Scaling of Reaction Zones in the A+B->0 Diffusion-Limited Reaction

We study reaction zones in three different versions of the A+B->0 system. For a steady state formed by opposing currents of A and B particles we derive scaling behavior via renormalization group analysis. By use of a previously developed analogy, these results are extended to the time-dependent case of an initially segregated system. We also consider an initially mixed system, which forms reaction zones for dimension d<4. In this case an extension of the steady-state analogy gives scaling results characterized by new exponents.Comment: 4 pages, REVTeX 3.0 with epsf, 2 uuencoded postscript figures appended, OUTP-94-33

Diffusion-Limited Annihilation with Initially Separated Reactants

A diffusion-limited annihilation process, A+B->0, with species initially separated in space is investigated. A heuristic argument suggests the form of the reaction rate in dimensions less or equal to the upper critical dimension $d_c=2$. Using this reaction rate we find that the width of the reaction front grows as $t^{1/4}$ in one dimension and as $t^{1/6}(\ln t)^{1/3}$ in two dimensions.Comment: 9 pages, Plain Te

Renormalization Group Study of the A+B->0 Diffusion-Limited Reaction

The $A + B\to 0$ diffusion-limited reaction, with equal initial densities $a(0) = b(0) = n_0$, is studied by means of a field-theoretic renormalization group formulation of the problem. For dimension $d > 2$ an effective theory is derived, from which the density and correlation functions can be calculated. We find the density decays in time as a,b \sim C\sqrt{\D}(Dt)^{-d/4} for $d < 4$, with \D = n_0-C^\prime n_0^{d/2} + \dots, where $C$ is a universal constant, and $C^\prime$ is non-universal. The calculation is extended to the case of unequal diffusion constants $D_A \neq D_B$, resulting in a new amplitude but the same exponent. For $d \le 2$ a controlled calculation is not possible, but a heuristic argument is presented that the results above give at least the leading term in an $\epsilon = 2-d$ expansion. Finally, we address reaction zones formed in the steady-state by opposing currents of $A$ and $B$ particles, and derive scaling properties.Comment: 17 pages, REVTeX, 13 compressed figures, included with epsf. Eq. (6.12) corrected, and a moderate rewriting of the introduction. Accepted for publication in J. Stat. Phy

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

Cluster persistence in one-dimensional diffusion--limited cluster--cluster aggregation

The persistence probability, $P_C(t)$, of a cluster to remain unaggregated is studied in cluster-cluster aggregation, when the diffusion coefficient of a cluster depends on its size $s$ as $D(s) \sim s^\gamma$. In the mean-field the problem maps to the survival of three annihilating random walkers with time-dependent noise correlations. For $\gamma \ge 0$ the motion of persistent clusters becomes asymptotically irrelevant and the mean-field theory provides a correct description. For $\gamma < 0$ the spatial fluctuations remain relevant and the persistence probability is overestimated by the random walk theory. The decay of persistence determines the small size tail of the cluster size distribution. For $0 < \gamma < 2$ the distribution is flat and, surprisingly, independent of $\gamma$.Comment: 11 pages, 6 figures, RevTeX4, submitted to Phys. Rev.

Formation of Liesegang patterns: A spinodal decomposition scenario

Spinodal decomposition in the presence of a moving particle source is proposed as a mechanism for the formation of Liesegang bands. This mechanism yields a sequence of band positions x_n that obeys the spacing law x_n~Q(1+p)^n. The dependence of the parameters p and Q on the initial concentration of the reagents is determined and we find that the functional form of p is in agreement with the experimentally observed Matalon-Packter law.Comment: RevTex, 4 pages, 4 eps figure

Reaction Front in an A+B -> C Reaction-Subdiffusion Process

We study the reaction front for the process A+B -> C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular its unusual behavior at the center of the reaction zone