The parity conserving branching-annihilating random walk (pc-BARW) model is a reaction-diffusion system on a lattice where particles can branch into $m$ offsprings with even $m$ and hop to neighboring sites. If two or more particles land on the same site, they immediately annihilate pairwise. In this way the number of particles is preserved modulo two. It is well known that the pc-BARW with $m=2$ in 1 spatial dimension has no phase transition (it is always subcritical), if the hopping is described by a continuous time random walk. In contrast, the $m=2$ 1-d pc-BARW has a phase transition when formulated in discrete time, but we show that the continuous time limit is non-trivial: When the time step $δt\to 0$, the branching and hopping probabilities at the critical point scale with different powers of $δt$. These powers are different for different microscopic realizations. Although this phenomenon is not observed in some other reaction-diffusion systems like, e.g. the contact process, we argue that it should be generic and not restricted to the 1-d pc-BARW model.
We study the single-species diffusion-annihilation process with a time-dependent reaction rate, lambda(t)=lambda_0 t^-omega. Scaling arguments show that there is a critical value of the decay exponent omega_c(d) separating a reaction-limited regime for omega > omega_c from a diffusion-limited regime for omega < omega_c. The particle density displays a mean-field, omega-dependent, decay when the process is reaction limited whereas it behaves as for a constant reaction rate when the process is diffusion limited. These results are confirmed by Monte Carlo simulations. They allow us to discuss the…
We study various combinations of active diffusion with branching, as an extension of standard reaction-diffusion processes. We concentrate on the selection of the asymptotic wavefront speed for thermal run-and-tumble and for thermal active Brownian processes in general spatial dimensions. Comparing 1D active branching processes with a passive counterpart (which has the same effective diffusion constant and reproduction rate), we find that the active process has a smaller propagation speed. In higher dimensions, a similar comparison yields the opposite conclusion.
We investigate the persistence probability in the Voter model for dimensions d\geq 2. This is achieved by mapping the Voter model onto a continuum reaction-diffusion system. Using path integral methods, we compute the persistence probability r(q,t), where q is the number of ``opinions'' in the original Voter model. We find r(q,t)\sim exp[-f_2(q)(ln t)^2] in d=2; r(q,t)\sim exp[-f_d(q)t^{(d-2)/2}] for 2<d<4; r(q,t)\sim exp[-f_4(q)t/ln t] in d=4; and r(q,t)\sim exp[-f_d(q)t] for d>4. The results of our analysis are checked by Monte Carlo simulations.
Reaction-diffusion equations deliver a versatile tool for the description of reactions in inhomogeneous systems under the assumption that the characteristic reaction scales and the scales of the inhomogeneities in the reactant concentrations separate. In the present work we discuss the possibilities of a generalization of reaction-diffusion equations to the case of anomalous diffusion described by continuous-time random walks with decoupled step length and waiting time probability densities, the first being Gaussian or Levy, the second one being an exponential or a power-law lacking the first …
In a two species reaction diffusion system,we show that it is possible to generate a set of wavelength doubling bifuractions leading to spatially chaotic state.The wavelength doubling bifurcations are preceded by a symmetry breaking transition which acts as a precursor.
A new asymptotic method is presented for the analysis of the traveling waves in the one-dimensional reaction-diffusion system with the diffusion with a finite velocity and Kolmogorov-Petrovskii-Piskunov kinetics. The analysis makes use of the path-integral approach, scaling procedure and the singular perturbation techniques involving the large deviations theory for the Poisson random walk. The exact formula for the position and speed of reaction front is derived. It is found that the reaction front dynamics is formally associated with the relativistic Hamiltonian/Lagrangian mechanics.
Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grunwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to other. In continuum limit, the suggested lattice diffusion equations with non-integer order differences give the diffusion equations with the Grunwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grunwald-Letni…
We formulate the generalized master equation for a class of continuous time random walks in the presence of a prescribed deterministic evolution between successive transitions. This formulation is exemplified by means of an advection-diffusion and a jump-diffusion scheme. Based on this master equation, we also derive reaction-diffusion equations for subdiffusive chemical species, using a mean field approximation.
We propose the deterministic rate equation of three-species in the reaction - diffusion system. For this case, our purpose is to carry out the decay process in our three-species reaction-diffusion model of the form $A+B+C\to D$. The particle density and the global reaction rate are also shown analytically and numerically on a two-dimensional square lattice with the periodic boundary conditions. Especially, the crossover of the global reaction rate is discussed in both early-time and long-time regimes.
We study long-time properties of reversible reaction-diffusion systems of type A + B <-> C by means of perturbation expansion in powers of 1/t (inverse of time). For the case of equal diffusion coefficients we present exact formulas for the asymptotic forms of reactant concentrations and a complete, recursive expression for an arbitrary term of the expansions. Taking an appropriate limit we show that by studying reversible reactions one can obtain "singular" solutions typical of irreversible reactions.
We present new numerical schemes to integrate stochastic partial differential equations which describe the spatio-temporal dynamics of reaction-diffusion (RD) problems under the effect of internal fluctuations. The schemes conserve the nonnegativity of the solutions and incorporate the Poissonian nature of internal fluctuations at small densities, their performance being limited by the level of approximation of density fluctuations at small scales. We apply the new schemes to two different aspects of the Reggeon model namely, the study of its non-equilibrium phase transition and the dynamics o…
The reversible A <-> B reaction-diffusion process, when species A and B are initially mixed and diffuse with different diffusion coefficients, is investigated using the boundary layer function method. It is assumed that the ratio of the characteristic time of the reaction to the characteristic time of diffusion is taken as a small parameter of the task. It was shown that diffusion-reaction process can be considered as a quasi-equilibrium process. Despite this fact the contribution of the reaction in changes of the species concentration is comparable with the diffusion contributions. Moreover t…
We consider solvability of the generalized reaction-diffusion equation with both space- and time-dependent diffusion and reaction terms by means of the similarity method. By introducing the similarity variable, the reaction-diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable reaction-diffusion systems. Several representative examples of exactly solvable reaction-diffusion equations are presented.
The problem of velocity selection of reaction-diffusion fronts has been widely investigated. While the mean field limit results are well known theoretically, there is a lack of analytic progress in those cases in which fluctuations are to be taken into account. Here, we construct an analytic theory connecting the first principles of the reaction-diffusion process to an effective equation of motion via field-theoretic arguments, and we arrive at the results already confirmed by numerical simulations.
A fractional reaction-diffusion equation is derived from a continuous time random walk model when the transport is dispersive. The exit from the encounter distance, which is described by the algebraic waiting time distribution of jump motion, interferes with the reaction at the encounter distance. Therefore, the reaction term has a memory effect. The derived equation is applied to the geminate recombination problem. The recombination is shown to depend on the intrinsic reaction rate, in contrast with the results of Sung et al. [J. Chem. Phys. {\bf 116}, 2338 (2002)], which were obtained from t…
The Doi formalism treats a reaction-diffusion process as a quantum many-body problem. We use this second quantized formulation as a starting point to derive a numerical scheme for simulating $X\to 0$ reaction-diffusion processes, following a well-established time discretization procedure. In the case of a reaction zone localized in the configuration space, this formulation provides also a systematic way of designing an optimized, multiple time step algorithm, spending most of the computation time to sample the configurations where the reaction is likely to occur.
We consider similarity solutions of the generalized convection-diffusion-reaction equation with both space- and time-dependent convection, diffusion and reaction terms. By introducing the similarity variable, the reaction-diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable convection-diffusion-reaction systems. Some representative examples of exactly solvable systems are presented. We also describe how an equivalent convection-diffusion-reac…
Daily, are reported systems in nature that present anomalous diffusion phenomena due to irregularities of medium, traps or reactions process. In this scenario, the diffusion with traps or localised--reactions emerge through various investigations that include numerical, analytical and experimental techniques. In this work, we construct a model which involves a coupling of two diffusion equations to approach the random walkers in a medium with localised reaction point (or controlled diffusion). We present the exact analytical solutions to the model. In the following, we obtain the survival prob…
We investigate reversible diffusion-influenced reactions of an isolated pair in two dimensions. To this end, we employ convolution relations that permit deriving the survival probability of the reversible reaction directly in terms of the survival probability of the irreversible reaction. Furthermore, we make use of the mean reaction time approximation to write the irreversible survival probability in restricted spaces as a single exponential. In this way, we obtain exact and approximative expressions in the time domain for the reversible survival probability for three different two dimensiona…
In this work we are concerned with generating solutions of a class of Convection-Diffusion-Reaction equation from the solutions of another CDR equation through the Darboux transformations. The method is elucidated by cases with certain types of the reaction coefficients. We have also discussed briefly supersymmetric pairs of Convection-Diffusion-Reaction equations connected through similarity solutions.
We investigate the influence of a stochastically fluctuating step-barrier potential on bimolecular reaction rates by exact analytical theory and stochastic simulations. We demonstrate that the system exhibits a new resonant reaction behavior with rate enhancement if an appropriately defined fluctuation decay length is of the order of the system size. Importantly, we find that in the proximity of resonance the standard reciprocal additivity law for diffusion and surface reaction rates is violated due to the dynamical coupling of multiple kinetic processes. Together, these findings may have impo…
We study diffusion of particles on the surface of a sphere toward a partially reactive circular target with partly reversible binding kinetics. We solve the coupled diffusion-reaction equations and obtain the exact expressions for the time-dependent concentration of particles and the total diffusive flux. Explicit asymptotic formulas are derived in the small target limit. This study reveals the strong effects of reversible binding kinetics onto diffusion-mediated reactions that may be relevant for many biochemical reactions on cell membranes.
The mean-field reaction-diffusion equations of the diffusive pair-annihilation and triplett-annihilation processes are considered. A direct lower bound on the time-dependent mean particle-density is derived. The results are applied to the mean-field theory of the diffusive pair-contact process.
