osek, B.J. Matkowsky and Z. Schuss,
{\em Berichte Der Bunsen-Gesellschaft F\"ur Physikalische
Chemie} (1991).
\bibitem{anis} M.M. K\l osek, B.J. Matkowsky, A. Nitzan, M.
Ratner, Z. Schuss, (in preparation).
\bibitem{FV} M.I. Freidlin and A.D. Wentzell, {\em Random
Perturbations of Dynamical Systems}, Springer-Verlag, N.Y.
1984.
\bibitem{Ludwig} D. Ludwig, {\em SIAM Rev.} {\bf 17} (4), 605,
(1975).
\bibitem{79} Z. Schuss and B.J. Matkowsky, {\em SIAM J. Appl. Math.}
{\bf 36}, 604 (1979).
\bibitem{MangelLudwig} M. Mangel, D. Ludwig, {\em SIAM J. Appl.
Math.} {\bf 33} (2), 256 (1977).
\bibitem{Mangel} M. Mangel, {\em SIAM J. Appl. Math.} {\bf 36}
(3), 544 (1979).
\bibitem{Direct} T. Naeh, M.M. K\l osek, B. J. Matkowsky, and
Z. Schuss, {\em SIAM J. Appl. Math.} {\bf 50}, 595 (1990).
\bibitem{Risken} H. Risken, {\em The Fokker-Planck Equation},
Springer-Verlag, N.Y. 1984.
\bibitem{Cartling} B. Cartling, {\em J. Chem. Phys.} {\bf 87},
2638 (1987).
\bibitem{book} Z. Schuss, {\em Theory and Applications of Stochastic
Differential Equations}, J. Wiley \& Sons, N.Y. 1980.
(1975).
\bibitem{Brinkman} H.C. Brinkman, {\em Physica} (Utrecht)
{\bf 22}, 149 (1956).
\bibitem{LandauerSwanson} R. Landauer and J.A. Swanson, {\em Phys.
Rev.} {\bf 121}, 1668 (1961).
\bibitem{Langer} J.S. Langer, {\em Ann. Phys.} (N.Y.) {\bf 54},
258 (1969).
\bibitem{PLL} B.Z. Bobrovsky and Z. Schuss, {\em SIAM J. Appl.
Math.} {\bf 42}, 174 (1982).
\bibitem{MSchar} B.J. Matkowsky and Z. Schuss, {\em SIAM J.
Appl. Math.} {\bf 42}, 822 (1982).
\bibitem{MST} B.J. Matkowsky, Z. Schuss, and C. Tier, {\em
J. Stat. Phys.} {\bf  35} (3,4), 443 (1984).
\bibitem{Day} M.V. Day, {\em SIAM J. Math. Anal.} {\bf 20}, 222
(1989), {\em J. Math. Anal. Appl.} {\bf 147}, 134 (1990),
preprint (1991).
\bibitem{Weiss} I. Oppenheim, K.E. Shuler, and G.H. Weiss, {\em
Stochastic Processes in Chemical Physics: The Master Equation},
The MIT Press, Cambridge MA 1977.
\bibitem{Gardiner} C.W. Gardiner, {\em Handbook of Stochastic
Methods}, Springer Verlag, N.Y. 1982.
\bibitem{ProtterWeinberger} M.H. Protter and H.F. Weinberger,
{\em Maximum Principles in Differential Equations},
Prentice-Hall, Englewood Cliffs 1967.
\bibitem{MS} B.J. Matkowsky and  Z. Schuss, {\em SIAM J. Appl.
Math.} {\bf 33}, 365 (1977).
\bibitem{GrahamTel} R. Graham and T. Tel, {\em Phys. Rev. Lett.}
{\bf 52}, 9 (1984) (see also {\em Phys. Rev.} A {\bf 33}, 1322 (1986)).
\bibitem{Hagan} C.R. Doering, P.S. Hagan, C.D. Levermore, {\em
Phys. Rev. Lett.} {\bf 59}, 2129 (1987), A.W. Yu, G.P.
Agrawal, R. Roy, {\em J.Stat. Phys.} {\bf 54}, 1223 (1989).
\bibitem{RyterJSP} D. Ryter, {\em J. Stat. Phys.} {\bf 49},
751 (1987).
\bibitem{RyterPhysica} D. Ryter, {\em Physica} {\bf 142}A, 103
(1987).
\bibitem{MSB} B.J. Matkowsky, Z. Schuss, and E. Ben-Jacob,
{\em SIAM J. Appl. Math.} {\bf 42}, 835 (1982).
\bibitem{Glasstone} S. Glasstone, K.J. Laidler, H. Eyring, {\em
The Theory of Rate Processes}, McGraw-Hill, N.Y. 1941.
\bibitem{Karlin} S. Karlin and H.M. Taylor, {\em A Second Course in
Stochastic Processes}, Academic Press, N.Y. 1981.
\bibitem{Feller} W. Feller, {\em An Introduction to
Probability Theory and its Applications} I, Third Edition,
J. Wiley \& Sons, N.Y. 1968.
\bibitem{Truhlar} M.M. Kreevoy and D.G. Truhlar, in {\em
Investigation of Rates and Mechanisms of Reactions}, vol. 6,
4th ed. (C.F. Bernasconi, Ed.), J. Wiley, N.Y. 1986.
\bibitem{Benson}  S.H. Benson, {\em The Foundations of
Chemical Kinetics}, McGraw-Hill, N.Y. 1960.
\bibitem{Haken} H. Haken, {\em Rev. Mod. Phys.} {\bf 47}, 67
(1975).
\bibitem{green} E. Ben-Jacob, D.J. Bergman, B.J. Matkowsky, Z.
Schuss, {\em Phys. Rev.} A, {\bf 26}, 2805 (1982).
\bibitem{SQUID} E. Ben-Jacob, D.J. Bergman, Y. Imry,
B.J. Matkowsky, Z.  Schuss, {\em J. Appl. Phys.} {\bf 54},
6533 (1983).
\bibitem{Viterbi} A.J. Viterbi, {\em Principles of Coherent
Communications}, McGraw-Hill, N.Y. 1966.
\bibitem{BZ} A.M. Berezhkovskii and V.Y. Zitserman, {\em Physica}
{\bf 166}A, 585 (1990).
\bibitem{BZ1} A.M. Berezhkovskii and V.Y. Zitserman, {\em Chem.
Phys. Lett.} {\bf 172}, 235 (1990).
\bibitem{JCP} M.M. K\l osek, B.J. Matkowsky, A. Nitzan, M.
Ratner, Z. Schuss, {\em J. Chem. Phys.}  {\bf 95} (2) (1991).
\end{thebibliography}
\end{document}




