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SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 7, No. 1, pp. 207–219

Numerical Experiments on Noisy Chains: From Collective Transitions to Nucleation-Diffusion∗ Mario Castro† and Grant Lythe‡ Abstract. We consider chains of particles with nearest-neighbor coupling, independently subjected to noise, all initially in the same well of a symmetric double-well potential. If there are sufficiently few particles, transitions from one well to another are “collective”; i.e., all particles remain close together as they make the passage from one well to the other. In longer chains, only a fraction of the particles make an initial transition, creating a nucleated region that may grow or collapse by diffusion of its boundaries. Numerical experiments are used to explore the change of the scaling of the passage time as a function of the length of the chain, which distinguishes the two regimes. A suitable relationship between the noise amplitude, coupling, and number of particles in the chain yields convergence to the continuum φ4 or Allen–Cahn stochastic partial differential equations in one space dimension. We estimate the characteristic width of newly nucleated regions and construct a numerical effective potential describing the dynamics in the nucleation-diffusion regime. Key words. stochastics, nucleation, passage time AMS subject classifications. 60H15, 82C05 DOI. 10.1137/070695514

1. Introduction. Many spatially extended systems exhibit two locally stable states that coexist in the sense that, at any one time, different parts of the system are in different states. Nucleation events are fluctuation-driven transitions of part of the system from one state to another, creating domains in which the system is in one state. The domain boundaries are coherent structures [1, 2] that subsequently move about due to fluctuations [3]. The model system we study here is homogeneous and symmetric; i.e., there is no preferred part of the system, and, in the long run, each of the two states is present in equal proportion on average. These properties also mean that there is no a priori natural lengthscale of a newly nucleated region. Behind the apparent simplicity of the energy landscape lies a challenging problem of determining the most probable nucleation pathways [4, 5, 6]. In this article, we consider the model system of a chain of overdamped particles, each coupled to its two neighbors, subject to noise and to the double-well potential with minima at φ = ±1: (1.1)

1 1 V (φ) = − φ2 + φ4 . 2 4

∗ Received by the editors June 26, 2007; accepted for publication (in revised form) by A. Hagberg November 19, 2007; published electronically March 19, 2008. This work has been partially supported by the DGI of the Ministerio de Educaci´ on y Ciencia, Spain, through grant FIS2006-12253-C06-06. http://www.siam.org/journals/siads/7-1/69551.html † GISC, Escuela T´ecnica Superior de Ingenier´ıa (ICAI), Universidad Pontificia Comillas, E-28015 Madrid, Spain ([email protected]). ‡ Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, U.K. ([email protected]).

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The position of the ith particle at time t is a real-valued random variable Φt (i), and the stochastic differential equation for the ith particle is   (1.2) dΦt (i) = Φt (i) − Φ3t (i) + k(Φt (i − 1) + Φt (i + 1) − 2Φt (i)) dt + (2/β)1/2 dBt (i), where E(dBt (i)dBt (j)) = δi−j dt. The index i runs from 1 to N , and we shall always use periodic boundaries. In our numerical experiments, the whole chain is initially in the left-hand well of the potential. We record the first t > 0 at which the chain is in the right-hand well. As a function of N , with k and β fixed, we find that the mean of this time, which we call the complete passage time, increases exponentially until a certain value and then increases less rapidly. This change corresponds to a transition from “collective” behavior, where all the particles surmount the potential barrier together, to “nucleation-diffusion” behavior, where only a subset of the chain makes the initial transition and the domain subsequently grows by diffusion of its boundaries. A Java applet that permits interactive numerical experiments is at http://www.maths.leeds.ac.uk/Applied/stochastic/chain.htm. Although a discrete system is an appropriate model in many situations, an important reason for interest in (1.2) is as a finite-difference approximation of a continuum stochastic partial differential equation (SPDE) in one space dimension. Let (1.3)

k = Δx−2 ,

N=

L , Δx

and

β=

Δx . Θ

As Δx → 0, the limit of the set of equations (1.2) is the overdamped φ4 SPDE [7, 8, 1, 9] (1.4)

∂2 ∂ φt (x) = φ (x) + φt (x) − φ3t (x) + (2Θ)1/2 ξ t (x), ∂t ∂x2 t

where φt (iΔx) = Φt (i) and x ∈ [0, L]. The last term in (1.4) is space-time white noise: (1.5)

E(ξ t (x)ξ t (x )) = δ(x − x )δ(t − t ),

where Θ is thought of as proportional to temperature. We will often refer to (1.3) as it provides scaling relations among the parameters of the problem and it will help us to understand the nucleation process in terms of adimensional relations. If, instead of (1.3), we let N = Δx−1 with β and k fixed then, as Δx → 0, we obtain the stochastic Allen–Cahn equation: (1.6)

∂ ∂2 φt (x) = 2 2 φt (x) + φt (x) − φ3t (x) + σ ξ t (x), ∂t ∂x

where 2 = k/N 2 , σ = (2Δx/β)1/2 , and x ∈ [0, 1]. In the SPDEs, a configuration is a continuous function of x, φt (x), obtained by fixing t in one realization. At most values of x, φt (x) is close to either −1 or +1. A narrow region where the configuration crosses through 0 from below is called a kink; one where it crosses from above is called an antikink. The width of a kink in the φ4 SPDE is order 1; in the Allen–Cahn case it is proportional to  [10, 11, 12]. We shall concentrate on the φ4 SPDE below. After a sufficiently long time, in both the continuum SPDEs and the discrete system, a statistically steady state is attained and maintained by a balance between continual nucleation of new domains and the diffusion and annihilation of existing ones [13, 14, 3]. Many steadystate quantities, such as the mean number of kinks per unit length, can be calculated from

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the invariant density of the SPDE, by evaluating the partition function [15, 16, 17]. Further insight has recently been obtained by demonstrating the equivalence between the invariant density of paths of the SPDE, on the spatial domain [0, L], and the density of paths of a suitable bridge process [18, 19], with time in the interval [0, L]. Simple models, in which kinks and antikinks are nucleated with a fixed separation, diffuse and annihilate on collision and give insight into the dynamics that produces and maintains the stationary density [20, 21, 22, 23]. If kink-antikink dynamics in our symmetric system is described in terms of a potential as a function of separation, a nucleation event is a fluctuationinduced escape from a well to a flat region. B¨ uttiker and Christen calculated the nucleation rate by introducing a parameter analogous to a nucleus size s [20]. The mean kink lifetime is proportional to s and the nucleation rate is inversely proportional to s, so that the steady-state kink density is independent of s. More detailed models also calculate the distribution of kink lifetimes [3, 21, 23]. The initial separation at the instant of nucleation is an input parameter in these simple models; in order to calculate the appropriate value in the SPDE dynamics, it is necessary to return the focus to the details of the nucleation process. 1.1. The energy landscape. In chains sufficiently short that transitions are collective, the complete passage time is most conveniently calculated by considering the energy function of the discretized system: (1.7)

E(Φt (1), . . . , Φt (N )) =

N   i=1

 1 2 V (Φt (i)) + k (Φt (i) − Φt (i − 1)) . 2

The initial condition is at the energy minimum: E(−1, . . . , −1) = −N/4. If the transition is collective, crossing the saddle point on the energy surface at the origin E(0, . . . , 0) = 0 involves surmounting an energy barrier 14 N ; the mean time for such a transition is proportional to exp( 41 βN ). On the other hand, if only a part of the system makes the initial transition, the energy barrier is less than 14 N ; the nucleation event produces a region with two boundaries. In the latter case, which we call the “nucleation-diffusion regime,” it is convenient to calculate the nucleation rate working in the continuum limit, (1.4), where the analogue of (1.7) is the energy functional [4]  2   1 ∂ (1.8) E[φt ] = φ dx. V (φt ) + 2 ∂x t

The energy of a kink is Ek = E[ψ] = 8/9 [1]. A kink or antikink has energy E0 ; a transition 0 creates one of each and therefore has energy barrier 2E0 and characteristic time exp( 2E Θ ) [24, 25, 26, 27, 15, 4]. The numerical value of the kink energy depends on the precise potential used, but the qualitative dynamics requires only a double-well potential with wells of equal depth. It is also possible to study SPDEs where V (φ) has two (or more) wells of unequal depth, so that there are stable and metastable states. Then, one can base calculations of nucleation rates on the idea of a critical nucleus or droplet [27, 28, 29, 30, 31], an extremum of (1.8), whose length diverges as the asymmetry between wells vanishes. In an interval containing only one kink, centered at Xt , the configuration can be written as φt (x) = ψ(x − Xt ) + (Θ/Ek )1/2 χ(x − Xt ), where ψ is a smooth function, satisfying

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MARIO CASTRO AND GRANT LYTHE

V  (ψ) + ψ  (x) = 0, with ψ(x) → ±1 as x → ±∞, that gives the shape of an unperturbed kink. The fluctuating stochastic field χ then has stationary statistical properties [32, 33]. The idea that collective transitions are favored in small domains but another mechanism is at work in larger domains has also emerged from numerical and analytical studies of minimumaction transition paths [4, 34, 6, 5]. Optimal transitions consist of nucleation events followed by propagation of domain walls, with behavior depending on L and on the (periodic, Dirichlet, or Neumann) boundary conditions. In this work we locate, by simple numerical experiments, the crossover from collective transitions to nucleation-diffusion behavior. In the latter regime, the characteristic width of newly nucleated regions is b = 8E0 and the complete passage time is proportional to exp(2E0 /Θ). This value of b, obtained from the slope of the logarithm of the complete passage time versus Θ−1 , is consistent with numerical observations of the critical value of L at which the crossover from collective to nucleation-diffusion behavior is found, and with the long-held hypothesis that the separation between a kink and an antikink at nucleation is several times the kink width. Systematic computational studies of the φ4 SPDE, in the limit L → ∞, require low temperatures in order to unambiguously identify kinks; they are computationally costly because the steady-state density of kinks decreases exponentially with temperature (necessitating very long chains) and the equilibration time increases exponentially with temperature (necessitating very long runs). Our aim in this paper is complementary to such studies. We focus on the dynamics of small- to medium-length chains in order to distinguish the regimes of collective transitions and of nucleation-diffusion. Although a first-principles theory of nucleation is the most difficult theoretical challenge, numerical studies that focus only on the measurement of the nucleation rate have the advantage of not needing to first attain a steady state. In section 2 we define the complete passage time, our adopted measure of the mean time for the whole chain to make the transition from one well to another. The key observation is that there is a critical number of particles in the chain, below which the complete passage time increases exponentially and above which it increases more slowly. In the course of deriving theoretical expressions for the complete passage time, we are led to consider the short-range interaction between kinks and antikinks. This is examined in a second set of numerical experiments, differing from the first in the initial conditions, which now have a kink and an antikink relatively close together. Also, in section 2, we use numerical results to make an estimate of the typical separation, b, of a kink-antikink pair at nucleation. In section 3 we report on numerical experiments where the distribution of the “center-of-mass” of the chain is measured by means of long numerical runs and displayed in terms of an “effective potential” that has the B¨ uttiker–Christen form of two wells separated by a long flat region. 2. Complete passage time. Our first set of numerical experiments measures the time taken for all particles to make the transition from one minimum to the other. We choose the initial condition Φ0 (i) = −1, i = 1, . . . , N , and denote  N  (2.1) h = inf t > 0 : Φt (i) = N . i=1

The complete passage time, τ , is defined as the mean of h: τ = E(h).

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Figure 1. Realizations of the coupled-particle system with β = 6 and k = 1, illustrating the two routes to a complete passage of the chain from one well to another. Upper timeseries: N = 5, collective regime; Lower timeseries: N = 50, nucleation-diffusion regime.

In Figure 1, the sum i Φt (i) is plotted as a function of time for two realizations. In the top part of the figure, N = 5, β = 6, and k = 1. The transition from one well to the other is collective; all particles in the chain make the transition close together. The second realization, shown in the lower part of the figure, has N = 50, β = 6, and k = 1. Several episodes are visible where the sum increases and then falls back to its starting level. These correspond to nucleation-diffusion episodes: a group of nearby particles makes the transition to the upper well, creating a region with two boundaries. The boundaries diffuse until the region either disappears or encompasses the whole chain. Figure 2 shows τ versus N for k = 1 and four values of β. A data point typically corresponds to 103 realizations. We observe, in numerical experiments of this type, that the critical number of particles, at which the crossover from collective to nucleation-diffusion behavior is found, is independent of β if k is fixed. In Figure 3, three graphs of τ against N are displayed with different values of k; all have β = 6. The critical number of particles is seen to be an increasing function of k. The existence of the SPDE √ limit implies that the passage time should approach a limit as k → ∞ (Δx → 0), with L = N/ k fixed. In Figure 4, the mean passage time is plotted √ against L, using the scaled variables (1.3). Each data set has the same value of Θ = (β k)−1 . The figure shows a much greater degree of universality than we expected: the data corresponding to large and small values of k are difficult to distinguish on the scale of the figure. 2.1. Analytical expressions. We develop our analytical approximations in the continuum limit using the scaled variables (1.3). Then the complete passage time, τ (Θ, L), is a function of the temperature and the length of the domain. Transitions are collective in sufficiently short chains because there is a single, well-defined, saddle point on the energy surface, with the whole chain at 0, separating two global minima,

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MARIO CASTRO AND GRANT LYTHE

Figure 2. Complete passage time versus number of particles for k = 1. The critical number of particles is independent of β. (Statistical errors are approximately the symbol sizes.)

Figure 3. Complete passage time versus number of particles for β = 6. The critical number of particles is an increasing function of k.

with the whole chain at ±1. The probability per unit time of a transition over the saddle point is calculated by constructing a nonequilibrium steady-state density which is the equilibrium density multiplied by a function of the distance along a line connecting the minima via the saddle [24, 25, 26]. As Θ → 0 and L → 0, τ (Θ, L) → τc (Θ, L), where     1L −1 . (2.2) τc (Θ, L) = Ac (L) + O(Θ ) exp 4Θ

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Figure 4. Complete passage time versus L for Θ = 18 . The values of k = 1, k = 4, and k = 16 correspond L to Δx = 1, Δx = 0.5, and Δx = 0.25. The solid line is 4π exp(− √L2 ) exp( 14 Θ ), valid for collective transitions.

The exponent in (2.2) is the difference between the energy at the saddle point and that at the minima; the prefactor Ac (L) is sometimes referred to as a frequency factor. Its square is the ratio of the products of the eigenvalues corresponding to the N stable directions at a minimum and the N − 1 stable directions at the saddle point. Interestingly, the ratio diverges at L = 2π [34, 12, 35], corresponding to a breakdown of the collective-transition hypothesis. For our purposes, let us write     f (L) −1 (2.3) τ (Θ, L) = A(L) + O(Θ ) exp , 4Θ with A(L) and f (L) to be determined, in the collective regime, the nucleation-diffusion regime, and the crossover region between the two. According to (2.2), as L → 0, f (L)/L → 1. We conjecture that, as L increases, f (L) → b and A(L) → A∞ for constants b and A∞ to be determined. This choice of functional form will be justified a posteriori by comparison with numerical data. Next, consider the situation for L > b. Here, we make the hypothesis that nucleation events, equally likely to happen anywhere, always produce a region of width b. In other words, there is a constant probability per unit length and time, Γ = (bτn (Θ, b))−1 , of a nucleation event. Let the probability that a region of width x grows to encompass the whole domain be q(x, L). Then 1 b (τn (Θ, b) + τd (Θ, b, L)) , L > b, τ (Θ, L) = L q(b, L) where τd (Θ, b, L) is the mean time spent during a diffusion episode. It is easy to calculate an upper limit on τd (Θ, b, L) by ignoring the short-range kink-antikink attraction, assuming that each diffuses with diffusivity Θ/E0 [32, 36]:

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τd (Θ, b, L)
0, reaches L before 0. Thus, at first approximation, q(x, L)  q0 (x, L), where (2.5)

q0 (x, L) =

x . L

In order to ascertain the effect on q(x, L) of the short-term interaction between a kink and an antikink, we performed a set of numerical experiments with initial conditions corresponding to a domain of width x:  1, i ≤ n, Φ0 (i) = −1, n < i ≤ N, √

where n = x/ k. Realizations were stopped the first time that either N i=1 Φt (i) = 0 or

N i=1 Φt (i) = N . The probability of the latter outcome is plotted as a function of L in Figure 5. Based on these numerical data, we devised the following approximation: (2.6)

q(b, L) 

b−a . L − 2a

The intuitive motivation for the form (2.6) is that two domain walls experience a strong attraction at separations less than a. The lines in Figure 5 are (2.5) and (2.6), using the least-squares best fit value of a, which yielded a = 5.4 ± 0.1. With (2.6), we have the following expression for the complete passage time in the nucleation-diffusion regime as Θ → 0:   1 − 2a/L 1b (2.7) τ (Θ, L) → A∞ exp . 1 − a/b 4Θ

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Figure 5. The effect of the short-range kink-antikink attraction. The probability that an initial region of width x = 10 grows to encompass the entire domain of length L is plotted against L for k = 1, k = 4, and k = 9. In all cases Θ = 0.01. The dotted lines are (2.5), which would hold if there were no attraction, and (2.6).

2.3. Estimating the width b. By comparing the expression (2.7) with numerical data, we can estimate the value of the parameter b, the characteristic width of nucleation regions. In particular, (2.8)

A∞ τ (Θ, L) → exp 1 − a/b



1b 4Θ

 ,

b  L  L∗ .

That is, the logarithm of the complete passage time is a linear function of Θ−1 . The slope is proportional to b as L/b increases. Figure 6 contains four plots summarizing numerical data with L = 5, L = 10, L = 20, and L = 30. The numerical runs were carried out with k = 4. For sufficiently small L, transitions are collective and (2.2) holds, so that the slope of a graph of ln τ versus Θ−1 is proportional to L. The first set of numerical data, with L = 5, falls into this regime. As a function of L, the slope does not increase indefinitely but approaches a well-defined limit. A least-squares fit of the numerical data for large L gives b = 7.4 ± 0.1. This value is consistent with the knee in our numerical curves of complete passage time versus L and with the following argument. L The complete passage time is proportional to exp( 14 Θ ) for sufficiently small L and pro2E0 portional to exp( Θ ) for long chains. With the ansatz τ ∝ exp( 14 f (L) Θ ), • f (L)/L → 1 as L → 0, and • f (L) → 8E0 when b  L  L∗ , thus identifying the characteristic mean width of newly nucleated regions as b = 8E0 . (In the Allen–Cahn scaling (1.6), the characteristic width of a newly nucleated region is 8E0 .)

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Figure 6. Estimating the width, b, from numerical data. For sufficiently small L, the complete passage L time of a domain length L is proportional to exp( 14 Θ ); when L  b, it is proportional to exp( 14 Θb ). The solid line, a least-squares fit to the L = 40 data, has slope corresponding to b = 7.4.

3. Effective potential. Chains, whatever their length, eventually reach a stationary state where transitions are equally frequent in either direction. In the final set of numerical experiments reported here, we exploit the stationarity of the long-term dynamics by considering the “center-of-mass” process, defined as (3.1)

N 1  φt = Φt (i), N i=1

and we construct its stationary density. In practice, the time for the system to explore all regions of state space thoroughly is several orders of magnitude greater than the complete passage time. The stationary density of the center-of-mass is denoted (3.2)

ρ(φ) = lim

t→∞

d P[φt < φ]; dx

the effective potential, Veff (φ), is defined via ρ(φ) = e−Veff (φ) . At parameter values such that transitions are collective, the effective potential is not dissimilar to V (φ). In the regime of nucleation-diffusion behavior, on the other hand, a long plateau appears in the effective potential, terminated at either end by a narrow well. See Figure 7. The wells are approximately quadratic near the minima, at φ = ±(1 − 12 EΘ0 ) [37, 38, 39]; the width of each well is (Θ/(3N ))1/2 [37]. The effective potential takes a pleasingly universal form under the scaling (1.3). In Figure 8, we plot numerical results, all obtained on chains with L = 100 and Θ = 18 . The three values of k correspond to N = 100, N = 200, and N = 400.

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Figure 7. Effective potential: Numerical results with k = 1. Solid lines: β = 8. Dotted lines: β = 10. On the left, N = 5 and transitions are collective. On the right, N = 100 and the chain exhibits nucleation-diffusion behavior.

Figure 8. The effective potential for three values of k, with Θ = 18 and L = 100. The wide central plateau corresponds to the freedom of a kink-antikink separation to wander over the majority of the length of the chain without noticeably affecting the total energy.

The effective potentials, in Figure 8 and on the right in Figure 7, bear a striking resemblance to those of B¨ uttiker and Christen [14, 20], introduced to provide an intuitive understanding of the dynamics of the nucleation-diffusion regime. A nucleation event is imagined as an exit from a potential well to a long flat region, which corresponds to the freedom of the

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kink and antikink, once they have attained a certain separation, to wander over large parts of the chain without affecting the total energy of the configuration. The potential well at the opposite end of the flat region corresponds to the possibility that the nucleated region grows to encompass the whole domain; the kink and antikink meet again and annihilate. Our numerical experiments construct, in a systematic manner, a potential with these useful illustrative properties.

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