Repartition of the quasi-stationary distribution and first exit point density for a double-well potential

Repartition of the quasi-stationary distribution and first exit point density for a double-well potential

Dorian Le Peutrecand Boris Nectoux Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France. E-mail: dorian.lepeutrec@math.u-psud.frInstitut für Analysis und Scientific Computing, E101-TU Wien, Wiedner Hauptstr. 8, 1040 Wien, Austria. E-mail: boris.nectoux@asc.tuwien.ac.at
Abstract

Let be a smooth function and be the stochastic process solution to the overdamped Langevin dynamics

Let be a smooth bounded domain and assume that is a double-well potential with degenerate barriers. In this work, we study in the small temperature regime, i.e. when , the asymptotic repartition of the quasi-stationary distribution of in within the two wells of . We show that this distribution generically concentrates in precisely one well of when but can nevertheless concentrate in both wells when admits sufficient symmetries. This phenomenon corresponds to the so-called tunneling effect in semiclassical analysis. We also investigate in this setting the asymptotic behaviour when of the first exit point distribution from of when is distributed according to the quasi-stationary distribution.

1 Setting and results

1.1 Quasi-stationary distribution and purpose of this work

Let be the stochastic process solution to the overdamped Langevin dynamics in :

(1)

where is the potential (chosen in all this work), is the temperature and is a standard -dimensional Brownian motion. Let be a bounded open and connected subset of and introduce

the first exit time from . A quasi-stationary distribution for the process (1) on is a probability measure on such that, when , it holds for any time and any Borel set ,

From [2, 6, 12, 15], there exists a probability measure supported in such that for any probability measure on : when , one has for any borel set ,

(2)

It follows from (2) that is the unique quasi-stationary distribution for the process (1) on .

In molecular dynamics, the quasi-stationary distribution  is used to quantify the metastability of the subdomain of as follows: for a probability measure  supported in , the domain is said to be metastable for the initial condition  if, when , the convergence in (2) is much quicker than the average exit time from . When  is metastable, it is thus relevant to study the exit event of the process (1) from  starting from , i.e. when . This is used in several algorithms aiming at accelarating the sampling of the exit even from a metastable domain, see for instance [1, 17, 12, 14]. The study of the metastability is a very active field of science research which is at the heart of the numerical challenges observed in molecular dynamics. We refer in particular to [13] for an overview on this topic.

In this work, we study the repartition when of the quasi-stationary distribution  within the wells of a double-well potential  with degenerate barriers (see the assumption [H-Well] below). We show in particular that  generically concentrates in one well (see Theorem 1 below) but can also concentrate in both wells when the function is (nearly) even (see Theorems 2 and 3 below). According to the analysis led in [3], the second phenomenon can only appear when the potential function admits degenerate deepest barriers. It is particularly unstable (see Remark 4 below) and arises from a strong tunneling effect between the wells. The asymptotic behaviour of the law of when is also investigated in order to discuss the metastability of for deterministic initial conditions within the wells.

1.2 Double-well potential

We assume more generally from now on that is a oriented compact and connected Riemannian manifold of dimension with boundary . The basic assumption in this work is the following:

  • [H-Well]: The function belongs to , on , and and are Morse functions. Moreover, the function has only two local minima and in which satisfy

    Finally, the open set has precisely two connected components, denoted by and , such that for all ,

Under the assumption [H-Well], the potential function has precisely two wells, namely the open sets and . This double-well potential is moreover said to have degenerate barriers since the depths of and are the same and equal (see Figure 1)

(3)

Let us also recall that a function  is a Morse function if all its critical points are non degenerate. This implies in particular that has a finite number of critical points.

Figure 1: A one dimensional example where [H-Well] is satisfied.

When replacing the assumption by in [H-Well] (i.e. when the barriers are not degenerate), it is proved in [3, Proposition 10] that the quasi-stationary distribution concentrates in when . This work aims precisely at studying the degenerate case which introduces some additional technical difficulties, see the next section for some explanation.

Let us assume from now on that the assumption [H-Well] is satisfed. The set of saddle points of  of index  in  is denoted by . Let us also define

and

According to the terminology of [8, Section 5.2], we call the elements of the generalized saddle points for the Witten Laplacian acting on -forms with tangential Dirichlet boundary conditions on . Note that does not have any saddle point on (since there) but that extending by outside (which is consistent with zero boundary Dirichlet conditions), the elements of are geometrically saddle points (since for such an element , is a local minimum of and a local maximum of , where is the straight line passing through and orthogonal to at ).

Notice that from the assumption [H-Well], one has for all :

Let us define, for , by

(4)

One defines furthermore by

where ( meaning ). From [3, Proposition 20], it holds

and one orders so that

where . Note finally the relation

(5)

See Figures 2 and 3 for a schematic representation of the potential under [H-Well] when and when .

Figure 2: Schematic representation of the connected components and of  when the assumption [H-Well] is satisfied. In this representation, , , and . Thus, , and .

Figure 3: Schematic representation of the connected components and of  when the assumption [H-Well] is satisfied. In this representation, , , and . Thus, , , and .

1.3 Results

Preliminary spectral analysis

Let be the infinitesimal generator of the diffusion (1),

where is the Hodge Laplacian on and the gradient associated with the metric tensor on . Let moreover be the differential operator on with domain

The operator is self-adjoint, positive, and has compact resolvent. Moreover, its smallest eigenvalue is positive, non degenerate and any eigenfunction associated with  has a sign on (see for instance [5, Section 6]). Let be an eigenfunction associated with . According to [12], the quasi-stationary distribution is then given by

(6)

where is the Lebesgue measure on . We assume furthermore from now on that

(7)

In view of (6), in order to study the asymptotic behaviour of when , we look for an accurate approximation of . This is delicate since exponentially small eigenvalues of the same order are into play. Indeed, according to [3, Theorem 5], under [H-Well], it holds

and there exists such that for every small enough,

where denotes the second smallest eigenvalue of . This makes in particular difficult to properly estimate by simply projecting a well chosen quasi-mode on since the quality of such an approximation is typically bounded from above by the quotient which does not tend to when . To overcome this difficulty, the key point relies on the fact that we are able to precisely analyse the restriction of  to the eigenspace associated with and . Indeed, this eigenspace has dimension two and the remaining eigenvalues of are bounded from below by 111They are actually bounded from below by some positive constant.. More precisely, we have according to [8, Theorem 3.2.3] the

Lemma 1.

Let us assume that the hypothesis [H-Well] is satisfied. Then, there exists such that for all ,

where is the orthogonal projector on the vector space associated with the eigenvalues of in .

Remark 1.

As a consequence of Lemma 1, there exists such that for every , the second smallest eigenvalue of is non degenerate.

Moreover, it follows from the general analysis led in [3] that the matrix of  satisfies Proposition 1 below. Before stating it, let us introduce the following notation. For , one writes if there exist and such that for all ,

(8)

In addition, for , one says that admits a full asymptotic expansion in , and one writes , if there exists a sequence such that for any , it holds in the limit :

(9)
Proposition 1.

Let us assume that the hypothesis [H-Well] is satisfied. Then, there exists such that for every , there exists an orthonormal basis of such that the matrix of the restriction of to in  has the form:

(10)

where is defined in (3),

  • satisfies in the limit :

    (11)

    for some independent of and where the symbol is defined in (8),

  • there exist two sequences  and such that for , in the limit :

    (12)

    where the symbol is defined in (9) and

    (13)

    Moreover, when , one has for every ,

    (14)

    where is the negative eigenvalue of . Finally, the sequence (resp. ) only depends on the values of the derivatives of at and on (resp. of the derivatives of at and on ).

Proposition 1 will be proven in Section 2.1. It permits to reduce the study of the asymptotic repartition of within the wells and to linear algebra considerations in dimension two. Then, when , the study of the asymptotic concentration of the law of (which occurs on a subset of , see [3, Definition 1] for a precise definition) follows from the analysis made in [3] and based on the following formula [12]: for any , it holds

(15)

where the notation stands for the expectation when .

Results when  concentrates in precisely one well when

Let us define here the following assumption:

  • [H1]: The assumption [H-Well] is satisfied, there exists such that

    either (16)
    or (17)

    and it holds

Note that the assumption [H1] is generic (given an arbitrary function satisfying [H-Well]) according to the following:

  • when and the asymptotic expansion in of and in (12) differ (i.e. when ), the assumption [H1] is satisfied and there exists such that when (see indeed (11)),

    (18)
  • when the assumption [H1] is, according to (11) and (12), equivalent to , where and are defined in (13). In this case, when :

    (19)

Our main result under the generic assumption [H1] is the following.

Theorem 1.

Let us assume that the hypotheses [H-Well] and [H1] together with (16) are satisfied. Let be the quasi-stationary distribution of the process (1) on (see (6)). Let be an open neighborhood of and be an open neighborhood of such that . Then, there exists such that in the limit :

(20)

where for ,

(21)

Moreover, for any and for any family of disjoint open neighborhoods of in , there exists such that in the limit :

(22)

and

(23)

In addition, when, for some , is around , one has when :

(24)

where, for and , the constant is defined by

(25)
Remark 2.

When [H-Well] and [H1] are satisfied, one also obtains from Proposition 1 sharp asymptotic estimates on the two smallest eigenvalues of when , see indeed (55) and (56).

From Theorem 1, when [H-Well] holds and [H1] is satisfied with (16), the quasi-stationary distribution concentrates when in and more precisely around any arbitrary small neighborhood of . Moreover, when , the law of concentrates when on with an explicit repartition given by (25). Adapting the proof of [3, Proposition 11] by using (20) and (21), one can also show that when , the law of concentrates when on with the same repartition as when . This exhibits a metastable behavior for such initial conditions. Moreover, when on , it follows from [3, Theorem 2] that when , the law of concentrates when on with the repartition given by (25) (with ). This exhibits a non metastable behavior for such initial conditions.

To connect with the literature dealing with semiclassical Schrödinger operators of the form on manifolds without boundary (where is a potential function independent of ), one can say in this situation that the tunneling effect between the two wells is too weak to mix their respective properties and that these two wells are hence somehow independent, that is, in the terminology of [11, 10], weakly resonant or non resonant. We also refer to [7] for an overview on this topic for semiclassical Schrödinger operators (see in particular pp. 41–42 there). Notice lastly that (21) shows that some tunneling effect of order appears nevertheless when (see indeed (19)), contrary to the case when and do not have the same asymptotic expansion, see (18). As expected, when , the independence between the two wells in this case is hence generically weaker.

Results when concentrates in both wells when

Let us define here the following assumption:

  • [H2]: The assumption [H-Well] is satisfied. Moreover, there exists such that for all , it holds

Let us exhibit situations where the assumption [H2] is satisfied.

  • When , the assumption [H2] is satisfied if and only if and . This equivalence follows from (11) and (12). Therefore, when , using (13) and (14), the assumption [H2] is satisfied if and only if

    (26)

    and

    (27)

    Moreover, it holds in this case:

  • Let us assume that is an even function as defined by (34) below. Then, from Theorem 3 below, the assumption [H2] is satisfied (see indeed Remark 7).

Remark 3.

When , we are not able to explicit assumptions on which imply [H2] except in the symmetric situation described in Theorem 3. Note in particular that when and [H2] holds, one has when : (which follows from [H2], (12) and the fact that , see (11)) and thus:

(28)

Moreover, it also holds in this case (see (68)).

Remark 4.

The assumption [H2] is non generic, that is unstable with respect to perturbations of the potential  in the following sense. For any satisfying [H2], it follows from (26)—(28) that there exists an arbitrary small perturbation such that satisfies [H1]. Then, according to Theorem 1, the quasi-stationary distribution for the potential concentrates when in precisely one of the wells or .

The following result shows that when [H2] is satisfied, the quasi-stationary distribution  concentrates when in the two wells  and .

Theorem 2.

Let us assume that the hypotheses [H-Well] and [H2] are satisfied. Let be the quasi-stationary distribution of the process (1) on (see (6)). Let be an open neighborhood of and be an open neighborhood of such that . Then, there exists such that in the limit :

(29)

where, for ,

(30)

where, defining by ,

(31)

Moreover, for any and for any family of disjoint open neighborhoods of in , there exists such that in the limit :

(32)

Lastly, when, for some , is around , one has when :

(33)

where is defined in (31) and is defined in (25).

Remark 5.

When [H-Well] and [H2] are satisfied, one also gives sharp asymptotic estimates on the two smallest eigenvalues of when , see indeed (68), (69) and (70).

When [H-Well] and [H1] hold, Theorem 2 implies that the quasi-stationary distribution  concentrates when in and , and more precisely around any arbitrary small neighborhood of and . Note also that when , the coefficient (31) specifying the repartition of within the wells equals according to (27). Moreover, when the law of concentrates when on with an explicit repartition given by (25). In addition, when on , it follows from [3, Theorem 2] that when , , the law of concentrates when