Soliton-dynamical approach to a noisy Ginzburg-Landau model

Abstract

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 ...

Correlations, soliton modes, and non-Hermitian linear mode transmutation in the one-dimensional noisy Burgers equation

“…Here the analysis, corroborating recent numerical optimization studies, is simpler because the soliton excitations are topological and have a fixed amplitude [59].…”

Kardar-Parisi-Zhang equation in the weak noise limit: Pattern formation and upper critical dimension

“…See also an application to a nonlinear finite-time-singularity model in Refs. [91,92] and to an extended system, the noise-driven Ginzburg-Landau model, in Refs [93,94].…”

Patterns in the Kardar-Parisi-Zhang equation

“…In higher d the scaling results based on the weak noise method are still subject to scrutiny. Finally, we mention that the weak noise method has also been applied to the noise-driven Ginzburg-Landau equation, a finite-time-singularity model, and DNA bubble dynamics [9].…”