Computing Transition Rates for the 1-D Stochastic Ginzburg–Landau–Allen–Cahn Equation for Finite-Amplitude Noise with a Rare Event Algorithm

Abstract

In this article we compute and analyse the transition rates and duration of reactive trajectories of the stochastic 1-D Allen–Cahn equations for both the Freidlin–Wentzell regime (weak noise or temperature limit) and finite-amplitude white noise, as well as for small and large domain. We demonstrate that extremely rare reactive trajectories corresponding to direct transitions between two metastable states are efficiently computed using an algorithm called adaptive multilevel splitting. This algorithm is dedicated to the computation of rare events and is able to provide ensembles of reactive trajectories in a very efficient way. In the small noise limit, our numerical results are in agreement with large-deviation predictions such as instanton-like solutions, mean first passages and escape probabilities. We show that the duration of reactive trajectories follows a Gumbel distribution like for one degree of freedom systems. Moreover, the mean duration growths logarithmically with the inverse temperature. The prefactor given by the potential curvature grows exponentially with size. The main novelty of our work is that we also perform an analysis of reactive trajectories for large noises and large domains. In this case, we show that the position of the reactive front is essentially a random walk. This time, the mean duration grows linearly with the inverse temperature and quadratically with the size. Using a phenomenological description of the system, we are able to calculate the transition rate, although the dynamics is described by neither Freidlin–Wentzell or Eyring–Kramers type of results. Numerical results confirm our analysis.

Appendices

Appendices

We first develop the calculation of the approximation of the negative eigenvalue of the Hessian of the potential V We then give here some details on the AMS algorithm, first we provide a detailed description of the algorithm itself, then some basic applications like the computation of mean first passage time. We end with a practical discussion on some numerical convergence issues.

1.1 Appendix 1: Approximating the Eigenvalue $\lambda _s$

In this first appendix, we detail the calculation of an approximation of the first eigenvalue $\lambda _s$ of the Hessian of V, presented in Sect. 3.2.2. This eigenvalue controls the prefactor of the mean first passage time (Eq. 14).

The eigenvalues $\lambda _i$ and eigenmodes $\Phi _i$ are the solution of the problem

$$\begin{aligned} \underbrace{ -\partial _x^2 \Phi _i + (3A_{s,0}^2-1)}_{H_{s,0}}\Phi _i=\lambda _{i,A_{s,0}} \Phi _{i,A_{s,0}}\,, \end{aligned}$$

(33)

with boundary conditions $\Phi _i(0)=\Phi _i(L)=0$. The eigenvalue problem can be posed either at the saddle, for $A_s$, or in the minimum $A_{0}$. The operator $H_{s,0}=\nabla ^2 V|_{A_{s,0}}$ is the hessian of the potential V. It is self adjoint on the set of function (regular enough) that cancel out on both ends of the domain $x=0,L$. It is well known that in the limit of infinite size $L\rightarrow \infty$ (see [36] and references within): we have

$$\begin{aligned} A_s=A_{1,\infty }=\tanh \left( \frac{x-\frac{L}{2}}{\sqrt{2}}\right) \,,\quad \Phi _\infty =1-\tanh ^2\left( \frac{x-\frac{L}{2}}{\sqrt{2}}\right) \,,\quad \lambda _\infty =0\,. \end{aligned}$$

(34)

This eigenmode is a goldstone mode, related to the translational invariance of the front in an infinite domain [58]. We denote hyperbolic tangent $\mathcal {H}_{u}(x)\equiv \tanh ((x-u)/\sqrt{2})$, which will simplify our notations.

How $\lambda _s$ converges toward 0 in a finite domain of size L ? In order to address that question, we will provide an upper bound for $\lambda _s$. For that matter, we use the fact that $H_s$ is self-adjoint. Its Rayleigh quotient

$$\begin{aligned} R(H_s,f)=\frac{\int _0^L fH_s(f)\, \mathrm{d}x}{\int _0^L f^2 \, \mathrm{d}x}\,,\, f(0)=f(L)=0\,, \end{aligned}$$

(35)

then verifies the min-max theorem, which states that $R(H_s,f)$ is bounded by the lowest ($\lambda _s$) and highest eigenvalue of $H_s$. Moreover, we have that $R(H_s,\Phi )=\lambda _s$ and that given $\Psi$ an approximation of $\Phi$, $R(H_s,\Psi )$ will converge (from above) toward $\lambda _s$ as $\Psi$ converges toward $\Phi$.

We will now tailor an analytical approximation of $\Phi$ and $A_1$ so as to give an upper bound on $\lambda _s$. This upper bound will serve as an approximation. The lowest eigenvalue $\lambda _s$ and the eigenmode $\Phi$ correspond to the unstable direction of V at the saddle: small motion of the position of the front near L / 2. If the field A is slightly displaced away from the saddle $A_1$ along the unstable direction, the front (located at position y) is displaced by a small $\delta x=y-L/2$. We then have $A\approx A_1+\delta x\Phi$. In order to obtain an approximation of $\Phi$, we start from an analytical approximation K(x, y) of the field A(x) when a front is located at position y

$$\begin{aligned} K(x,y)\equiv \mathcal {H}_0(x) \mathcal {H}_L(x) \mathcal {H}_y(x)\,. \end{aligned}$$

(36)

Performing the expansion gives us

$$\begin{aligned} \approx K\left( x, \frac{L}{2} +\delta x\right) =K\left( x,\frac{L}{2} \right) +\delta x \frac{\partial K}{\partial y}\left( x,\frac{L}{2} \right) +O(\delta x^2)\,. \end{aligned}$$

(37)

This leads us to choose

$$\begin{aligned} \Psi \equiv -\sqrt{2}\frac{\partial K}{\partial y}\left( x,\frac{L}{2} \right) =-\mathcal {H}_0(x) \mathcal {H}_{L}(x)\left( 1- \mathcal {H}_{\frac{L}{2}}(x)^2\right) \,. \end{aligned}$$

(38)

The minus sign and the square root of 2 simply ensure that $\Psi$ is a positive function whose maximum is 1 at $x=L/2$. In order to give a good analytical approximation of the ratio of integrals $I=R(H_s,\Psi )$, we set

$$\begin{aligned} \tilde{A}_1\equiv \mathcal {H}_0(x)+\mathcal {H}_{L}(x)-\mathcal {H}_{\frac{L}{2}}(x) \,,\, \tilde{I}\equiv \frac{\displaystyle \int _0^L \Phi _\infty \left( -\partial _x^2 + (3\tilde{A}_{1}^2-1) \right) \Phi _\infty \, \mathrm{d}x}{\int _0^L \Phi _\infty ^2 \, \mathrm{d}x}\,. \end{aligned}$$

(39)

Note that $\Phi _\infty$ does not cancel out at $x=0,L$, nor is $\tilde{A}_1$ strictly the saddle point of V. As a consequence $\tilde{I}$ is not strictly speaking a Rayleigh quotient, nor should we a priori expect to verify exactly its properties. The ratio $\tilde{I}$ is a good approximation of I because $\Phi _\infty ^2$ like $\Psi ^2$ decrease like $1/\cosh ^4$ away from L / 2, up to corrections on the boundaries (Fig. 13). Any relative contribution to the integrals near $x=0$ and $x=L$, where the errors are made, are exponentially small by a factor $\exp ^4(-L/(2\sqrt{2}))=\exp (-L\sqrt{2})$ relatively to the contributions from the neighbourhood of $x=L/2$. Meanwhile, the relative difference between $\Phi _\infty$ and $\Psi$, or between $\tilde{A}_1$ and $A_1$ are also exponentially small near L / 2. Note however, that it is fundamental that $A_1$ has the proper boundary conditions at $x=0$ and $x=L$, as it is they that will contribute to the exponential decay of the eigenvalue. The quality of this approximation depends on how large L is. For $L=10$, $\exp (-L\sqrt{2})\simeq 7\times 10^{-7}$: the system is quickly in the range of validity.

The denominator is calculated in the same way as $\Delta V_1$, it is equal to $(4\sqrt{2})/3$. In order to calculate the numerator, denoted J, we first note that

$$\begin{aligned} \tilde{H}\Phi _\infty =-3\left( 1-\mathcal {H}_{\frac{L}{2}}(x)^2\right) \left( \underbrace{\mathcal {H}_0(x)+\mathcal {H}_L(x) }_{\equiv S}\right) \left( -\underbrace{\left( \mathcal {H}_0(x)+\mathcal {H}_L(x)\right) }_{=S} +2\mathcal {H}_{\frac{L}{2}}(x) \right) \,. \end{aligned}$$

(40)

We process the sum

$$\begin{aligned} S=2e^{-\frac{L}{\sqrt{2}}}\frac{\exp \left( \frac{2x-L}{\sqrt{2}}\right) -\exp \left( -\frac{2x-L}{\sqrt{2}}\right) }{1+\exp \left( \sqrt{2}(x-L)\right) +\exp \left( -\sqrt{2}x\right) +\exp \left( -\sqrt{2}L\right) }\,. \end{aligned}$$

(41)

In the neighbourhood of L / 2, only 1 has a significant contribution in the denominator, so that this sum can be approximated by the numerator in $\tilde{H}\Phi _\infty$. Note that it is negligible compared to $\tanh ((x-L/2)/\sqrt{2})$: Both cancel out in L / 2, while only the hyperbolic tangent is of order 1 in the direct neighbourhood. Now J reads

$$\begin{aligned} J=-12e^{-\frac{L}{\sqrt{2}}}\int _{0}^L \left( 1-\mathcal {H}_{\frac{L}{2}}(x)^2\right) ^2\mathcal {H}_{\frac{L}{2}}(x)\left( \exp \left( \frac{2x-L}{\sqrt{2}}\right) -\exp \left( -\frac{2x-L}{\sqrt{2}}\right) \right) \, \mathrm{d}x \end{aligned}$$

(42)

We eventually note that the only contributions to the integral are around L / 2, so that we can let the bounds go to infinity. This gives us $J=-32\sqrt{2}e^{-L/\sqrt{2}}$, then

$$\begin{aligned} \tilde{I}=-24e^{-\frac{L}{\sqrt{2}}}\,. \end{aligned}$$

(43)

This is an approximation at first order in $\exp (-L/\sqrt{2})$: the corrections are exponentially small as L takes large values that can however be considered in AMS simulations. A numerical calculation of the integral in I confirms this assertion.

Fig. 13

Logarithm of the eigenmode and its approximations in a domain of size $L=20$ (Color figure online)

1.2 Appendix 2: Algorithm Description

The general setting is to have a Markov process describing our system, i.e. $(X_t^{x_0})_{t \ge 0}$ with $X_0^{x_0} = x_0 \in \mathcal{E}$. The phase space $\mathcal{E}$ of the system can be either finite-dimensional of infinite-dimensional. We assume that there are two sets $\mathcal{A}$ and $\mathcal{B}$ with $\mathcal{A} \cap \mathcal{B} = \emptyset$. The goal is to estimate the probability $\alpha$ to reach $\mathcal{B}$ starting from the initial condition $x_0$ before going to $\mathcal{A}$. If $\tau _\mathcal{C}$ denote the stopping time defined as $\tau _\mathcal{C}(x) = \inf \{ t \ge 0, X_t^x \in \mathcal{C}, \}$ for a given Borel set $\mathcal{C}$, the problem translates into computing

$$\begin{aligned} \alpha \equiv \alpha (x_0) = \Pr (\tau _\mathcal{B}(x_0) < \tau _\mathcal{A}(x_0)). \end{aligned}$$

(44)

A reactive trajectory is a particular realisation of (44). Note that this probability, when seen as a function of $x_0$, takes values zero if $x_0 \in \mathcal{A}$ and one if $x_0 \in \mathcal{B}$. It is called committor or importance function or equilibrium potential in mathematics. In particular, for diffusive processes, it solves the backward Fokker–Planck equation with boundary conditions $\alpha (x)=0, x\in \mathcal{A}$ and $\alpha (x)=1,x\in \mathcal{B}$. The algorithm not only gives a pointwise estimate of the backward Fokker–Planck but also provides the ensemble associated with the probability. It is clear that for high-dimensional systems, access to the Fokker–Planck equation or doing direct Monte-Carlo simulations when $\alpha$ is too small is, most of the time, out of question.

The main idea is to introduce a scalar quantity which measures how far a trajectory is escaping from the set $\mathcal{A}$ before returning to $\mathcal{A}$. There is no unique way to measure how far a trajectory is escaping from the return set $\mathcal{A}$, or put in another way there is no unique order relation in the space of trajectories. However, a natural choice is to consider the following quantity

$$\begin{aligned} Q(x) = \sup _{t \in [0,\tau _\mathcal{A}]} \Phi (X_t^x),~ \Phi : \mathbb {E} \rightarrow \mathbb {R}. \end{aligned}$$

(45)

The function $\Phi$ which is often renormalized to be in [0, 1] is called reaction coordinate or observable and is such that $\Phi (\mathcal{A}) = 0$ and $\Phi (\mathcal{B}) = 1$. Note that the choice of $\Phi$ might become crucial and we will comment it hereafter. The algorithm is doing the following (see also Fig. 1):

Pseudo-code Let N be a fixed number, and $\Phi$ given.

• Initialization Set the counter $\mathcal{K} = 0$. Draw N i.i.d. trajectories indexed by $1 \le i \le N$, all starting at $x_0$ until they either reach $\mathcal{A}$ or $\mathcal{B}$. Compute $Q_i(x_0)$ for $1 \le i \le N$. In the following we will write $Q_i$ these quantities since their initial restarting conditions might change. DO WHILE $\left( {\displaystyle \min _i Q_i \le b}\right)$

  • Selection find

    $$\begin{aligned} i^\star = argmin_i ~Q_i. \end{aligned}$$

    Let $\widehat{Q} \equiv Q_{i^\star }$.

  • Branching select an index I uniformly in $\{1,\ldots N\} \setminus \{i^\star \}$. Find $t^\star$ such that

    $$\begin{aligned} t^\star = \inf _{t} \{ \Phi \left( \{ X_t \}_{I}\right) \ge \widehat{Q} \} \end{aligned}$$

    Compute the new trajectory $i^\star$ with initial condition $x^\star \equiv \{X_{t^\star } \}_I$ until it reaches either $\mathcal{A}$ or $\mathcal{B}$. Compute the new value $Q_{i^\star }(x^\star )$.

    $$\begin{aligned} \mathcal{K} \leftarrow \mathcal{K} + 1 \end{aligned}$$

  END WHILE

The performance of the algorithm depends rather crucially on the choice of $\Phi$. In fact, it is possible to show that when $\Phi (x) = \alpha (x)$ [see Eq. (44)] or when the system has only one degree of freedom ($\mathcal{E} = \mathbb {R}$), the number of iterations $\mathcal{K}$ when the algorithm has stopped has a Poissonian distribution [45, 46]

$$\begin{aligned} \mathcal{K} \sim \mathrm{Poisson}(-N \ln \alpha ). \end{aligned}$$

(46)

In other words, the quantity $\frac{K}{N}$ yields an estimate of $-\ln \alpha$ (see [47] for particular SDEs). Moreover,

$$\begin{aligned} \widehat{\alpha }= \left( 1 - \frac{1}{N}\right) ^\mathcal{K} \end{aligned}$$

(47)

gives an estimate of $\alpha$ itself. In some cases, one can demonstrate a central limit theorem for $\widehat{\alpha }$, with a variance scaling like $1/\sqrt{N}$

$$\begin{aligned} \sqrt{N}\left( \frac{\alpha -\widehat{\alpha }}{\alpha }\right) \xrightarrow [N\rightarrow \infty ]{\mathcal {D}}\mathcal {N}(0,|ln(\alpha )|). \end{aligned}$$

(48)

Some sets of hypotheses lead to a bias behaving as $\frac{1}{N}$ [41, 48]. With more loose hypotheses or in more complex cases, the variance still scales like $1/\sqrt{N}$, but can be larger than what is predicted [47].

Note that for arbitrary $\Phi$ and in dimension larger than 1, the statistical behavior of the algorithm is a difficult mathematical question and it is still open [45]. In particular, it is unclear whether these estimates are biased or not, even in the limit of $N \rightarrow \infty$. In practice, a “bad” reaction coordinate which differs significantly from the optimal committor, has a fat-tailed distribution although its variance still behaves like $\frac{1}{\sqrt{N}}$. Therefore, the larger N, the better precision one obtains. Overall, the algorithm numerically exhibits very robust results. There are other formulation of the algorithm, in particular, it is possible to kill a given proportion of the N trajectories at each algorithmic step (typically, if one kills p trajectories at each step followed by replacement, an estimate of the probability becomes $(1-\frac{p}{N})^\mathcal{K}$). This version has the advantage to be computationally more efficient when parallelized [47].

1.3 Appendix 3: Practical Applications for Allen–Cahn Equation

1.3.1 Appendix 3.1: Reaction Coordinates

In our case the set $\mathcal{A}$ is a neighborhood of the stationary solution $A_0^+$ and the set $\mathcal{B}$ a neighborhood of $A_0^-$. The choice of the reaction coordinate is often dictated by the problem itself, a natural one is to choose a quantity which tells how far one is from $A_0^+$. We use the following reaction coordinates

$$\begin{aligned} \Phi _l(A) = \frac{1}{2} \left( 1-\frac{\overline{A}}{\overline{A_0^+}} \right) ,~ \Phi _n(A) = \Phi _l(A) + \left| \frac{1}{\sqrt{L}} \frac{{<}A,A_1{>}}{||A_1||} \right| , \end{aligned}$$

(49)

where $\overline{A} = \int _0^L A(x)~dx$ and ${<}A,B{>} = \int _0^L A(x)B(x)dx, ||A||^2=\int _0^L A^2(x)dx$. These reaction coordinates have the property that they vanish at $A_0^+$ and are equal to 1 at $A_0^-$. The second reaction coordinate $\Phi _n$ gives some additional weights to the fronts present in A. In practice, we chose

$$\begin{aligned} \partial \mathcal {A}=\{A \backslash \Phi _n(A)=0 \}\,,\, \partial \mathcal {B}=\{A \backslash \Phi _n(A)=1 \}\,,\,\mathcal {C}=\{A \backslash \Phi _n(A)=0.05 \}\,. \end{aligned}$$

(50)

For domain of large size ($L=60$), both values $\Phi _n=0.05$ and $\Phi _n=0.025$ were tested. The two definitions led to differences in computed quantities ($\widehat{\alpha }$, $\tau$) clearly smaller than their variance. For this reason, we consider than this change of definition of $\mathcal {C}$ does not play any role on the final result. However, this change plays a clear role in the computation time of the generation of initial conditions distributed on $\mathcal {C}$, especially when $\beta$ or L are large. For this reason, $\Phi _n=0.025$ was systematically used for computations at $L=80$ and $L=100$.

1.3.2 Appendix 3.2: Statistics from the Output

There are two important quantities which can be obtained from AMS output:

• Distribution of duration of reactive trajectories In [47], a numerical analysis shows the convergence of this distribution. In the case of SPDE and Allen–Cahn for example, there is no theoretical results on this distribution. In our case, one can check for instance the rate of growth of these lengths with $\beta$. More generally, since AMS is providing an ensemble of reactive trajectories, it is possible to perform many a-posteriori statistics which become accurate as N is larger.

• Mean first passage time In order to compute this quantity, one defines a isosurface $\mathcal{C}$ surrounding the set $\mathcal{A}$. One generates an ensemble of N trajectories which all start at $A_0^+$. One then estimates the mean time $\tau _1$ it takes to cross the surface $\mathcal{C}$ and $\tau _2$ the mean duration of nonreactive trajectories (that is, the length of trajectories between the hitting time of $\mathcal{C}$ and the return time to $\mathcal{A}$) [62]. Assuming that there are a mean number $<n>$ of nonreactive trajectories, the mean first passage time T is

  $$\begin{aligned} T = <n>(\tau _1+\tau _2) + (\tau _1+\tau _r), \end{aligned}$$

  (51)

  where $\tau _r$ the mean duration of reactive trajectories found by the algorithm. The number of failed attempts $<n>$ is in fact related to $\alpha$ by $\alpha = \frac{1}{<n> + 1}$.

1.3.3 Appendix 3.3: Convergence Issues

There are two sets of parameters to take into account: first, the SPDE discretisation with timestep dt and spatial resolution dx, second, the number N of the trajectories in the algorithm and the choice of the reaction coordinate $\Phi$. We first illustrate the convergence properties of the SPDE itself. We use a test case of $L=7.07$ using $N=20~000$ trajectories and the norm reaction coordinate $\Phi _n$ (which yields better results than $\Phi _n$ [47]). We increase the spatial resolution from $dx=\frac{1}{6}$ to $dx=\frac{1}{18}$ and decrease the timestep dt accordingly. Note that, in order to ensure the stability of the numerical integration of the diffusive part, the constraint is to have $D > \frac{dt}{dx^2}$. The choice of dt in addition to this constraint crucially depends on the accuracy of hitting times (of $\mathcal{A}$ and isosurfaces of the reaction coordinate $\Phi$) needed in the algorithm. It is well-known that these hitting times converge like $\sqrt{dt}$ [63] bringing a limit to the accuracy of the solution and the probability $\alpha$ found by the algorithm as well (see [47]).

Estimates of $\ln \alpha$ (Fig.14a) and the average duration $\tau _r$ of reactive trajectories (Fig.14b) are obtained for the different parameter cases. The result is that increasing the spatial resolution does not improve the accuracy of $\ln \alpha$ whereas it improves the accuracy on the average duration $\tau _r$ of reactive trajectories (Fig. 14b). Nevertheless, the qualitative behavior of $\tau _r$ is robust to the parameter dx and dt. In this article, we have decided to keep the values $dx = \frac{1}{6}$ and $dt = 10^{-2}$ in all the simulations.

Fig. 14

a Logarithm of the escape probability as a function of $\beta$ for $L=7.07$, $N=20{,}000$, $\Phi _n$ and three different values of dt and dx. b Average duration of reactive trajectories, in the same setting, as a function of $\ln (\beta )$ for the same three resolutions. c Comparison of the rescaled variance $\sigma _0$ of the escape probability as a function of the logarithm of the escape probability, for the two reaction coordinates $\Phi _l$ and $\Phi _n$. Both Allen–Cahn equation and the two-saddle model (see Eq. (53)) are shown (Color figure online)

We now discuss the effects of the number N of initial trajectories and the choice of the reaction coordinate $\Phi$ in the quality of the results. In practice, for most choices of $\Phi$, the results converge in the limit $N \rightarrow \infty$. However, the bias and variance of the escape probability estimate can behave rather differently depending on $\beta$ and the choice of $\Phi$ [47]. We look here at the relative variance

$$\begin{aligned} \sigma _0 = \sigma \frac{\sqrt{N}}{\alpha \sqrt{-\ln \alpha }} \end{aligned}$$

(52)

as a function of $\beta$ and for the two reaction coordinates $\Phi _l$ and $\Phi _n$. Here $\sigma$ is the variance of the estimate of $\alpha$ obtained by the algorithm. Note that in dimension 1 or when $\Phi$ is the committor, we have $\sigma _0 = 1$ ([41, 45]). In order to simplify a bit the discussion, we also perform an analysis on the following system with two degrees of freedom.

$$\begin{aligned} d(x,y) = -\nabla V dt + \sqrt{\frac{2}{\beta }} (dW_1,dW_2),\quad V(x,y) = \frac{x^4}{4}-\frac{x^2}{2} -0.3\left( \frac{y^4}{4}-\frac{y^2}{2} + x^2 y^2\right) . \end{aligned}$$

(53)

The main interest of this reduced model is that it displays a similar behavior of $\sigma _0$ as the escape probability $\alpha$ becomes small. The reaction coordinates for (53) are $\Phi _l(x,y) = \frac{x+1}{2}$ and $\Phi _n(x,y) = \frac{1}{2} \sqrt{(x+1)^2+y^2/2}$. We used the same definitions of $\mathcal {A}$, $\mathcal {B}$ and $\mathcal {C}$ as Eq. (50). Note that this constitutes a slight change in the definition of $\mathcal {A}$ and $\mathcal {B}$ compared to what was used in [47]. Figure 14c shows the behavior of $\ln \alpha$ as a function of $\beta$ for both Allen–Cahn equation and the reduced model.

For the linear reaction coordinate $\Phi _l$, the variance of the estimate distribution increases with $\beta$ and has a bias going to zero when $N \rightarrow \infty$. This bias increases when $\beta \rightarrow \infty$. For the Euclidian reaction coordinate $\Phi _n$, one obtains better results in the small model with a smaller variance and bias. These results appear to be generic among models [47] and is the subject of further theoretical investigations. Based on these results, we perform all the Allen–Cahn simulations using the reaction coordinate $\Phi _n$ in (49). These are done with a rather small number of trajectories $N=1000$, when displaying qualitative properties, and $N = 20~000$ for more quantitative results.