Exact Results for a Three-Body Reaction-Diffusion System

A system of particles hopping on a line, singly or as merged pairs, and annihilating in groups of three on encounters, is solved exactly for certain symmetrical initial conditions. The functional form of the density is nearly identical to that found in two-body annihilation, and both systems show non-mean-field, ~1/t**(1/2) instead of ~1/t, decrease of particle density for large times.Comment: 10 page

Particle Dynamics in a Mass-Conserving Coalescence Process

We consider a fully asymmetric one-dimensional model with mass-conserving coalescence. Particles of unit mass enter at one edge of the chain and coalescence while performing a biased random walk towards the other edge where they exit. The conserved particle mass acts as a passive scalar in the reaction process $A+A\to A$, and allows an exact mapping to a restricted ballistic surface deposition model for which exact results exist. In particular, the mass- mass correlation function is exactly known. These results complement earlier exact results for the $A+A\to A$ process without mass. We introduce a comprehensive scaling theory for this process. The exact anaytical and numerical results confirm its validity.Comment: 5 pages, 6 figure

Symmetry and species segregation in diffusion-limited pair annihilation

We consider a system of q diffusing particle species A_1,A_2,...,A_q that are all equivalent under a symmetry operation. Pairs of particles may annihilate according to A_i + A_j -> 0 with reaction rates k_{ij} that respect the symmetry, and without self-annihilation (k_{ii} = 0). In spatial dimensions d > 2 mean-field theory predicts that the total particle density decays as n(t) ~ 1/t, provided the system remains spatially uniform. We determine the conditions on the matrix k under which there exists a critical segregation dimension d_{seg} below which this uniformity condition is violated; the symmetry between the species is then locally broken. We argue that in those cases the density decay slows down to n(t) ~ t^{-d/d_{seg}} for 2 < d < d_{seg}. We show that when d_{seg} exists, its value can be expressed in terms of the ratio of the smallest to the largest eigenvalue of k. The existence of a conservation law (as in the special two-species annihilation A + B -> 0), although sufficient for segregation, is shown not to be a necessary condition for this phenomenon to occur. We work out specific examples and present Monte Carlo simulations compatible with our analytical results.Comment: latex, 19 pages, 3 eps figures include

Model of Cluster Growth and Phase Separation: Exact Results in One Dimension

We present exact results for a lattice model of cluster growth in 1D. The growth mechanism involves interface hopping and pairwise annihilation supplemented by spontaneous creation of the stable-phase, +1, regions by overturning the unstable-phase, -1, spins with probability p. For cluster coarsening at phase coexistence, p=0, the conventional structure-factor scaling applies. In this limit our model falls in the class of diffusion-limited reactions A+A->inert. The +1 cluster size grows diffusively, ~t**(1/2), and the two-point correlation function obeys scaling. However, for p>0, i.e., for the dynamics of formation of stable phase from unstable phase, we find that structure-factor scaling breaks down; the length scale associated with the size of the growing +1 clusters reflects only the short-distance properties of the two-point correlations.Comment: 12 page

Contest based on a directed polymer in a random medium

We introduce a simple one-parameter game derived from a model describing the properties of a directed polymer in a random medium. At his turn, each of the two players picks a move among two alternatives in order to maximize his final score, and minimize opponent's return. For a game of length $n$, we find that the probability distribution of the final score $S_n$ develops a traveling wave form, ${\rm Prob}(S_n=m)=f(m-v n)$, with the wave profile $f(z)$ unusually decaying as a double exponential for large positive and negative $z$. In addition, as the only parameter in the game is varied, we find a transition where one player is able to get his maximum theoretical score. By extending this model, we suggest that the front velocity $v$ is selected by the nonlinear marginal stability mechanism arising in some traveling wave problems for which the profile decays exponentially, and for which standard traveling wave theory applies

Condensation phase transitions of symmetric conserved-mass aggregation model on complex networks

We investigate condensation phase transitions of symmetric conserved-mass aggregation (SCA) model on random networks (RNs) and scale-free networks (SFNs) with degree distribution $P(k) \sim k^{-\gamma}$. In SCA model, masses diffuse with unite rate, and unit mass chips off from mass with rate $\omega$. The dynamics conserves total mass density $\rho$. In the steady state, on RNs and SFNs with $\gamma>3$ for $\omega \neq \infty$, we numerically show that SCA model undergoes the same type condensation transitions as those on regular lattices. However the critical line $\rho_c (\omega)$ depends on network structures. On SFNs with $\gamma \leq 3$, the fluid phase of exponential mass distribution completely disappears and no phase transitions occurs. Instead, the condensation with exponentially decaying background mass distribution always takes place for any non-zero density. For the existence of the condensed phase for $\gamma \leq 3$ at the zero density limit, we investigate one lamb-lion problem on RNs and SFNs. We numerically show that a lamb survives indefinitely with finite survival probability on RNs and SFNs with $\gamma >3$, and dies out exponentially on SFNs with $\gamma \leq 3$. The finite life time of a lamb on SFNs with $\gamma \leq 3$ ensures the existence of the condensation at the zero density limit on SFNs with $\gamma \leq 3$ at which direct numerical simulations are practically impossible. At $\omega = \infty$, we numerically confirm that complete condensation takes place for any $\rho > 0$ on RNs. Together with the recent study on SFNs, the complete condensation always occurs on both RNs and SFNs in zero range process with constant hopping rate.Comment: 6 pages, 6 figure

Approach to Asymptotic Behaviour in the Dynamics of the Trapping Reaction

We consider the trapping reaction A + B -> B in space dimension d=1, where the A and B particles have diffusion constants D_A, D_B respectively. We calculate the probability, Q(t), that a given A particle has not yet reacted at time t. Exploiting a recent formulation in which the B particles are eliminated from the problem we find, for t -> \infty, $Q(t) \sim \exp[-(4/\sqrt{\pi})(\rho^2 D_Bt)^{1/2} - (C \rho^2 D_A t)^{1/3} + ...]$, where $\rho$ is the density of B particles and $C \propto D_A/D_B$ for $D_A/D_B << 1$.Comment: 8 pages, 2 figures; minor change

The A+B -> 0 annihilation reaction in a quenched random velocity field

Using field-theoretic renormalization group methods the long-time behaviour of the A+B -> 0 annihilation reaction with equal initial densities n_A(0) = n_B(0) = n_0 in a quenched random velocity field is studied. At every point (x, y) of a d-dimensional system the velocity v is parallel or antiparallel to the x-axis and depends on the coordinates perpendicular to the flow. Assuming that v(y) have zero mean and short-range correlations in the y-direction we show that the densities decay asymptotically as n(t) ~ A n_0^(1/2) t^(-(d+3)/8) for d<3. The universal amplitude A is calculated at first order in \epsilon = 3-d.Comment: 19 pages, LaTeX using IOP-macros, 5 eps-figures. It is shown that the amplitude of the density is universal, i.e. independent of the reaction rat

Exact Solutions of Anisotropic Diffusion-Limited Reactions with Coagulation and Annihilation

We report exact results for one-dimensional reaction-diffusion models A+A -> inert, A+A -> A, and A+B -> inert, where in the latter case like particles coagulate on encounters and move as clusters. Our study emphasized anisotropy of hopping rates; no changes in universal properties were found, due to anisotropy, in all three reactions. The method of solution employed mapping onto a model of coagulating positive integer charges. The dynamical rules were synchronous, cellular-automaton type. All the asymptotic large-time results for particle densities were consistent, in the framework of universality, with other model results with different dynamical rules, when available in the literature.Comment: 28 pages in plain TeX + 2 PostScript figure

Diffusion-Limited Coalescence with Finite Reaction Rates in One Dimension

We study the diffusion-limited process $A+A\to A$ in one dimension, with finite reaction rates. We develop an approximation scheme based on the method of Inter-Particle Distribution Functions (IPDF), which was formerly used for the exact solution of the same process with infinite reaction rate. The approximation becomes exact in the very early time regime (or the reaction-controlled limit) and in the long time (diffusion-controlled) asymptotic limit. For the intermediate time regime, we obtain a simple interpolative behavior between these two limits. We also study the coalescence process (with finite reaction rates) with the back reaction $A\to A+A$, and in the presence of particle input. In each of these cases the system reaches a non-trivial steady state with a finite concentration of particles. Theoretical predictions for the concentration time dependence and for the IPDF are compared to computer simulations. P. A. C. S. Numbers: 82.20.Mj 02.50.+s 05.40.+j 05.70.LnComment: 13 pages (and 4 figures), plain TeX, SISSA-94-0