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We present the exact solution for the full dynamics of a nonequilibrium spin chain and its dual reaction-diffusion model, for arbitrary initial conditions. The spin chain is driven out of equilibrium by coupling alternating spins to two…

Statistical Mechanics · Physics 2007-05-23 M. Mobilia , B. Schmittmann , R. K. P. Zia

The scaling exponent and scaling function for the 1D single species coagulation model $(A+A\rightarrow A)$ are shown to be universal, i.e. they are not influenced by the value of the coagulation rate. They are independent of the initial…

Condensed Matter · Physics 2009-10-22 Klaus Krebs , Markus Pfannmueller , Horatiu Simon , Birgit Wehefritz

One-dimensional non-equilibrium models of particles subjected to a coagulation-diffusion process are important in understanding non-equilibrium dynamics, and fluctuation-dissipation relation. We consider in this paper transport properties…

Statistical Mechanics · Physics 2015-06-18 Jean-Yves Fortin

By considering the master equation of the totally asymmetric exclusion process on a one-dimensional lattice and using two types of boundary conditions (i.e. interactions), two new families of the multi-species reaction-diffusion processes,…

Statistical Mechanics · Physics 2013-01-15 Yaghoob Naimi , Frinaz Roshani

We establish the existence of solutions to a class of non-linear stochastic differential equation of reaction-diffusion type in an infinite-dimensional space, with diffusion corresponding to a given transition kernel. The solution obtained…

Probability · Mathematics 2021-08-10 Conrado da Costa , Bernardo Freitas Paulo da Costa , Daniel Valesin

We consider an interacting particle system $(\eta_t)_{t\geq 0}$ with values in $\{0,1\}^{\mathbb{Z}}$, in which each vacant site becomes occupied with rate 1, while each connected component of occupied sites become vacant with rate equal to…

Probability · Mathematics 2007-05-23 Xavier Bressaud , Nicolas Fournier

We consider diffusion-limited reactions A_i + A_j -> 0 (1 <= i < j <= q) in d space dimensions. For q > 2 and d >= 2 we argue that the asymptotic density decay for such mutual annihilation processes with equal rates and initial densities is…

Statistical Mechanics · Physics 2009-11-07 Olivier Deloubriere , Henk Hilhorst , Uwe C. Tauber

We consider the finite volume approximation of a reaction-diffusion system with fast reversible reaction. We deduce from a priori estimates that the approximate solution converges to the weak solution of the reaction-diffusion problem and…

Analysis of PDEs · Mathematics 2008-08-05 R. Eymard , D. Hilhorst , M. Olech

The crossing probability in the time direction is defined for an off-equilibrium reaction-diffusion system as the probability that the system of size L is still active at time t, in the finite-size scaling limit. Exact results are obtained…

Statistical Mechanics · Physics 2007-05-23 L. Turban

We consider a one-dimensional system with particles having either positive or negative velocity, which annihilate on contact. To the ballistic motion of the particle, a diffusion is superimposed. The annihilation may represent a reaction in…

Statistical Mechanics · Physics 2016-03-02 Soham Biswas , Hernán Larralde , Francois Leyvraz

Phenomenological scaling arguments suggest the existence of universal amplitudes in the finite-size scaling of certain correlation lengths in strongly anisotropic or dynamical phase transitions. For equilibrium systems, provided that…

Statistical Mechanics · Physics 2008-11-26 Malte Henkel , Ulrich Schollwöck

We present in this work a very short proof for the existence, uniqueness and smoothness in dimensions $d\leq 3$ of the system of reaction diffusion $ \partial\_t a\_i - d\_i \Delta a\_i = (-1)^i (a\_1 a\_3 - a\_2 a\_4)$, where $a\_i \geq 0$…

Analysis of PDEs · Mathematics 2026-05-25 Hector Bouton , Laurent Desvillettes , Helge Dietert

We solve an one-dimensional stochastic model of interacting particles on a chain. Particles can have branching and coagulation reactions, they can also appear on an empty site and disappear spontaneously. This model which can be viewed as…

Statistical Mechanics · Physics 2009-11-07 Mauro Mobilia , Pierre-Antoine Bares

The most general exclusion single species one dimensional reaction-diffusion models with nearest-neighbor interactions which are both autonomous and can be solved exactly through full interval method are introduced. Using a generating…

Statistical Mechanics · Physics 2012-04-27 Mohammad Khorrami , Amir Aghamohammadi

Exact solutions for nonlinear Arrhenius reaction-diffusion are constructed in $n$ dimensions. A single relationship between nonlinear diffusivity and the nonlinear reaction term leads to a nonclassical Lie symmetry whose invariant solutions…

Analysis of PDEs · Mathematics 2016-02-17 P. Broadbridge , B. H. Bradshaw-Hajek , D. Triadis

In reaction-diffusion models of annihilation reactions in low dimensions, single-particle dynamics provides a bottleneck for reactions, leading to an anomalously slow approach to the empty state. Here, we construct a reaction model with a…

Statistical Mechanics · Physics 2024-09-26 Enrique Rozas Garcia , Alfred Weddig Karlsson , Johannes Hofmann

We study diffusion-limited coalescence, A+A<-->A$, in one dimension, and derive an exact solution for the steady state in the presence of a trap. Without the trap, the system arrives at an equilibrium state which satisfies detailed balance,…

Statistical Mechanics · Physics 2009-10-31 Daniel ben-Avraham

We show that, for any spatially discretized system of reaction-diffusion, the approximate solution given by the explicit Euler time-discretization scheme converges to the exact time-continuous solution, provided that diffusion coefficient…

Optimization and Control · Mathematics 2019-12-12 Adrien Le Coënt , Laurent Fribourg

We study the kinetics of ballistic annihilation for a one-dimensional ideal gas with continuous velocity distribution. A dynamical scaling theory for the long time behavior of the system is derived. Its validity is supported by extensive…

Statistical Mechanics · Physics 2009-10-30 Pierre-Antoine Rey , Michel Droz , Jaroslaw Piasecki

Reaction-diffusion systems with reversible reactions generically display power-law relaxation towards chemical equilibrium. In this work we investigate through numerical simulations aging processes that characterize the non-equilibrium…

Statistical Mechanics · Physics 2011-06-10 Nasrin Afzal , Justin Waugh , Michel Pleimling
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