Single-Species Reactions on a Random Catalytic Chain

We present an exact solution for a catalytically-activated annihilation A + A \to 0 reaction taking place on a one-dimensional chain in which some segments (placed at random, with mean concentration p) possess special, catalytic properties. Annihilation reaction takes place, as soon as any two A particles land from the reservoir onto two vacant sites at the extremities of the catalytic segment, or when any A particle lands onto a vacant site on a catalytic segment while the site at the other extremity of this segment is already occupied by another A particle. We find that the disorder-average pressure $P^{(quen)}$ per site of such a chain is given by $P^{(quen)} = P^{(lan)} + \beta^{-1} F$, where $P^{(lan)} = \beta^{-1} \ln(1+z)$ is the Langmuir adsorption pressure, (z being the activity and \beta^{-1} - the temperature), while $\beta^{-1} F$ is the reaction-induced contribution, which can be expressed, under appropriate change of notations, as the Lyapunov exponent for the product of 2 \times 2 random matrices, obtained exactly by Derrida and Hilhorst (J. Phys. A {\bf 16}, 2641 (1983)). Explicit asymptotic formulae for the particle mean density and the compressibility are also presented.Comment: AMSTeX, 17 pages, 1 figure, submitted to J. Phys.

Kinetic description of diffusion-limited reactions in random catalytic media

We study the kinetics of bimolecular, catalytically-activated reactions (CARs) in d-dimensions. The elementary reaction act between reactants takes place only when these meet in the vicinity of a catalytic site; such sites are assumed to be immobile and randomly distributed in space. For CARs we develop a kinetic formalism, based on Collins-Kimball-type ideas; within this formalism we obtain explicit expressions for the effective reaction rates and for the decay of the reactants' concentrations.Comment: 15 pages, Latex, two figures, to appear in J. Chem. Phy

On the distribution of surface extrema in several one- and two-dimensional random landscapes

We study here a standard next-nearest-neighbor (NNN) model of ballistic growth on one- and two-dimensional substrates focusing our analysis on the probability distribution function $P(M,L)$ of the number $M$ of maximal points (i.e., local ``peaks'') of growing surfaces. Our analysis is based on two central results: (i) the proof (presented here) of the fact that uniform one--dimensional ballistic growth process in the steady state can be mapped onto ''rise-and-descent'' sequences in the ensemble of random permutation matrices; and (ii) the fact, established in Ref. \cite{ov}, that different characteristics of ``rise-and-descent'' patterns in random permutations can be interpreted in terms of a certain continuous--space Hammersley--type process. For one--dimensional system we compute $P(M,L)$ exactly and also present explicit results for the correlation function characterizing the enveloping surface. For surfaces grown on 2d substrates, we pursue similar approach considering the ensemble of permutation matrices with long--ranged correlations. Determining exactly the first three cumulants of the corresponding distribution function, we define it in the scaling limit using an expansion in the Edgeworth series, and show that it converges to a Gaussian function as $L \to \infty$.Comment: 25 pages, 12 figure

Random patterns generated by random permutations of natural numbers

We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time $n$, whose moves to the right or to the left are induced by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site $X$ at time $n$, obtain the probability measure of different excursions and define the asymptotic distribution of the number of "U-turns" of the trajectories - permutation "peaks" and "through". In the second part, we focus on some statistical properties of surfaces obtained by randomly placing natural numbers $1,2,3, >...,L$ on sites of a 1d or 2d square lattices containing $L$ sites. We calculate the distribution function of the number of local "peaks" - sites the number at which is larger than the numbers appearing at nearest-neighboring sites - and discuss some surprising collective behavior emerging in this model.Comment: 16 pages, 5 figures; submitted to European Physical Journal, proceedings of the conference "Stochastic and Complex Systems: New Trends and Expectations" Santander, Spai

Reversible Diffusion-Limited Reactions: "Chemical Equilibrium" State and the Law of Mass Action Revisited

The validity of two fundamental concepts of classical chemical kinetics - the notion of "Chemical Equilibrium" and the "Law of Mass Action" - are re-examined for reversible \textit{diffusion-limited} reactions (DLR), as exemplified here by association/dissociation $A+A \rightleftharpoons B$ reactions. We consider a general model of long-ranged reactions, such that any pair of $A$ particles, separated by distance $\mu$, may react with probability $\omega_+(\mu)$, and any $B$ may dissociate with probability $\omega_-(\lambda)$ into a geminate pair of $A$s separated by distance $\lambda$. Within an exact analytical approach, we show that the asymptotic state attained by reversible DLR at $t = \infty$ is generally \textit{not a true thermodynamic equilibrium}, but rather a non-equilibrium steady-state, and that the Law of Mass Action is invalid. The classical picture holds \text{only} in physically unrealistic case when $\omega_+(\mu) \equiv \omega_-(\mu)$ for any value of $\mu$.Comment: 4 page

Influence of auto-organization and fluctuation effects on the kinetics of a monomer-monomer catalytic scheme

We study analytically kinetics of an elementary bimolecular reaction scheme of the Langmuir-Hinshelwood type taking place on a d-dimensional catalytic substrate. We propose a general approach which takes into account explicitly the influence of spatial correlations on the time evolution of particles mean densities and allows for the analytical analysis. In terms of this approach we recover some of known results concerning the time evolution of particles mean densities and establish several new ones.Comment: Latex, 25 pages, one figure, submitted to J. Chem. Phy

Corrections to the Law of Mass Action and Properties of the Asymptotic t=∞t = \infty State for Reversible Diffusion-Limited Reactions

On example of diffusion-limited reversible $A+A \rightleftharpoons B$ reactions we re-examine two fundamental concepts of classical chemical kinetics - the notion of "Chemical Equilibrium" and the "Law of Mass Action". We consider a general model with distance-dependent reaction rates, such that any pair of $A$ particles, performing standard random walks on sites of a $d$-dimensional lattice and being at a distance $\mu$ apart of each other at time moment $t$, may associate forming a $B$ particle at the rate $k_+(\mu)$. In turn, any randomly moving $B$ particle may spontaneously dissociate at the rate $k_-(\lambda)$ into a geminate pair of $A$s "born" at a distance $\lambda$ apart of each other. Within a formally exact approach based on Gardiner's Poisson representation method we show that the asymptotic $t = \infty$ state attained by such diffusion-limited reactions is generally \textit{not a true thermodynamic equilibrium}, but rather a non-equilibrium steady-state, and that the Law of Mass Action is invalid. The classical concepts hold \text{only} in case when the ratio $k_+(\mu)/k_-(\mu)$ does not depend on $\mu$ for any $\mu$.Comment: 30 pages, 2 figure

Equilibrium Properties of A Monomer-Monomer Catalytic Reaction on A One-Dimensional Chain

We study the equilibrium properties of a lattice-gas model of an $A + B \to 0$ catalytic reaction on a one-dimensional chain in contact with a reservoir for the particles. The particles of species $A$ and $B$ are in thermal contact with their vapor phases acting as reservoirs, i.e., they may adsorb onto empty lattice sites and may desorb from the lattice. If adsorbed $A$ and $B$ particles appear at neighboring lattice sites they instantaneously react and both desorb. For this model of a catalytic reaction in the adsorption-controlled limit, we derive analytically the expression of the pressure and present exact results for the mean densities of particles and for the compressibilities of the adsorbate as function of the chemical potentials of the two species.Comment: 19 pages, 5 figures, submitted to Phys. Rev.