Reaction-Diffusion Model for A + A Reaction

Lindenberg, Katja; Argyrakis, Panos; Kopelman, Raoul

doi:10.1021/j100019a041

Cited by 41 publications

(33 citation statements)

References 36 publications

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“…In contrast, the nucleation rate is proportional to ρ 3 k in systems where nucleation does not occur in pairs [18,19]. The latter scaling was incorrectly predicted for the φ 4 system [8], from an estimate of the annihilation rate that does not take into account paired nucleation.…”

Section: Fig 4 Recombination Time Density Only Recombinant Historimentioning

confidence: 99%

Dynamics of Kinks: Nucleation, Diffusion, and Annihilation

Habib

Lythe

2000

Phys. Rev. Lett.

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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.

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Section: Fig 4 Recombination Time Density Only Recombinant Historimentioning

confidence: 99%

Dynamics of Kinks: Nucleation, Diffusion, and Annihilation

Habib

Lythe

2000

Phys. Rev. Lett.

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“…Although the stochastic approaches currently in use differ in their results from rate equations in the limit of a small number of reactive species per grain, these approaches still incorporate an approximation that may lead to error. The reasons for the possible error have been studied in some detail in the statistical physics literature (Lin et al 1996;Hinrichsen 2000;Krug 2003;Lindenberg et al 1995;Weiss & Rubin 1983;Simon 1995), but for our purposes here, one can simplify the problem in the following manner. The basic common assumption to the previous methods utilized is that the rate of reaction is determined by the rate of hopping (or tunneling) of a hydrogen atom from one site to a nearest neighbor site multiplied by the probability of finding a reactant partner in this site.…”

Section: Introductionmentioning

confidence: 99%

Continuous-time random-walk simulation of H$\mathsf{_{2}}$ formation on interstellar grains

Chang

Cuppen

Herbst

2005

A&A

168

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Abstract. The formation of molecular hydrogen via the recombination of hydrogen atoms on interstellar grain surfaces has been investigated anew. A detailed Monte Carlo procedure known as the continuous-time random-walk method has been used. This Monte Carlo approach has two advantages over the stochastic master equation method: it treats random walk on a surface correctly, and it can easily be used for inhomogeneous surfaces. The recombination efficiency for H 2 formation as a function of surface temperature and grain size has been calculated for a variety of grain surfaces with a flux of hydrogen atoms representative of diffuse interstellar clouds. The surfaces studied include homogeneous olivine and amorphous carbon, characterized by single energies for the diffusion barrier and binding energy of H atoms, inhomogeneous versions of these two surfaces with distributions of H-atom diffusion barriers and binding energies, and a variety of mixed surfaces of olivine and carbon. For the homogeneous surfaces, we confirm that the temperature range for efficient formation of H 2 is very small. At temperatures near peak efficiency, there is little dependence on grain size. At temperatures higher than those of peak efficiency, the Monte Carlo procedure exhibits smaller efficiencies for molecular hydrogen formation than the master equation method in the limit of large grain sizes. For various types of inhomogeneous and mixed surfaces, the major effect we find is an increase in the temperature range over which the efficiency of molecular hydrogen formation is high. Efficient formation of H 2 in diffuse interstellar clouds now seems possible with inhomogeneous grains.

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“…For instance, in the A + B reaction the typical reaction term simply involves the product of the local concentrations of reactants, −ka(r, t)b(r, t) [1]. In the A+A reaction one has to be slightly more careful because in writing, for example, −ka 2 (r, t) one must be careful not to include spurious self-reaction contributions [5,8]. Once the A + A exact models were developed that did not require one to explicitly write a reaction term, it was possible to analyze the accuracy of the approximate truncations [43].…”

Section: Introductionmentioning

confidence: 99%

Reaction front in anA+B→Creaction-subdiffusion process

2004

Self Cite

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We study the reaction front for the process A + B → C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular its unusual behavior at the center of the reaction zone.

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Reaction-Diffusion Model for A + A Reaction

Cited by 41 publications

References 36 publications

Dynamics of Kinks: Nucleation, Diffusion, and Annihilation

Dynamics of Kinks: Nucleation, Diffusion, and Annihilation

Continuous-time random-walk simulation of H$\mathsf{_{2}}$ formation on interstellar grains

Reaction front in anA+B→Creaction-subdiffusion process

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