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A revised analysis of the exit problem at weak noise, and a simpler computation of the quasipotential with two variables

A new approach for the weak noise analysis of exit problems removes an intrinsic contradiction of an existing method. It applies for both the mean time and the location of the exits; novel outcomes mainly concern the exits from entire domains of attraction. Moreover, the involved quasipotential is obtained without use of a Hamiltonian system in the case of two variables.


I. Introduction
Noise-induced exits from a domain (in presence of a dynamical system) are described by the Feynman-Kac theory [1] . Its application in practical problems is however so intricate that alternative methods were presented, especially for weak noise and for domains with an attractor A of the dynamical system. With unit diffusion and with a gradient drift field this was accomplished in [2] . A more general theory was put forward in [3,4] with ideas of [5], but the application to the Kramers problem [4,6] reveals some deficiencies: i) dependence of the rate on an inaccessible part of the potential ii) inconsistent approaches for the rate and for the locations [7] of the exits iii) different methods according to the friction range.
The present paper treats the non-gradient drift fields (and a general diffusion) in a novel way, and eliminates these shortcomings. It is based on the Green's function of the Fokker-Planck operator, with sources near A . As it turns out, it differs from the existing one by the need of evaluating a backward equation in terms of an "associated" drift, instead of the actual one. This drift is obtained by reversing the conservative contribution (determined by the quasipotential), which leaves the attractor unchanged. For the final result that substitution is rather insignificant, when the part of the boundary with the most likely exits is attracted to A , and when the quasipotential is regular there. However, for exits from a whole basin of attraction, the boundary integral for the rate now accounts for the asymmetric arrivals of the (actual) trajectories near the most likely exit point. This asymmetry is implied by the twist of the associated drift against the separatrix , and it removes the flaws mentioned above.
The associated drift naturally arises when the quasipotential is evaluated by the Hamiltonian method for solving the "eikonal" or Freidlin equation, since it is given by one group of the canonical equations. A further new finding states that the Hamiltonian system is not really required in the case of two variables: the quasipotential can then be computed without an extension of the variable space, and this more direct method immediately yields the abovementioned twist responsible for the asymmetry of the exits.
It will further be argued that the existing method is rather suited for transition rates in bistable models, which are not always simply related with the exit rates.
The paper is organized as follows: The exposition of the background and the Green's function approach for the exit problem will be presented for n variables. The remainder with 2  n includes the new construction of the quasipotential (generating function  , numerical aspects, expansions at equilibrium points), followed by the asymptotic solution of the new backward equation (in standard variables, according to the rank of the diffusion matrix). Finally, the Kramers problem will be revisited, in order to exhibit the theoretical progress of the novel method, as well as to put forward some new results concerning the exit rate for any friction and the location of the exit points.

II. Background
The autonomous dynamical system with variables n i x i ,..., with smooth functions i a is supposed to have at least one compact attractor A .
The negative divergence (contraction) of the drift field a  will be denoted by (

2.2)
White Gaussian noise is further supposed to interfere in such a way that the pdf  obeys a Fokker-Planck equation [8,9] with the symmetric diffusion matrix , with smooth elements and with 0 : asymptotically for small  ( is the "quasipotential" or "eikonal" [11][12][13] ).

The Associated System
As (2.6) shows, a reversal of does not affect  , and (2.9) is also unchanged. The corresponding "associated" drift is It is easily seen from (2.6) that The associated drift thus follows the projection of the Hamiltonian curves into the x  -space.
In [14] the associated system was introduced without the eikonal form of 0 w , and for a general Fokker-Planck operator L . The main difference is that there  [9] , and in the Kramers problem it entails the "outer normal expansion" of [7] .

III. The Exit Problem
It is now supposed that a domain  of the x  -space contains one attractor A , and that the boundary   is smooth and absorbing. The problem is when and where particles driven by weak noise arrive on   after a start near A . A natural tool for analyzing this problem is the Green's function ) and with a starting point 0 x  near A (here  denotes the n-dimensional deltafunction).
Since 0 w is concentrated near A , the relevant information is already contained in The righthand side of (3.1) describes the insertion of one particle per unit time at 0 x  . The mean lifetime of trajectories starting near A is therefore while the density of the exit points is given by the modulus of G D  on   .
It is now useful to introduce the function , and by (2.16) it follows that Mind that this involves the associated drift. The solution of (3.4) yields both T and the location of the exits.
Weak noise allows further simplifications. Note that the steady particle flux on   is Dividing this by T , and introducing ) ( for the exit rate .
The unnormalized density of the exit points is The method is completed by the homogeneous equation for q now with the inner boundary value 1  q near A , which follows by (3.3) with  is the probability to reach the attractor region from x  without a visit on   , and under the associated dynamics.
These findings agree with the existing approach [3,4] up to the crucial twiddle on  L : that approach (which is not based on a Green's function) just uses the actual drift a  for computing Q or q . Clearly, the methods are equivalent when a a    , i.e. in [2] , It is remarkable that in bistable models with two variables the existing method (with the separatrix as   ) yields the rate of the transitions to the other attractor region.
This was shown in the Chapt. This means that the presence of the twiddle on  L distinguishes between the exit and the transition rates. Mind that a first arrival on the separatrix is followed either by a return, or by a transition to the other attractor. When these continuations are equally probable, the exits are just twice as frequent as the transitions. An example for this special case is the Kramers problem with a smooth threshold and with an appreciable friction, as will be shown below in the Chapter VIII.

IV. Qualitative Impacts of the Modification
Here we briefly exhibit the major changes in the planar case, with arguments to be substantiated below.

a) Exits from an entire basin of attraction
The drift a  usually has a hyperbolic point on the boundary   of the basin  , and 0 w on   takes a (narrow) maximum there. If  exists there and is regular, ã  is locally hyperbolic as well. Except for the situation in [2], the local eigendirections of these drifts do not coincide ( ã  does not parallel   ), so that ã  points into  on one side of the fixpoint, and outwards on the other side. As a consequence, q  changes rapidly along to zero. The density of the exit points is therefore skew [16], and the integrand in (3.5) is strongly asymmetric. In the existing method, however, Since   is smooth, it parallels c a  at P, so that the normal drift 0 

A generating function for 
The case of two variables allows some substantial simplifications, also for the computation of  and of b a, .
The orthogonality of c a  and   , see (2.6) , suggests to introduce This must be a gradient. The condition The last property allows an easy computation of  itself along the characteristics: from (2.14) and (5.5) it follows that The quasipotential  is thus indeed obtained along characteristics in the plane, and (2.12) is not invoked.

Remarks:
1. Once  is known, (5.5) can also be integrated along any path. An advantage of (5.7) is that it saves the storing of  .
2. The "time" in characteristic equations is merely an auxiliary variable, and its sign can be chosen so that the integration goes "upwards" (i.e. starting from an attractor) or "downwards" (starting from a hyperbolic point or a repellor).

When
, and thus the bound

Expansions at an equilibrium point
: .
An immediate consequence is (5.10) (In the special case , which locally satisfies (2.7) .
The emphasis lies on the analogue of M for the associated drift (to be inferred from (5.3) ), the eigenvalues of both M and M are the same. A more explicit form of (5.11) is A (and also the one towards the outer attractor, possibly at infinity); these provide the respective differences in  by a single integration, and thus the leading contribution (i.e. the Arrhenius factor) of the exit rates from there. Remarks: is exceptional, as (5.6) leaves  undetermined. The eikonal form is then only possible when also but both S and M remain unspecified. According to [17] it is then also possible that several characteristics link the hyperbolic point with the same attractor.
2. The corresponding expansion in the Hamiltonian approach involves a 4 4  matrix with four eigenvalues and eigenvectors, see [18] .
Since a numerical integration cannot be started on an equilibrium point, but only in its neighbourhood, it is essential to evaluate the local   as well. This can be done by taking the derivatives in y x, of (5.6) . With is also the Kramers model ( [4,6] and the Chap.VIII below), even with a state-dependent In contrast to "detailed balance" [9] neither (5.14) nor (5.15) is based on time reversal.

VI. The Function
) , ( y x q

For Regular Diffusion
The aim is to solve (3.6) in the plane and asymptotically for small  . This is conveniently done in standardized variables so that I D  when the original D is regular everywhere (see [19] for the transformation). Then (3.6) reads Note the analogy with (6.5) .

Attracted exit region
Here it is supposed that at the point P , where 0 w is maximum on   , the drift a  points is not involved, the result is the same as for the existing approach.
It is further easily seen that the density of the exit points has its maximum at P .

Exits near a transition point
It is now supposed that  is the whole domain attracted to A by a  , and that P ( ) 0 , 0 (  say) is an equilibrium point on   , where 0 w on   is maximum ( minimum).
The condition (6.7) is not met for small | | s . Yet one may modify (6.6) by The results is ) ( is the missing inner boundary value in (7.3) , and therefore it multiplies the above solution. The evaluation of ) (x g is a nontrivial task, which will only be accomplished here for the Kramers model.
When the maximum point of 0 w on   is attracted, it follows that in analogy with Chapt. VI . The results are thus again the same as for the existing method, up to the factor ) (x g , which at attracted points tends to 1 when 0   . It is worth noting that C cancels in the integrand of (3.5) since C D  .
For the exits on a separatrix with a hyperbolic point (where , is again crucial. It is essentially specified by at t x  are given by This yields , the separatrix is tangent to the  y axis). It is further easily seen that Remark: , it seems natural to expect that  is not be involved, so that (7.6) applies with (7.9) . In the Kramers case this will indeed be shown below, by use of the explicit ) ( (7.13)

VIII. The Kramers Problem
The very well-known Kramers problem [4,6] is revisited here in some detail. The purpose is to show how the deficiencies of the existing method (as mentioned in the Introduction) are now resolved. This includes the explicit determination of ) (x g , and thereby a correction of the exit point density. The usual linearization at the saddlepoint will be avoided, so that the results hold for each friction  .
The model describes a massive particle moving in a potential ) (x U with a threshold , as well as the unstable manifold (see (7.11) The separatrix ) (x v s is depicted in [4] and [15] ; note that in view of (8.1) , and is thus given by In the matrix (7.1) , and the steady-state pdf ) , ( According to (7.4) with ' in view of (8.1) . The negative sign of  is due to the fact that is the probability to reach the region of the attractor A from 0  x in the associated dynamics, more precisely the maximum probability with respect to v .
At 0  x the nonabsorbed trajectories have a negative v and continue to 0 is determined by the corresponding backward equation Note that this does not depend on  . The simultaneous validity for the associated system is due to the fact that the only parameters are  and  (and it can be verified more formally).
The pertinent approximation for q is now It depends on  via s v only, and it will be confirmed in the Appendix. Accordingly

Result for the exit points
It is natural to consider the density with respect to x (the arclength on the separatrix involves the sum of terms with different units). Up to normalization it is given by . The normalizing prefactor is 1 1 .

The exit rate
In view of (3.5) the rate e r is now given by  .

Concluding remarks
1) For the exit problem it is required that the quasipotential  exists and is differentiable at the preferred exit point. Whether possible singularities at different places can be relevant remains to be investigated.