Diffusion-Limited One-Species Reactions in the Bethe Lattice

Journal of Physics A: Mathematical and General, 2002

We study the 1D kinetics of diffusion-limited coalescence and annihilation with back reactions and different kinds of particle input. By considering the changes in occupation and parity of a given interval, we derive sets of hierarchical equations from which exact expressions for the lattice coverage and the particle concentration can be obtained. We compare the mean-field approximation and the continuum approximation to the exact solutions and we discuss their regime of validity.

Arxiv preprint cond-mat/ …, 1996

We discuss a reaction-diffusion model in one dimension subjected to an external driving force. Each lattice site may be occupied by at most one particle. The particles hop with rates (1 ± η)/2 to the right or left nearest neighbour site if it is vacant, and annihilate with rate one if it is occupied. We compute the long time behaviour of the space dependent average density in states where the initial density profiles are step functions. We also compute the exact time dependence of the particle density for uncorrelated random initial conditions. The representation of the uncorrelated random initial state and also of the step function profile in terms of free fermions allows for the calculation of time-dependent higher order correlation functions. We outline the procedure using a field theoretic approach. Finally, we show how this gives rise to predictions on experiments in TMMC exciton dynamics.

Journal of Physics A: Mathematical and General, 2005

The close similarity between the hierarchies of multiple-point correlation functions for the diffusion-limited coalescence and annihilation processes has caused some recent confusion, raising doubts as to whether such hierarchies uniquely determine an infinite particle system. We elucidate the precise relations between the two processes, arriving at the conclusion that the hierarchy of correlation functions does provide a complete representation of a particle system on the line. We also introduce a new hierarchy of probability density functions, for finding particles at specified locations and none in between. This hierarchy is computable for coalescence, through the method of empty intervals, and is naturally suited for questions concerning the ordering of particles on the line.

Journal of Physics A: Mathematical and General, 1995

... Printed in the UK Diffusion-limited coalescence with finite reaction rates in one dimension Dexin Zhongt and Daniel ben-Amaham$ Clarkson InStiNte for Statistical Physics (CISP). Physics Departmenf Clarkson University, Potsdam, NY 13699-5820, USA Received 14 July 1994 ...

The Journal of Chemical Physics, 2001

We study the dynamics of diffusing particles in one space dimension with annihilation on collision and nucleation (creation of particles) with constant probability per unit time and length. The cases of nucleation of single particles and nucleation in pairs are considered. A new method of analysis permits exact calculation of the steady-state density and its time evolution in terms of the three parameters describing the microscopic dynamics: the nucleation rate, the initial separation of nucleated pairs, 1 and the diffusivity of a particle. For paired nucleation at sufficiently small initial separation the nucleation rate is proportional to the square of the steady-state density. For unpaired nucleation, and for paired nucleation at sufficiently large initial separation, the nucleation rate is proportional to the cube of the steady-state density.

Annals of Physics, 1994

The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schrödinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions. Since many one-dimensional quantum chains are integrable, this opens a new field of applications. At the same time physical intuition and probabilistic methods bring new insight into the understanding of the properties of quantum chains. A simple example is the asymmetric diffusion of several species of particles which leads naturally to Hecke algebras and qdeformed quantum groups. Many other examples are given. Several relevant technical aspects like critical exponents, correlation functions and finite-size scaling are also discussed in detail.

2017

Diffusion-coagulation can be simply described by a dynamic where particles perform a random walk on a lattice and coalesce with probability unity when meeting on the same site. Such processes display non-equilibrium properties with strong fluctuations in low dimensions. In this work we study this problem on the fully-connected lattice, an infinite-dimensional system in the thermodynamic limit, for which mean-field behaviour is expected. Exact expressions for the particle density distribution at a given time and survival time distribution for a given number of particles are obtained. In particular we show that the time needed to reach a finite number of surviving particles (vanishing density in the scaling limit) displays strong fluctuations and extreme value statistics, characterized by a universal class of non-Gaussian distributions with singular behaviour.

Physical Review A, 1992

We report on the spatial distribution of particles in the reaction A +B~0. For the spatial dimension d 4, this process exhibits anomalously slow kinetics which stems from the formation of a mosaic of continuously growing domains which contain only one of the two species. We investigate the temporal evolution of the distribution of domain sizes, as well as the distribution of interparticle distances between closest-neighbor particles, both between the same and opposite species. Our results are considerably richer than might at first be expected. The average distance between closest-neighbor AB pairs scales differently than the corresponding distance between same-species pairs. The full distribution of A A separations is found to reflect the competing influences of these two length scales. Many of our observations can be accounted for in terms of simple scaling arguments. Rather surprisingly, many of our results are drastically altered if one of the species is immobile. The spatial distribution of the immobile reactant exhibits a self-similar character, leading to complex behavior for the moments of the interparticle distance distribution.

Single shock in reaction-diffusion models on the Bethe lattice with nearest neighbor interaction is introduced. The evolution equation of the system can be solved through the full interval method in closed form exactly. Comparing the results of analytical and numerical approaches is studies. The stationary and dynamics solution of such models are discussed. It is shown that in the stationary configuration, all sites are occupied. PACs: 02.50.Ga; 02.60.Cb; 02.70.Bf; 05.40.-a; 07.50Tp.

1999

The probabilistic dynamics of a pair of particles which can mutually annihilate in the course of their random walk on a lattice is considered and analytically found for d=1 and d=2. In view of available recent experiments achieved on the femtosecond scale, emphasis is put on the necessity of a full continuous-time, discrete-space solution at all times. Quantities of physical interest are calculated at any time, including the total pair survival probability N(t) and the twoparticle correlation function. As a by-product, the lattice version allows for a precise regularization of the continuous-space framework, which is ill-conditionned for d 2; this being done, formal generalization to any real dimensionality can be straightforwardly performed.