Spatial Organization in the Reaction A + B --> inert for Particles with a Drift

We describe the spatial structure of particles in the (one dimensional) two-species annihilation reaction A + B --> 0, where both species have a uniform drift in the same direction and like species have a hard core exclusion. For the case of equal initial concentration, at long times, there are three relevant length scales: the typical distance between similar (neighboring) particles, the typical distance between dissimilar (neighboring) particles, and the typical size of a cluster of one type of particles. These length scales are found to be generically different than that found for particles without a drift.Comment: 10 pp of gzipped uuencoded postscrip

Partially asymmetric exclusion models with quenched disorder

We consider the one-dimensional partially asymmetric exclusion process with random hopping rates, in which a fraction of particles (or sites) have a preferential jumping direction against the global drift. In this case the accumulated distance traveled by the particles, x, scales with the time, t, as x ~ t^{1/z}, with a dynamical exponent z > 0. Using extreme value statistics and an asymptotically exact strong disorder renormalization group method we analytically calculate, z_{pt}, for particlewise (pt) disorder, which is argued to be related to the dynamical exponent for sitewise (st) disorder as z_{st}=z_{pt}/2. In the symmetric situation with zero mean drift the particle diffusion is ultra-slow, logarithmic in time.Comment: 4 pages, 3 figure

Exact solutions for a mean-field Abelian sandpile

We introduce a model for a sandpile, with N sites, critical height N and each site connected to every other site. It is thus a mean-field model in the spin-glass sense. We find an exact solution for the steady state probability distribution of avalanche sizes, and discuss its asymptotics for large N.Comment: 10 pages, LaTe

Kinetics of A+B--->0 with Driven Diffusive Motion

We study the kinetics of two-species annihilation, A+B--->0, when all particles undergo strictly biased motion in the same direction and with an excluded volume repulsion between same species particles. It was recently shown that the density in this system decays as t^{-1/3}, compared to t^{-1/4} density decay in A+B--->0 with isotropic diffusion and either with or without the hard-core repulsion. We suggest a relatively simple explanation for this t^{-1/3} decay based on the Burgers equation. Related properties associated with the asymptotic distribution of reactants can also be accounted for within this Burgers equation description.Comment: 11 pages, plain Tex, 8 figures. Hardcopy of figures available on request from S

A multi-species asymmetric simple exclusion process and its relation to traffic flow

Using the matrix product formalism we formulate a natural p-species generalization of the asymmetric simple exclusion process. In this model particles hop with their own specific rate and fast particles can overtake slow ones with a rate equal to their relative speed. We obtain the algebraic structure and study the properties of the representations in detail. The uncorrelated steady state for the open system is obtained and in the ($p \to \infty)$ limit, the dependence of its characteristics on the distribution of velocities is determined. It is shown that when the total arrival rate of particles exceeds a certain value, the density of the slowest particles rises abroptly.Comment: some typos corrected, references adde

Correlation functions and queuing phenomena in growth processes with drift

We suggest a novel stochastic discrete growth model which describes the drifted Edward-Wilkinson (EW) equation $\partial h /\partial t = \nu \partial_x^2 h - v\partial_x h +\eta(x,t)$. From the stochastic model, the anomalous behavior of the drifted EW equation with a defect is analyzed. To physically understand the anomalous behavior the height-height correlation functions $C(r)=$ and $G(r)=$ are also investigated, where the defect is located at $x_0$. The height-height correlation functions follow the power law $C(r)\sim r^{\alpha'}$ and $G(r)\sim r^{\alpha''}$ with $\alpha'=\alpha''=1/4$ around a perfect defect at which no growth process is allowed. $\alpha'=\alpha''=1/4$ is the same as the anomalous roughness exponent $\alpha=1/4$. For the weak defect at which the growth process is partially allowed, the normal EW behavior is recovered. We also suggest a new type queuing process based on the asymmetry $C(r) \neq C(-r)$ of the correlation function around the perfect defect

Asymptotic behavior of A + B --> inert for particles with a drift

We consider the asymptotic behavior of the (one dimensional) two-species annihilation reaction A + B --> 0, where both species have a uniform drift in the same direction and like species have a hard core exclusion. Extensive numerical simulations show that starting with an initially random distribution of A's and B's at equal concentration the density decays like t^{-1/3} for long times. This process is thus in a different universality class from the cases without drift or with drift in different directions for the different species.Comment: LaTeX, 6pp including 3 figures in LaTeX picture mod

Diffusion-Annihilation in the Presence of a Driving Field

We study the effect of an external driving force on a simple stochastic reaction-diffusion system in one dimension. In our model each lattice site may be occupied by at most one particle. These particles hop with rates $(1\pm\eta)/2$ to the right and left nearest neighbouring site resp. if this site is vacant and annihilate with rate 1 if it is occupied. We show that density fluctuations (i.e. the $m^{th}$ moments $\langle N^m \rangle$ of the density distribution at time $t$) do not depend on the spatial anisotropy $\eta$ induced by the driving field, irrespective of the initial condition. Furthermore we show that if one takes certain translationally invariant averages over initial states (e.g. random initial conditions) even local fluctuations do not depend on $\eta$. In the scaling regime $t \sim L^2$ the effect of the driving can be completely absorbed in a Galilei transformation (for any initial condition). We compute the probability of finding a system of $L$ sites in its stationary state at time $t$ if it was fully occupied at time $t_0 = 0$.Comment: 17 pages, latex, no figure

Exact Solution of Two-Species Ballistic Annihilation with General Pair-Reaction Probability

The reaction process $A+B->C$ is modelled for ballistic reactants on an infinite line with particle velocities $v_A=c$ and $v_B=-c$ and initially segregated conditions, i.e. all A particles to the left and all B particles to the right of the origin. Previous, models of ballistic annihilation have particles that always react on contact, i.e. pair-reaction probability $p=1$. The evolution of such systems are wholly determined by the initial distribution of particles and therefore do not have a stochastic dynamics. However, in this paper the generalisation is made to $p<1$, allowing particles to pass through each other without necessarily reacting. In this way, the A and B particle domains overlap to form a fluctuating, finite-sized reaction zone where the product C is created. Fluctuations are also included in the currents of A and B particles entering the overlap region, thereby inducing a stochastic motion of the reaction zone as a whole. These two types of fluctuations, in the reactions and particle currents, are characterised by the `intrinsic reaction rate', seen in a single system, and the `extrinsic reaction rate', seen in an average over many systems. The intrinsic and extrinsic behaviours are examined and compared to the case of isotropically diffusing reactants.Comment: 22 pages, 2 figures, typos correcte

Phases of a conserved mass model of aggregation with fragmentation at fixed sites

To study the effect of quenched disorder in a class of reaction-diffusion systems, we introduce a conserved mass model of diffusion and aggregation in which the mass moves as a whole to a nearest neighbour on most sites while it fragments off as a single monomer (i.e. chips off) from certain fixed sites. Once the mass leaves any site, it coalesces with the mass present on its neighbour. We study in detail the effect of a \emph{single} chipping site on the steady state in arbitrary dimensions, with and without bias. In the thermodynamic limit, the system can exist in one of the following phases -- (a) Pinned Aggregate (PA) phase in which an infinite aggregate (with mass proportional to the volume of the system) appears with probability one at the chipping site but not in the bulk. (b) Unpinned Aggregate (UA) phase in which \emph{both} the chipping site and the bulk can support an infinite aggregate simultaneously. (c) Non Aggregate (NA) phase in which there is no infinite cluster. Our analytical and numerical studies show that the system exists in the UA phase in all cases except in 1d with bias. In the latter case, there is a phase transition from the NA phase to the PA phase as density is increased. A variant of the above aggregation model is also considered in which total particle number is conserved and chipping occurs at a fixed site, but the particles do not interact with each other at other sites. This model is solved exactly by mapping it to a Zero Range Process. With increasing density, it exhibits a phase transition from the NA phase to the PA phase in all dimensions, irrespective of bias. Finally, we discuss the likely behaviour of the system in the presence of extensive disorder.Comment: RevTex, 19 pages including 11 figures, submitted to Phys. Rev.