Dynamics of Kinks: Nucleation, Diffusion, and Annihilation

Abstract

We investigate the nucleation, annihilation, and dynamics of kinks in a classical (1+1)-dimensional φ 4 field theory at finite temperature. From large scale Langevin simulations, we establish that the nucleation rate is proportional to the square of the equilibrium density of kinks. We identify two annihilation time scales: one due to kink-antikink pair recombination after nucleation, the other from non-recombinant annihilation. We introduce a mesoscopic model of diffusing kinks based on "paired" and "survivor" kinks/antikinks. Analytical predictions for the dynamical time scales, as well as the corresponding length scales, are in good agreement with the simulations.PACS numbers: 63.75.+z, 64.60.Cn Many extended systems have localized coherent structures that maintain their identity as they move, interact and are buffeted by local fluctuations. The statistical mechanics of these objects has diverse applications, e.g., in condensed matter physics [1], biology [2], and particle physics [3]. The model to be studied here is a kinkbearing φ 4 field theory in (1 + 1) dimensions, popular because its properties are representative of those found in many applications. Static equilibrium quantities of this theory, such as the kink density and spatial correlation functions, are now well understood and recent work has shown that theory and simulations are in good agreement [4][5][6]. However, dynamical processes, both close to and far out of equilibrium, are much less well understood. Questions include: What is the nucleation rate of kinkantikink pairs? How is an equilibrium population maintained? How do these dynamical processes depend on the temperature and damping? These questions, among others, are the subject of this Letter.We introduce and analyze below a simple model of kink diffusion and annihilation that predicts the nucleation rate and provides a picture of the physical situation, including the existence of multiple time and length scales. We also carry out high resolution numerical simulations. As one consequence of our work, we are able to settle a recent controversy as to whether the nucleation rate of kinks in an overdamped system is proportional to exp(−2E k β) [7] or exp(−3E k β) [8] in favor of the first result (E k is the kink energy and β = 1/k B T ).We consider the dynamics of the φ 4 field obeying the following dimensionless Langevin equation [4]:with the fluctuation-dissipation relation enforced byWe perform simulations on lattices typically of 10 6 sites, using a finite difference algorithm that has second-order convergence to the continuum [6]. Typical values of the grid spacing and time step are ∆x = 0.4 and ∆t = 0.01.

Soliton-dynamical approach to a noisy Ginzburg-Landau model

“…3 we have plotted S versus T for n = 1−6 domain walls using the parameter values in [4]. Choosing S nuc according to (13) we find excellent agreement with the numerical results. As also discussed in [4] we note that the switching scenario depends on T .…”

“…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]. In the present problem the noisy equation provides a generalization of the classical Kramers problem [14] to spatially extended systems.…”

Coarsening, nucleation, and the marked Brownian web

“…For more background on the non-noisy case in the contexts of zero-temperature Ising/Potts models and diffusion limited reactions, see [10,11,12]. The scaling limits for voter models, with or without noise, expressed in terms of Brownian webs with or without marks, should be the same scaling limits one gets for certain stochastic PDE models that arise in a variety of physical settings, e.g., those that describe nucleation, diffusion and annihilation of coherent structures (kinks) in a regime where they can be regarded as pointwise objects-see, e.g., [13] (also [14]) and references therein. This happens in the stochastic GinzburgLandau equation in the limit of small noise and large damping [15] or in a classical (1 + 1)-dimensional φ 4 field theory at finite (low) temperature [13].…”

Reaction front in anA+B→Creaction-subdiffusion process

“…It is simpler for the A + A problem because a number of appropriate non-chemical species can be identified that essentially undergo the simplest annihilation reaction or variants thereof. Examples include exciton annihilation experiments in one-dimensional pores and in effectively one-dimensional polymer wires [15], excited molecule naphthalene fusion and quenching experiments in one-dimensional pores [27], and kink-antikink simulations in one dimension [28]. Experimental observations of the A + B anomalies instead generally involve reaction fronts.…”