Equivalences between stochastic systems

Reaction-Limited Quantum Reaction-Diffusion Dynamics

We consider the quantum nonequilibrium dynamics of systems where fermionic particles coherently hop on a one-dimensional lattice and are subject to dissipative processes analogous to those of classical reaction-diffusion models. Particles can either annihilate in pairs, A+A→0, or coagulate upon contact, A+A→A, and possibly also branch, A→A+A. In classical settings, the interplay between these processes and particle diffusion leads to critical dynamics as well as to absorbing-state phase transitions. Here, we analyze the impact of coherent hopping and of quantum superposition, focusing on the so-called reaction-limited regime. Here, spatial density fluctuations are quickly smoothed out due to fast hopping, which for classical systems is described by a mean-field approach. By exploiting the time-dependent generalized Gibbs ensemble method, we demonstrate that quantum coherence and destructive interference play a crucial role in these systems and are responsible for the emergence of locally protected dark states and collective behavior beyond mean field. This can manifest both at stationarity and during the relaxation dynamics. Our analytical results highlight fundamental differences between classical nonequilibrium dynamics and their quantum counterpart and show that quantum effects indeed change collective universal behavior.

One-dimensional annihilating random walk with long-range interaction

We study the annihilating random walk with long-range interaction in one dimension. Each particle performs random walks on a one-dimensional ring in such a way that the probability of hopping toward the nearest particle is W=[1-ɛ(x+μ)^{-σ}]/2 (the probability of moving away from its nearest particle is 1-W), where x is the distance from the hopping particle to its nearest particle and ɛ, μ, and σ are parameters. For positive (negative) ɛ, a particle is effectively repulsed (attracted) by its nearest particle and each hopping is generally biased. On encounter, two particles are immediately removed from the system. We first study the survival probability and the mean spreading behaves in the long-time limit if there are only two particles in the beginning. Then we study how the density decays to zero if all sites are occupied at the outset. We find that the asymptotic behaviors are classified by seven categories: (i) σ>1 or ɛ=0, (ii) σ=1 and 2ɛ>1, (iii) σ=1 and 2ɛ=1, (iv) σ=1 and 2ɛ<1, (v) σ<1 and ɛ>0, (vi) σ=0 and ɛ<0, and (vii) 0<σ<1 and ɛ<0. The asymptotic behaviors in each category are universal in the sense that μ (and sometimes ɛ) cannot affect the asymptotic behaviors.

Scaling approach to related disordered stochastic and free-fermion models

Motivated by mapping from a stochastic system with spatially random rates, we consider disordered non-conserving free-fermion systems using a scaling procedure for the equations of motion. This approach demonstrates disorder-induced localization acting in competition with the asymmetric driving. We discuss the resulting implications for the original stochastic system.

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.

Static and dynamic phase transitions in multidimensional voting models on continua

A voting model (or a generalization of the Glauber model at zero temperature) on a multidimensional lattice is defined as a system composed of a lattice, each site of which is either empty or occupied by a single particle. The reactions of the system are such that two adjacent sites, one empty, the other occupied, may evolve to a state where both of these sites are either empty or occupied. The continuum version of this model in a D-dimensional region with a boundary is studied, and two general behaviors of such systems are investigated, the stationary behavior of the system, and the dominant way of relaxation of the system toward its stationary state. Based on the first behavior, a static phase transition (discontinuous changes in the stationary profiles of the system) is studied. Based on the second behavior, a dynamical phase transition (discontinuous changes in the relaxation times of the system) is studied. It is shown that the static phase transition is induced by the bulk reactions only, while the dynamical phase transition is a result of both bulk reactions and boundary conditions.

Perturbative calculation of one-point functions of one-dimensional single-species reaction-diffusion systems

Perturbations around autonomous one-dimensional single-species reaction-diffusion systems are investigated. It is shown that the parameter space corresponding to the autonomous systems is divided into two parts: In one part, the system is stable against the perturbations, in the sense that the largest relaxation time of the one-point functions changes continuously with perturbations. In the other part, however, the system is unstable against perturbations, so that any small perturbation drastically modifies the large-time behavior of the one-point functions.

General reaction-diffusion processes with separable equations for correlation functions

We consider general multispecies models of reaction diffusion processes and obtain a set of constraints on the rates which give rise to closed systems of equations for correlation functions. Our results are valid in any dimension and on any type of lattice. We also show that under these conditions the evolution equations for two point functions at different times are also closed. As an example we introduce a class of two species models that may be useful for the description of voting processes or the spreading of epidemics.

Dynamical phase transition of a one-dimensional kinetic Ising model with boundaries

The Glauber model on a one-dimensional lattice with boundaries (for the ferromagnetic and antiferromagnetic cases) is considered. The large-time behavior of the one-point function is studied. It is shown that at any temperature, the system shows a dynamical phase transition. The dynamical phase transition is controlled by the rate of spin flip at the boundaries, and is a discontinuous change of the derivative of the relaxation time towards the stationary configuration.

Correlation functions for diffusion-limited annihilation, A+A-->0

The full hierarchy of multiple-point correlation functions for diffusion-limited annihilation, A+A-->0, is obtained analytically and explicitly, following the method of intervals. In the long-time asymptotic limit, the correlation functions of annihilation are identical to those of coalescence, A+A-->A, despite differences between the two models in other statistical measures, such as the interparticle distribution function.

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.

Exactly solvable models through the empty-interval method

The most general one dimensional reaction-diffusion model with nearest-neighbor interactions that can be solved exactly through empty-interval method has been introduced. Assuming translationally invariant initial conditions, the probability that n consecutive sites are empty, E(n), has been exactly obtained. Here, however, we do not consider reactions changing two empty neighboring sites. In the thermodynamic limit, the large-time behavior of the system has also been investigated. Releasing translationally invariance, the evolution equation for the probability that n consecutive sites, starting from the site k, are empty, E(k,n), is obtained. In the thermodynamic limit, the large time behavior of the system is also considered. Finally, the continuum limit of the model is considered and the empty-interval probability function is obtained.

Phase transition in an asymmetric generalization of the zero-temperature q-state Potts model

An asymmetric generalization of the zero-temperature q-state Potts model on a one-dimensional lattice, with and without boundaries, has been studied. The dynamics of the particle number, and especially the large time behavior of the system, has been analyzed. In the thermodynamic limit, the system exhibits two kinds of phase transitions, a static and a dynamic phase transition.

Method of intervals for the study of diffusion-limited annihilation, A+A-->0

We introduce a method of intervals for the analysis of diffusion-limited annihilation, A+A-->0, on the line. The method leads to manageable diffusion equations whose interpretation is intuitively clear. As an example, we treat the following cases: (a) annihilation in the infinite line and in infinite (discrete) chains; (b) annihilation with input of single particles, adjacent particle pairs, and particle pairs separated by a given distance; (c) annihilation, A+A-->0, along with the birth reaction A-->3A, on finite rings, with and without diffusion.

Binary spreading process with parity conservation

Recently there has been a debate concerning the universal properties of the phase transition in the pair contact process with diffusion (PCPD) 2A-->3A, 2A-->0. Although some of the critical exponents seem to coincide with those of the so-called parity-conserving universality class, it was suggested that the PCPD might represent an independent class of phase transitions. This point of view is motivated by the argument that the PCPD does not conserve parity of the particle number. In the present work we question what happens if the parity conservation law is restored. To this end, we consider the reaction-diffusion process 2A-->4A, 2A-->0. Surprisingly, this process displays the same type of critical behavior, leading to the conclusion that the most important characteristics of the PCPD is the use of binary reactions for spreading, regardless of whether parity is conserved or not.

Exact solution of a class of one-dimensional nonequilibrium stochastic models

We consider various one-dimensional nonequilibrium models, namely, the diffusion-limited pair-annihilation and creation model (DPAC) and its unbiased version (the Lushnikov model), the DPAC model with particle injection, as well as (biased) diffusion-limited coagulation model (DC). We study the DPAC model using an approach based on a duality transformation and the generating function of the dual model. We are able to compute exactly the density and correlation functions in the general case with arbitrary initial states. Further, we assume that a source injects particles in the system. Solving, via the duality transformation, the equations of motion of the density, and the noninstantaneous two-point correlation functions, we see how the source affects the dynamics. Finally we extend the previous results to the DC model with help of a similarity transformation.

Phase transition in an asymmetric generalization of the zero-temperature Glauber model

An asymmetric generalization of the zero-temperature Glauber model on a lattice is introduced. The dynamics of the particle-density and especially the large-time behavior of the system is studied. It is shown that the system exhibits two kinds of phase transitions, a static one and a dynamic one.

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.

Diffusion in disordered lattices and related heisenberg ferromagnets

We study the diffusion of classical hard-core particles in disordered lattices within the formalism of a quantum spin representation. This analogy enables an exact treatment of noninstantaneous correlation functions at finite particle densities in terms of single spin excitations in disordered ferromagnetic backgrounds. Applications to diluted chains and percolation clusters are discussed. It is found that density fluctuations in the former exhibit a stretched exponential decay while an anomalous power law asymptotic decay is conjectured for the latter.

Exact density profile of a stochastic reaction-diffusion process

We calculate exactly the time dependent density profile of a one-dimensional stochastic reaction-diffusion process of hard-core particles subjected to the reactions AA <--> OO and AO <--> OA. The solution is based on the fundamental property that the evolution operator, defined over an appropriate vector space, transforms vectors with n kinks into vectors with n or n+2 kinks, only. In this space, a basis vector is represented by a string of plus and minus signs and a kink is defined as a pair of opposite signs. The exact time dependent profiles are calculated for the cases of uncorrelated initial states that are translational invariant as well as initial states that are inhomogeneous in space.