Reaction-diffusion equations for interacting particle systems

Masi, Anna De; Ferrari, Pablo A.; Lebowitz, Joel L.

doi:10.1007/bf01011311

Cited by 196 publications

(151 citation statements)

References 26 publications

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“…An other important representative of PC class appear among nonequilibrium Ising models, in which the steady state is generated by kinetic processes in connection with heat baths at different temperatures (DeMasi et al, 1985(DeMasi et al, , 1986Droz et al, 1989;Gonzaléz-Miranda et al, 1987;Wang and Lebowitz, 1988). The research of them have shown that phase transitions are possible even in 1d under nonequilibrium conditions (for a review see (Rácz, 1996)).…”

Section: The Nekim Modelmentioning

confidence: 99%

Universality classes in nonequilibrium lattice systems

2004

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This work is designed to overview our present knowledge about universality classes occurring in nonequilibrium systems defined on regular lattices. In the first section I summarize the most important critical exponents, relations and the field theoretical formalism used in the text. In the second section I briefly address the question of scaling behavior at first order phase transitions. In section three I review dynamical extensions of basic static classes, show the effect of mixing dynamics and the percolation behavior. The main body of this work is given in section four where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. In section five I continue overviewing such nonequilibrium classes but in coupled, multi-component systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion type of systems. However by mapping they can be related to universal behavior of interface growth models, which I overview in section six. Finally in section seven I summarize families of absorbing state system classes, mean-field classes and give an outlook for further directions of research.

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Section: The Nekim Modelmentioning

confidence: 99%

Universality classes in nonequilibrium lattice systems

2004

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“…1. BBM is a prototypical model of evolution, but has also been extensively used as a simple model for reaction-diffusion systems, disordered systems, nuclear reactions, cosmic ray showers, epidemic spreads amongst others [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. In one dimension, the position of the existing particles at time t constitute a set of strongly correlated variables that are naturally ordered according to their positions on the line with x 1 (t) > x 2 (t) > x 3 (t) .…”

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confidence: 99%

Universal Order and Gap Statistics of Critical Branching Brownian Motion

Ramola¹,

Majumdar²,

Schehr³

2014

Phys. Rev. Lett.

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We study the order statistics of one dimensional branching Brownian motion in which particles either diffuse (with diffusion constant D), die (with rate d) or split into two particles (with rate b). At the critical point b = d which we focus on, we show that, at large time t, the particles are collectively bunched together. We find indeed that there are two length scales in the system: (i) the diffusive length scale ∼ √ Dt which controls the collective fluctuations of the whole bunch and (ii) the length scale of the gap between the bunched particles ∼ D/b. We compute the probability distribution function P (g k , t|n) of the kth gap g k = x k −x k+1 between the kth and (k +1)th particles given that the system contains exactly n > k particles at time t. We show that at large t, it converges to a stationary distribution The statistics of the global maximum of a set of random variables finds applications in several fields including physics, engineering, finance and geology [1] and the study of such extreme value statistics (EVS) has been growing in prominence in recent years [2][3][4][5][6][7]. In many real world examples where EVS is important, the maximum is not independent of the rest of the set and there are strong correlations between near-extreme values. Examples can be found in meteorology where extreme temperatures are usually part of a heat or cold wave [8] and in earthquakes and financial crashes where extreme fluctuations are accompanied by foreshocks and aftershocks [9][10][11][12]. Nearextreme statistics also play a vital role in the physics of disordered systems where energy levels near the ground state become important at low but finite temperature [4]. In this context, the distribution of the kth maximum x k of an ordered set {x 1 > x 2 > x 3 ...} (order statistics [13]) and the gap between successive maxima g k = x k − x k+1 provides valuable information about the statistics near the extreme value. Such near-extreme distributions have recently been of interest in statistics [14] and physics [15,[17][18][19]. Although the order and gap statistics of independent identically distributed (i.i.d.) variables are fully understood [13], very few exact analytical results exist for strongly correlated random variables. In this context, random walks and Brownian motion offer a fertile arena where near-extreme distributions for correlated variables can be computed analytically [16][17][18][19].Another interesting system where order statistics plays an important role is the branching Brownian motion (BBM). In BBM, a single particle starts initially at the origin. Subsequently, in a small time interval dt, the particle splits into two independent offsprings with probability b dt, dies with probability d dt and with the remaining probability (1 − (b + d) dt) it diffuses with diffusion constant D. A typical realization of this process is shown in Fig. 1. BBM is a prototypical model of evolution, but has also been extensively used as a simple model for reaction-diffusion systems, disordered systems, nuclear reactio...

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“…Equation (7), on the other hand, appears to be an approximation to a stochastic integral equation of the type which occurs in central limit theorems for (spatially homogeneous) population dependent population processes (as discussed, for example in the book by Ethier and Kurtz [6]. The central limit theorem for the ddrc process will lead to a nonlinear stochastic partial differential equation in addition to the linear Ornstein Uhlenbeck process obtained by De Masi, Ferrari, and Lebowitz [5] or Kotelenez [19].…”

Section: Discussionmentioning

confidence: 96%

“…[9]). Fundamental results, including a law of large numbers and a central limit theorem for a reaction-diffusion, have been obtained by De Masi, Ferrari, and Lebowitz [5], based on the mathematical tools presented by Liggett [20], who also gives some background relevant to jump process models. The coupled diffusion models are characterized by a large number of independent Brownian motion particles with branching and anihilation at rates dependent upon the local particle density.…”

Section: Motivation and Relation To Other Workmentioning

confidence: 98%

A Branching Random Walk Model for Diffusion–Reaction–Convection

Hebert

1997

Advances in Mathematics

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A discrete stochastic model is introduced for populations which are diffusing, interacting, and drifting on an integer lattice in a finite-dimensional space. The model is used to construct a Markov process whose countable state space is a set of location maps, assigning individuals to their positions. The expected population sizes and population densities satisfy partial difference approximations to the nonlinear partial differential equations of diffusion reaction convection. Constructive convergence estimates give convergence results for the random process and its ensemble average. Academic Press

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Reaction-diffusion equations for interacting particle systems

Cited by 196 publications

References 26 publications

Universality classes in nonequilibrium lattice systems

Universality classes in nonequilibrium lattice systems

Universal Order and Gap Statistics of Critical Branching Brownian Motion

A Branching Random Walk Model for Diffusion–Reaction–Convection

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