Kinetics of A+B-->0 with driven diffusive motion

Irreversible bimolecular chemical reactions on directed scale-free networks

Kinetics of irreversible bimolecular chemical reactions A+A→0 and A+B→0 on directed scale-free networks with the in-degree distribution P(in)(k)∼k(-γ)(in) and the out-degree distribution P(out)(ℓ)∼ℓ(-γ)(out) are investigated. Since the correlation between k and ℓ of each node generally exists in directed networks, we control the correlation (kℓ) with the probability r∈[0,1] by two different algorithms for the construction of the directed networks, i.e., the so-called k and ℓ algorithms. For r=1, the k algorithm gives (kℓ)=(k(2)), whereas the ℓ algorithm gives (kℓ)=(ℓ(2). For r=0, (kℓ)=(k)(ℓ) for both algorithms. The kinetics of both reactions are analyzed using heterogeneous mean-field (HMF) theory and Monte Carlo simulations. The density of particles (ρ) algebraically decays in time t as ρ(t)∼t(-α). The kinetics of both reactions are determined by the same rate equation, dρ/dt=aρ(2)+bρ(θ-1), apart from coefficients. The exponent θ is determined by the algorithm: θ=γ(in) for the k algorithm (r≥0) and θ=γ(min) for the ℓ algorithm (r>0), where γ(min) is the smaller exponent between γ(in) and γ(out). For θ>3, one observes the ordinary mean-field kinetics, ρ∼1/t (α=1). In contrast, for θ<3, ρ(t) anomalously decays with α=1/(θ-2). The HMF predictions are confirmed by the simulations on quenched directed networks.

Bimolecular chemical reactions on weighted complex networks

We investigate the kinetics of bimolecular chemical reactions A+A→0 and A+B→0 on weighted scale-free networks (WSFNs) with degree distribution P(k)∼k^{-γ} . On WSFNs, a weight w{ij} is assigned to the link between node i and j . We consider the symmetric weight given as w{ij}=(k{i}k{j})^{μ} , where k{i} and k{j} are the degree of node i and j . The hopping probability T{ij} of a particle from node i to j is then given as T{ij}∝(k{i}k{j})^{μ} . From a mean-field analysis, we analytically show in the thermodynamic limit that the kinetics of A+A→0 and A+B→0 are identical and there exist two crossover μ values, μ{1c}=γ-2 and μ{2c}=(γ-3)/2 . The density of particles ρ(t) algebraically decays in time t as t^{-α} with α=1 for μ<μ{2c} and α=(μ+1)/(γ-μ-2) for μ{2c}≤μ<μ{1c} . For μ≥μ{1c} , ρ decays exponentially. With the mean-field rate equation for ρ(t) , we also analytically show that the kinetics on the WSFNs is mapped onto that on unweighted SFNs with P(k)∼k^{-γ^{'}} with γ^{'}=(μ+γ)/(μ+1) .

Irreversible nA+mB→0 reaction of driven hard-core particles

We investigate the kinetics of general two species annihilation nA+mB→0 of driven hard-core (HC) particles with N=n+m in one dimension. With uniform drift velocity, all particles are driven to the right. HC exclusion forbids the interchange of any particles and restricts the number of particles on a site to 0 or 1. The reaction is classified into two classes, the symmetric and the asymmetric reaction. The symmetric reaction means both nA+mB→0 and mA+nB→0 , while the asymmetric reaction means only nA+mB→0 for a given (n,m) pair. As N increases, the trains of particles causing the reaction rarely form. Hence, for sufficiently large N, particles are evenly distributed before the reaction, so one expects a crossover N(c) above which the kinetics follows the classical mean-field rate equation. We show the existence of N(c) and that the kinetics for N<N(c) is the same as that of A+B→0 of driven HC particles as in the reactions with the isotropic diffusion. However, compared to the isotropic cases, N(c) and the kinetics for N≥N(c) are shown to be completely changed by the interplay of the drift and HC exclusion, and strongly depend on the reaction symmetry. We also show that densities decay as t(-1/N) which cannot be explained by the classical mean-field rate equation. Instead the kinetics is explained analytically by a variant theory.

Distribution in flowing reaction-diffusion systems

A power-law distribution is found in the density profile of reacting systems A+B-->C+D and 2A-->2C under a flow in two and three dimensions. Different densities of reactants A and B are fixed at both ends. For the reaction A+B , the concentration of reactants asymptotically decay in space as x-1/2 and x-3/4 in two dimensions and three dimensions, respectively. For 2A , it decays as log(x)/x in two dimensions. The decay of A+B is explained considering the effect of segregation of reactants in the isotropic case. The decay for 2A is explained by the marginal behavior of two-dimensional diffusion. A logarithmic divergence of the diffusion constant with system size is found in two dimensions.

Reaction symmetry of irreversible reaction nA+mB-->0

We investigate the kinetics of irreversible reaction nA+mB-->0. When nA particles and mB particles encounter on the same site, the reaction takes place. We classify the reaction into two classes, the symmetric and the asymmetric reactions. The symmetric reaction means both nA+mB-->0 and mA+nB-->0, while the asymmetric reaction means only nA+mB-->0 for a given (n,m) pair. The kinetics of the reaction follows the fluctuation-dominated kinetics of A+B-->0 for N<Nc, where Nc is the crossover N(=n+m). For N>or=Nc, the kinetics follows a mean-field rate equation. For the asymmetric reaction, it was shown that Nc is 5. We numerically show that the reaction symmetry changes Nc in one dimension. We investigate the asymptotic scaling behaviors of density and various lengths characterizing the spatial organization of particles such as domain length and interparticle distance. Lengths exhibit much clear crossover to the mean-field region at the expected Nc of each reaction. From the scaling behavior of density and lengths, we show Nc=4 for the symmetric reaction in one dimension. Therefore the symmetry in the reaction changes the upper critical dimension, which implies that the kinetics cannot be described by a single theory.

Kinetics of the bosonic A+B-->0 reaction with on-site attractive interaction

We investigate kinetics of the uniformly driven bosonic A+B-->0 reaction with on-site attractive interaction in one dimension. In this model, n(i)(lambda) particles from a site with n(i) particles are driven to the right. When particles of opposite species occupy the same site, the reaction takes place instantaneously. Since n(i)(lambda)<n(i) for lambda<1, lambda controls the on-site attractive interaction between like particles. The lambda=0 case corresponds to the hard-core (HC) particle model. With equal initial densities of both species, we numerically confirm that the scaling behaviors of density and lengths are the same as those of the uniformly driven HC particle system. Especially the domain length l satisfies the power law l approximately t(2/3). The kinetics of the reaction is independent of lambda as long as lambda<1. The lambda -independent kinetics results from the lambda -independent collective motions of single species domains.

Continuously varying exponents in A+B-->0 reaction with long-ranged attractive interaction

We investigate kinetics of A+B-->0 reaction with long-range attractive interaction V(r) approximately -r(-2sigma) between A and B or with drift velocity v approximately r(-sigma) in one dimension, where r is the closest distance between A and B . It is analytically shown that dynamical exponents for density of particles (rho) and size of domains (l) continuously vary with sigma when sigma < sigma(c) = 1/2 , while that for the distance between adjacent opposite species (l(AB)) varies when sigma < sigma(c)AB = 7/6 . For sigma > sigma(c)AB diffusive motions dominate the kinetics. These anomalous behaviors with the two crossover values of sigma are supported by numerical simulations.

Anomalous kinetics of attractive A + B-->0 reactions

We investigate the kinetics of the A + B-->0 reaction with the attractive interaction between opposite species in one spatial dimension. The attractive interaction leads to isotropic diffusions inside segregated single species domains, and accelerates the reactions of opposite species at the domain boundaries. At equal initial densities of and , we analytically and numerically show that the density of particles (rho), the size of domains (l), the distance between the closest neighbor of same species (lAA), and the distance between adjacent opposite species (lAB) scale in time as rho approximately t(-1/3), lAA approximately t(1/3), and l approximately lAB approximately lAB(2/3), respectively. These dynamical exponents define critical behavior distinguished from the class of uniformly driven systems of hard-core particles.

Driven pair contact process with diffusion

The pair contact process with diffusion (PCPD) has been recently investigated extensively, but its critical behavior is not yet clearly established. By introducing biased diffusion, we show that the external driving is relevant and the driven PCPD exhibits a mean-field-type critical behavior even in one dimension. In systems which can be described by a single-species bosonic field theory, the Galilean invariance guarantees that the driving is irrelevant. The well-established directed percolation (DP) and parity-conserving (PC) classes are such examples. This leads us to conclude that the PCPD universality class should be distinct from the DP or the PC class. Moreover, it implies that the PCPD is generically a multispecies model and a field theory of two species is suitable for proper description.

Generating function, path integral representation, and equivalence for stochastic exclusive particle systems

We present the path integral representation of the generating function for classical exclusive particle systems. By introducing hard-core bosonic creation and annihilation operators and appropriate commutation relations, we construct the Fock space structure. Using the state vector, the generating function is defined and the master equation of the system is transformed into the equation for the generating function. Finally, the solution of the linear equation for the generating function is derived in the form of the path integral. Applying the formalism, the equivalence of reaction-diffusion processes of single species and two species is described.

Intermittency of height fluctuations in stationary state of the Kardar-Parisi-Zhang equation with infinitesimal surface tension in 1+1 dimensions

The Kardar-Parisi-Zhang (KPZ) equation with infinitesimal surface tension, dynamically develops sharply connected valley structures within which the height derivative is not continuous. We discuss the intermittency issue in the problem of stationary state forced KPZ equation in 1+1 dimensions. It is proved that the moments of height increments C(a) = <|h(x(1)) - h(x(2))|(a)> behave as |x(1) - x(2)|(xi(a)) with xi(a) = a for length scales |x(1) - x(2)|<<sigma . The length scale sigma is the characteristic length of the forcing term. We have checked the analytical results by direct numerical simulation.

Effect of spatial bias on the nonequilibrium phase transition in a system of coagulating and fragmenting particles

We examine the effect of spatial bias on a nonequilibrium system in which masses on a lattice evolve through the elementary moves of diffusion, coagulation, and fragmentation. When there is no preferred directionality in the motion of the masses, the model is known to exhibit a nonequilibrium phase transition between two different types of steady state, in all dimensions. We show analytically that introducing a preferred direction in the motion of the masses inhibits the occurrence of the phase transition in one dimension, in the thermodynamic limit. A finite-size system, however, continues to show a signature of the original transition, and we characterize the finite-size scaling implications of this. Our analysis is supported by numerical simulations. In two dimensions, bias is shown to be irrelevant.

Statistical theory for the Kardar-Parisi-Zhang equation in (1+1) dimensions

The Kardar-Parisi-Zhang (KPZ) equation in (1+1) dimensions dynamically develops sharply connected valley structures within which the height derivative is not continuous. We develop a statistical theory for the KPZ equation in (1+1) dimensions driven with a random forcing that is white in time and Gaussian-correlated in space. A master equation is derived for the joint probability density function of height difference and height gradient P(h-h*, partial differential(x)h,t) when the forcing correlation length is much smaller than the system size and much larger than the typical sharp valley width. In the time scales before the creation of the sharp valleys, we find the exact generating function of h-h* and partial differential(x)h. The time scale of the sharp valley formation is expressed in terms of the force characteristics. In the stationary state, when the sharp valleys are fully developed, finite-size corrections to the scaling laws of the structure functions left angle bracket(h-h*)(n)(partial differential(x)h)(m)right angle bracket are also obtained.

Autonomous multispecies reaction-diffusion systems with more-than-two-site interactions

Autonomous multispecies systems with more-than-two-neighbor interactions are studied. Conditions necessary and sufficient for the closedness of the evolution equations of the n-point functions are obtained. The average numbers of the particles at each site for one species and three-site interactions, and its generalization to the more-than-three-site interactions, are explicitly obtained. Generalizations of the Glauber model in different directions, using generalized rates, generalized numbers of states at each site, and generalized numbers of interacting sites, are also investigated.

Path-integral formulation of stochastic processes for exclusive particle systems

We present a systematic formalism to derive a path-integral formulation for hard-core particle systems far from equilibrium. Writing the master equation for a stochastic process of the system in terms of the annihilation and creation operators with mixed commutation relations, we find the Kramers-Moyal coefficients for the corresponding Fokker-Planck equation (FPE), and the stochastic differential equation (SDE) is derived by connecting these coefficients in the FPE to those in the SDE. Finally, the SDE is mapped onto field theory using the path integral, giving the field-theoretic action, which may be analyzed by the renormalization group method. We apply this formalism to a two-species reaction-diffusion system with drift, finding a universal decay exponent for the long-time behavior of the average concentration of particles in arbitrary dimension.

Multispecies reaction-diffusion systems

Multispecies reaction-diffusion systems, for which the time evolution equations of correlation functions become a closed set, are considered. A formal solution for the average densities is found. Some special interactions and the exact time dependence of the average densities in these cases are also studied. For the general case, the large-time behavior of the average densities has also been obtained.

Finite-size effects of two-particle diffusion-limited reactions

We have studied the finite-size effects of two-particle diffusion-limited reaction A+B-->0, on a one-dimensional lattice using the Monte Carlo method. The density at a finite lattice follows a power law, C(t) approximately t(-x) below the crossover time, and shows an exponential decay above the crossover time. The crossover time depends on the lattice size and the bias field. We found second-order correction terms of the density decay as C(t) approximately t(-1/4)[1+O(t(-1/8))] for the isotropic diffusion of particles and C(t) approximately t(-1/3)[1+O(t(-1/24))] for the maximum drift. We proposed the scaling function of the density given as C(t) approximately L(-1/2)f(o)(t/L(2))+L(-3/4)f(1)(t/L(2)) for the isotropic diffusion and C(t) approximately L(-1/2)f(o)(t/L(3/2))+L(-9/16)f(1)(t/L(3/2)) for the maximum drift where f(o) and f(1) are scaling functions.