We consider the general irreversible AϩB→2A autocatalytic reaction in one dimension, for which the corresponding diffusion constants D A and D B may differ. Contrary to mean-field-type predictions, the Monte Carlo simulations show that, as long as D A Ͼ0, only a unique, stable front propagates with constant velocity. When D A ϭ0 the behavior changes drastically: both the front's position and its characteristic width grow with t 1/2 . These findings are adequately described within a Smoluchowski-type approach. ͓S1063-651X͑97͒12610-1͔ PACS number͑s͒: 05.40.ϩj, 82.20.Mj, 82.65.Ϫi I. INTROUCTIONReaction kinetics in low dimensions were extensively investigated in the past two decades, since they differ significantly from the situation in high-dimensional spaces, and thus violate strongly the classical kinetics schemes based on the mass-action law ͓1,2͔. Remarkably, such violations did not find much attention in the society of scientists dealing with front propagation in autocatalytic reactions. The autocatalytic AϩB↔2A conversion ͑where both the direct and the back reaction follow the bimolecular scheme͒ can be described in the mean-field limit by the quadratic Fisher equation ͓Eq. ͑11.31͒ of Ref. ͓3͔͔, whose solution fronts may propagate with different velocities v; here the initial conditions determine whether a certain velocity is attained ͓3͔. We note that a mean-field-type description is not appropriate in low dimensions ͑dϭ1 and 2͒ where the reaction term depends strongly on particle correlations ͓4-7͔. The reaction AϩB↔2A was investigated analytically and via Monte Carlo simulations in Refs. ͓8, 9͔, where, based on ensembleaveraged quantities, in one and two dimensions strong deviations from the mean-field-type behavior were detected. A detailed study of the irreversible autocatalytic reaction A ϩB→2A in one dimension was provided in Ref. ͓10͔ for equal diffusion constants D A ϭD B . This reaction is the simplest model for infection spreading, by which ͑irreversibly infected͒ A particles infect at first encounter healthy B particles. Here we analyze the situation for general D A and D B .In fact, the question of the front structure falls into the class of spontaneous local ordering phenomena, which have found much attention in connection with the AϩA→0, A ϩA→A, and AϩB→0 reactions ͓11-22͔. Especially the last reaction shows ͑due to fluctuations effects͒ nontrivial large-scale spatial structures ͑clusters͒; these findings lay outside the classical kinetics scheme, and for their understanding need much more elaborated theories and extensive numerical studies. Note that a microscopic description of the front structure requires a knowledge of the local ordering of the A and the B particles near the front.In Ref. ͓10͔ only the case D A ϭD B was considered. Then the particles are mathematically equivalent, and the whole reaction consists solely in the renaming of B particles to A on encounter. This introduces an additional, rather unrealistic symmetry into the problem; here we consider the general situation of physic...

We introduce a fractional Fokker-Planck equation (FFPE) for Lévy flights in the presence of an external field. The equation is derived within the framework of the subordination of random processes which leads to Lévy flights. It is shown that the coexistence of anomalous transport and a potential displays a regular exponential relaxation toward the Boltzmann equilibrium distribution. The properties of the Lévy-flight FFPE derived here are compared with earlier findings for a subdiffusive FFPE. The latter is characterized by a nonexponential Mittag-Leffler relaxation to the Boltzmann distribution. In both cases, which describe strange kinetics, the Boltzmann equilibrium is reached, and modifications of the Boltzmann thermodynamics are not required.

We consider here the A + B -0 reaction between particles that diffuse, interact through short-range forces, and react on contact. In a plausible approximation the reaction can be described through a nonlinear diffusion equation, from which the scaling behavior of the respective A and B concentrations follows. We focus here on steady particle generation and obtain the exponents that govern the concentration's growth. Through explicit Monte Carlo simulations of the underlying stochastic process we obtain directly these exponents; furthermore we show that the assumption of particle segregation in clusters is correct, by computing the correlation length.

No abstract

No abstract