Rare Events in Stochastic Partial Differential Equations on Large Spatial Domains

Vanden‐Eijnden, Eric; Westdickenberg, María G.

doi:10.1007/s10955-008-9537-8

Cited by 20 publications

(30 citation statements)

References 46 publications

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“…With these notations, the nucleation rate formula of system (2.15) can be derived following [34,30] as (also see [27])…”

Section: Nucleation Rate Formulamentioning

confidence: 99%

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Nucleation Rate Calculation for the Phase Transition of Diblock Copolymers under Stochastic Cahn--Hilliard Dynamics

Li¹,

Zhang²,

Zhang³

2013

Multiscale Model. Simul.

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Abstract. We focus on the nucleation rate calculation for diblock copolymers by studying the two-dimensional stochastic Cahn-Hilliard dynamics with a Landau-Brazovskii energy functional. To do this, we devise the string method to compute the minimal energy path of nucleation events and the gentlest ascent dynamics to locate the saddle point on the path in Fourier space. Both methods are combined with the semi-implicit spectral method and hence are very effective. We derive the nucleation rate formula in the infinite-dimensional case and prove the convergence under numerical discretizations. The computation of the determinant ratio is also discussed for obtaining the rate. The algorithm is successfully applied to investigate the nucleation from the lamellar phase to the cylinder phase in the mean field theory for diblock copolymer melts. The comparison with projected stochastic Allen-Cahn dynamics is also discussed.Key words. Cahn-Hilliard equation, diblock copolymer, Landau-Brazovskii energy, string method, nucleation rate, functional determinant ratio AMS subject classification. 65C20 DOI. 10.1137/120876307 1. Introduction. In the past few decades, diblock copolymers [22,12] have been extensively investigated due to their interesting physical and chemical features as well as widespread applications in material science [13,6]. Roughly speaking, the studies of copolymers mainly focus on two kinds of issues: determining equilibrium phases and studying the dynamical behaviors. For the first issue, it has been known for a long time that the diblock copolymers admit various ordered phase separation in the microscale. Several kinds of stable ordered phases, such as the lamellar phase, cylinder phase, and spherical phase, etc., have been discovered in experiments, and lots of work has been done to determine the phase diagram [33,23]. For the second issue, people try to understand the dynamical behaviors of the diblock copolymers, for example, the interfacial motion [19,24], spinodal decomposition phenomena [28], and nucleation events between different kinds of ordered phases [31,21,5].In this paper, we study the nucleation rate calculation for diblock copolymers between the lamellar and cylinder phases with a Landau-Brazovskii energy functional, and the random fluctuation is assumed to be of stochastic Cahn-Hilliard type. In this model, it is supposed that diblock copolymer melts contain lots of chains, each of which contains monomers of types A and B linked together (see Figure 1.1). On each chain, the composition ratios of these two kinds of monomers are f A and f B , respectively, which satisfy f A + f B = 1. The order parameters φ A (r) and φ B (r) characterize

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“…With these notations, the nucleation rate formula of system (2.15) can be derived following [34,30] as (also see [27])…”

Section: Nucleation Rate Formulamentioning

confidence: 99%

“…The metastable states of dynamical system ∂φ ∂t = −P δF δφ (5.5) could be located using the semi-implicit spectral method [4]: [30,34], the nucleation rate of system (5.4) can be obtained as…”

Section: Comparison With the Projected Allen-cahn Dynamicsmentioning

confidence: 99%

Nucleation Rate Calculation for the Phase Transition of Diblock Copolymers under Stochastic Cahn--Hilliard Dynamics

Li¹,

Zhang²,

Zhang³

2013

Multiscale Model. Simul.

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“…Related results on the Allen-Cahn equation can be found in [11,43]. Notice, however, that in the Allen-Cahn case the result is much easier, since it is known a priori that the constants ±1 are the global minimizers of the energy, and the heteroclinic connection between these homogeneous states is well-understood.…”

Section: Introductionmentioning

confidence: 91%

“…In the next step one has to derive an estimate for the numerator in (43). Due to the strong Markov property, one obtains for arbitrary u 0 ∈ Γ(a) the identity Hence it suffices to bound the probability…”

Section: Exit From the Deterministic Attracting Setmentioning

confidence: 99%

Nucleation in the one-dimensional stochastic Cahn-Hilliard model

Blömker¹,

Gawron²,

Wanner³

2010

Discrete &Amp; Continuous Dynamical Systems - A

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Despite their misleading label, rare events in stochastic systems are central to many applied phenomena. In this paper, we concentrate on one such situationphase separation through homogeneous nucleation in binary alloys as described by the stochastic partial differential equation model due to Cahn, Hilliard, and Cook. We show that in the limit of small noise intensity, nucleation can be explained by the stochastically driven exit from the domain of attraction of an asymptotically stable homogeneous equilibrium state for the associated deterministic model. Furthermore, we provide insight into the subsequent nucleation dynamics via the structure of the attractor of the model in the absence of noise.

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“…For the simulations shown in the last section, we usually encounter many small droplets in the domain Ω, or equivalently, many droplets of size of order one on a large domain. Rather than trying to find equilibrium solutions which contain such a large number of droplets, we take the point of view used in [11,34] that understanding bulk effects of many droplets on a large domain can be accomplished by considering small domains which sustain either a single droplet or a small number of droplets. Applying the scaling argument again shows that this is equivalent to studying the fixed domain Ω = (0, 1) 2 , but for relatively small values of λ, meaning that this is computationally feasible.…”

Section: Equilibria For the Deterministic Systemmentioning

confidence: 99%

Degenerate Nucleation in the Cahn--Hilliard--Cook Model

Blömker¹,

Sander²,

Wanner³

2016

SIAM J. Appl. Dyn. Syst.

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Phase separation in metal alloys is an important pattern forming physical process with applications in materials science, both for understanding materials structure and for the design of new materials. The Cahn-Hilliard equation is a deterministic model for the dynamics of alloys which has proven to be fundamental for the understanding of several types of phase separation behavior. However, stochastic effects occur in any physical experiment, and thus need to be incorporated into models. While white noise is one standardly chosen option, it is not immediately clear in which way the noise characteristics affect the resulting patterns. In this paper we study the effects of not necessarily small colored noise on pattern formation in a stochastic Cahn-Hilliard model in the nucleation regime. More precisely, we focus on degenerate noise which acts either on isolated eigenmodes, or with a well-defined spatial wavelength. Our studies show that the types of resulting patterns depend critically on the spatial noise frequency, and we can explain via numerical continuation methods that if this frequency is too high, then pattern formation is significantly impaired by the underlying structure of the system. In addition, we provide rigorous bounds on the nucleation time frame in the degenerate stochastic setting.

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Rare Events in Stochastic Partial Differential Equations on Large Spatial Domains

Cited by 20 publications

References 46 publications

Nucleation Rate Calculation for the Phase Transition of Diblock Copolymers under Stochastic Cahn--Hilliard Dynamics

Nucleation Rate Calculation for the Phase Transition of Diblock Copolymers under Stochastic Cahn--Hilliard Dynamics

Nucleation in the one-dimensional stochastic Cahn-Hilliard model

Degenerate Nucleation in the Cahn--Hilliard--Cook Model

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