Fluctuations and correlations in a diffusion-reaction system: Unified description of internal fluctuations and external noise (original) (raw)
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Fluctuations and correlations in a diffusion-reaction system: Exact hydrodynamics
Journal of Statistical Physics, 1991
The one-dimensional single-species diffusion-limited-coagulation process with irreversible random particle input (A~A + A reversibly and B~A irreversibly), under the influence of external fluctuations in the system parameters, is formulated in terms of a closed and linear partial-differential equation. Our theoretical treatment includes both internal fluctuations and external noise simultaneously and without approximation, allowing investigation of the interplay of their effects on the macroscopic behavior of this diffusion-reaction system. For the reversible model with the rate of the A~A+ A reaction fluctuating between two values as a Markov stochastic process, we solve the system exactly. We observe that spatially homogeneous macroscopic fluctuations in the system parameters can induce microscopic spatial correlations in the nonequilibrium steady state. Direct Monte Carlo simulations of the microscopic dynamics are presented, confirming the theoretical analysis and directly illustrating the externalnoise-induced spatial correlations.
Role of fluctuations for inhomogeneous reaction-diffusion phenomena
Physical Review A, 1991
Although Auctuations have been known to change the long-time behavior of homogeneous diA'usionreaction phenomena dramatically in dimensions d~4, simulations of reaction fronts in two-dimensional 3 +8~C inhomogeneous systems have only shown marginal departure from mean-field behavior. We perform cellular-automata simulations of the one-dimensional case and find that the width 8 (t) of the reaction front behaves as t '-', in contrast to mean-field behavior t'. We develop a scaling theory to obtain inequalities for the exponents in the more general mechanism n 3+mB-+C. Heuristic arguments about the range of fluctuations imply that the mean-field behavior should be correct in dimensions larger than an upper critical dimension d"p =2, irrespective of the values of n and m. This leads us to reinterpret the two-dimensional data obtained previously in terms of a logarithmic correction to meanfield behavior.
Effect of spatial concentration fluctuations on effective kinetics in diffusion-reaction systems
Water Resources Research, 2012
The effect of spatial concentration fluctuations on the reaction of two solutes, A þ B * C, is considered. In the absence of fluctuations, the concentration of solutes decays as A det ¼ B det $ t À1. Contrary to this, experimental and numerical studies suggest that concentrations decay significantly slower. Existing theory suggests a t Àd/4 scaling in the asymptotic regime (d is the dimensionality of the problem). Here we study the effect of fluctuations using the classical diffusion-reaction equation with random initial conditions. Initial concentrations of the reactants are treated as correlated random fields. We use the method of moment equations to solve the resulting stochastic diffusion-reaction equation and obtain a solution for the average concentrations that deviates from tAˋ1tot À1 to tAˋ1tot Àd/4 behavior at characteristic transition time t Ã. We also derive analytical expressions for t à as a function of Damköhler number and the coefficient of variation of the initial concentration.
The statistical mechanics of the coagulation–diffusion process with a stochastic reset
Journal of Physics A: Mathematical and Theoretical, 2014
The effects of a stochastic reset, to its initial configuration, is studied in the exactly solvable one-dimensional coagulation-diffusion process. A finite resetting rate leads to a modified non-equilibrium stationary state. If in addition the input of particles at a fixed given rate is admitted, a competition between the resetting and the input rates leads to a non-trivial behaviour of the particle-density in the stationary state. From the exact inter-particle probability distribution, a simple physical picture emerges: the reset mainly changes the behaviour at larger distance scales, while at smaller length scales, the non-trivial correlation of the model without a reset dominates.
Journal of Statistical Physics, 1997
We investigate with the help of analytical and numerical methods the reaction A +A → A on a one-dimensional lattice opened at one end and with an input of particles at the other end. We show that if the diffusion rates to the left and to the right are equal, for large x, the particle concentration c(x) behaves like A s x −1 (x measures the distance to the input end). If the diffusion rate in the direction pointing away from the source is larger than the one corresponding to the opposite direction the particle concentration behaves like A a x −1/2 . The constants A s and A a are independent of the input and the two coagulation rates. The universality of A a comes as a surprise since in the asymmetric case the system has a massive spectrum.
Kinetic theory for spatial correlation in nonequilibrium reaction–diffusion systems
Physica A: Statistical Mechanics and its Applications, 2000
A kinetic model of a reaction-di usion system by which uctuation around a nonequilibrium steady state can be analyzed without resorting to a local equilibrium assumption is discussed. The spatial correlation function of concentration uctuations is analytically calculated, and it is shown that (i) the result is consistent with the previous studies for su ciently slow chemical reactions, and (ii) how the correlation function is modiÿed when chemical reactions are fast.
Journal of Statistical Mechanics: Theory and Experiment, 2010
The long-time dynamics of reaction-diffusion processes in low dimensions is dominated by fluctuation effects. The one-dimensional coagulationdiffusion process describes the kinetics of particles which freely hop between the sites of a chain and where upon encounter of two particles, one of them disappears with probability one. The empty-interval method has, since a long time, been a convenient tool for the exact calculation of time-dependent particle densities in this model. We generalize the empty-interval method by considering the probability distributions of two simultaneous empty intervals at a given distance. While the equations of motion of these probabilities reduce for the coagulation-diffusion process to a simple diffusion equation in the continuum limit, consistency with the single-interval distribution introduces several nontrivial boundary conditions which are solved for the first time for arbitrary initial configurations. In this way, exact space-time-dependent correlation functions can be directly obtained and their dynamic scaling behaviour is analysed for large classes of initial conditions.