Growth laws for phase ordering

Bray, A. J.; Rutenberg, Andrew D.

doi:10.1103/physreve.49.r27

Cited by 144 publications

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“…The differences between local and global conservation laws can be clearly seen in the late time behavior after a quench, which must be governed by the same non-linear dynamics as the early-time behavior. As discussed in a unified treatment [5] of (3), and in agreement with previous results [2], the growth laws are L(t) ∼ t 1/3 for (locally) conserved scalar quenches, and L(t) ∼ t 1/2 for non-conserved and globally constrained quenches, where t is the time since the quench. L(t) also describes the radius of the spheres in the previous paragraph, evolving by (3), where t is the time to annihilation.…”

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Comment on “Theory of Spinodal Decomposition”

Rutenberg

1996

Phys. Rev. Lett.

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The basis of Goryachev's analysis [1] of conserved scalar phase-ordering dynamics, to apply only the global constraint ψdx = const., is incorrect. For physical conserved systems, which evolve by mass transport, the stronger local conservation law embodied by the continuity equationis the appropriate one to use [2]. Even as an approximation, the global constraint is inadequate [3]. The standard evolution equation for systems with conserved dynamics iswhere2 is the effective free energy. These dynamics satisfy the local conservation law (1), and are motivated phenomenologically by a current j = −∇δF/δψ. At very early times after a quench from a disordered state, gradients will be large and higher order gradient terms will be needed. Other disagreements with (2) can stem, for example, from hydrodynamic, thermal, and stress relaxation effects. These indicate important extensions needed to (2) and F [ψ], however the local conservation (1) will still apply in all of these cases.A special initial condition emphasizes the differences in the microscopic evolution of local vs. global conservation, where we only require that the dissipative dynamics be invariant under ψ → −ψ and that F [ψ] is minimized by ψ = ±1 everywhere. Consider two half spaces, antisymmetric about a static flat domain wall, one of which has ψ = 1 everywhere except for a small sphere where ψ = −1, the other of which has ψ = +1 and −1 respectively. For spheres far from the domain wall, under local conserved dynamics the total magnetization of each half space will be constant as the spheres evolve. However with only global conservation, always satisfied by the symmetry of the problem, the dynamics are identical to non-conserved dynamics and the magnetization of each half-space will evolve in time and will eventually saturate. This is clearly inconsistent with a local conservation law.The differences between the global constraint and a local conservation law is also made clear by a class of dynamics introduced by Onuki [4] that includes both cases. In Fourier space we havewhere σ = 2 is the locally conserved dynamics of (2), σ → 0 + imposes the global constraint discussed by Goryachev, and σ = 0 is non-conserved dynamics. The differences between local and global conservation laws can be clearly seen in the late time behavior after a quench, which must be governed by the same non-linear dynamics as the early-time behavior. As discussed in a unified treatment [5] of (3), and in agreement with previous results [2], the growth laws are L(t) ∼ t 1/3 for (locally) conserved scalar quenches, and L(t) ∼ t 1/2 for non-conserved and globally constrained quenches, where t is the time since the quench. L(t) also describes the radius of the spheres in the previous paragraph, evolving by (3), where t is the time to annihilation.We can also consider long-range interactions within the effective free-energy F [ψ]. These are relevant both for attractive [5] and for repulsive [6], or competing, interactions. The free-energy should enter into the dynamics the same way, indep...

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“…These are relevant both for attractive [5] and for repulsive [6], or competing, interactions. The free-energy should enter into the dynamics the same way, independently of any long-range interactions.…”

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Comment on “Theory of Spinodal Decomposition”

Rutenberg

1996

Phys. Rev. Lett.

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“…Γ is a kinetic transport coefficient setting the time scale. The Ginzburg-Landau equation in its deterministic form has been used both in the context of phase ordering kinetics [10] and in its complex form in the study of pattern formation [11]. In the noisy case for a finite system the equation has been studied in [12]; see also an analysis of the related φ 4 theory in [13].…”

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Soliton-dynamical approach to a noisy Ginzburg-Landau model

2003

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We present a dynamical description and analysis of non-equilibrium transitions in the noisy Ginzburg-Landau equation based on a canonical phase space formulation. The transition pathways are characterized by nucleation and subsequent propagation of domain walls or solitons. We also evaluate the Arrhenius factor in terms of an associated action and find good agreement with recent numerical optimization studies.PACS numbers: 05.45.Yv,64.60.Qb, 75.60.Jk Phenomena far from equilibrium are widespread including turbulence in fluids, interface and growth problems, chemical and biological systems, and problems in material science and nanophysics. Here the dynamics of complex systems driven by weak noise, corresponding to rare events, is of particular interest in the context of e.g., nucleation during phase transitions, chemical reactions, and conformational changes in macromolecules. The weak noise limit is associated with a long time scale corresponding to the separation in energy scales of the thermal energy and the energy barriers between metastable states; the transition takes place by sudden jumps between metastable states followed by long waiting times in the vicinity of the states. The fundamental issue is thus the determination of the transition pathways and the associated transition rates.A particularly interesting non equilibrium problem of relevance in the nanophysics of switches is the influence of thermal noise on two-level systems with spatial degrees of freedom, see [1,2,3]. In a recent paper by E, Ren, and Vanden-Eijden [4] this problem has been addressed using the Ginzburg-Landau equation driven by thermal noise. These authors implement a powerful numerical optimization technique for the determination of the space time configuration minimizing the FreidlinWentzell [5] action and in this way determine the orbits and their associated action yielding the switching probabilities in the long time-low temperature limit. The picture that emerges from this numerical study is that of noise-induced nucleation and subsequent propagation of domain walls across the sample yielding the switch between the two metastable states.In recent work we have addressed a related problem in nonequilibrium physics, namely the Kardar-Parisi-Zhang equation or the equivalent noisy Burgers equation describing for example a growing interface in a random environment. Using a canonical phase space method derived from the weak noise limit of the Martin-Siggia-Rose functional [6,7] or directly from the Fokker-Planck equation [8,9], we have in the one dimensional case analyzed the resulting coupled field equations minimizing the action. The picture that emerges is that the transition probabilities in the weak noise limit are associated with soliton propagation and nucleation resulting from soliton collisions.In this letter we apply a soliton approach in the canonical phase space formulation to the noisy GinzburgLandau equation and attempt to account for some of the numerical results of E et al. We thus give analytical arguments for the ...

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“…It is well known that the quench of the high-temperature phase of the system to its low temperature phase (T < T c ) always leads to competing domains and their average size grows with time as t α , where α depends on the type of dynamics (without conserved order parameter) that dominates the system evolution [15,16]. In case of Ising spin systems (Eq.…”

Section: The Initial Mean Field Magnetizationmentioning

confidence: 99%

Space Dependent Mean Field Approximation Modelling

et al. 2014

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It is shown that the self-consistency condition which is the basic equation for calculating the mean-field order parameter of any mean-field model Hamiltonian can be replaced by the standard Metropolis Monte Carlo scheme. The advantage of this method is its ease of implementation for both the homogeneous mean-field order parameter and the heterogeneous one. To be specific, the mean-field version of the Ising model spin system is discussed in detail and the resulting magnetization is the same as in the case of solving the respective mean-field self-consistency equation. In addition, it is shown that if a high temperature phase of such system is quenched below critical temperature then the mean field experienced by spins develops into a network of domains in analogous way as it happens with the spins in the case of the exact many-body Hamiltonian system and the coarsening processes start to take place. To show that the introduced Metropolis Monte Carlo method works also in case of the continuous variables the order parameter for the Maier-Saupe model for nematic liquid crystals has been calculated.

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Growth laws for phase ordering

Cited by 144 publications

References 39 publications

Comment on “Theory of Spinodal Decomposition”

Comment on “Theory of Spinodal Decomposition”

Soliton-dynamical approach to a noisy Ginzburg-Landau model

Space Dependent Mean Field Approximation Modelling

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