Dynamical critical exponent for two-species totally asymmetric diffusion on a ring (original) (raw)

Reaction-Diffusion Processes, Critical Dynamics, and Quantum Chains

Annals of Physics, 1994

The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schrödinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions. Since many one-dimensional quantum chains are integrable, this opens a new field of applications. At the same time physical intuition and probabilistic methods bring new insight into the understanding of the properties of quantum chains. A simple example is the asymmetric diffusion of several species of particles which leads naturally to Hecke algebras and qdeformed quantum groups. Many other examples are given. Several relevant technical aspects like critical exponents, correlation functions and finite-size scaling are also discussed in detail.

Critical properties of the reaction-diffusion model 2A[over ?]3A, 2A[over ?]0

Phys Rev E, 2001

The steady-state phase diagram of the one-dimensional reaction-diffusion model 2A→3A, 2A→0 is studied through the non-Hermitian density matrix renormalization group. In the absence of single-particle diffusion the model reduces to the pair-contact process, which has a phase transition in the universality class of directed percolation ͑DP͒ and an infinite number of absorbing steady states. When single-particle diffusion is added, the number of absorbing steady states is reduced to 2 and the model no longer shows DP critical behavior. The exponents ϭ ʈ / Ќ and ␤/ Ќ are calculated numerically. The value of ␤/ Ќ is close to the value of the parity conserving universality class, in spite of the absence of local conservation laws.

Critical properties of the reaction-diffusion model 2A-->3A, 2A-->0

Phys Rev E, 2001

The steady-state phase diagram of the one-dimensional reaction-diffusion model 2A-->3A, 2A-->0 is studied through the non-Hermitian density matrix renormalization group. In the absence of single-particle diffusion the model reduces to the pair-contact process, which has a phase transition in the universality class of directed percolation (DP) and an infinite number of absorbing steady states. When single-particle diffusion is added, the number of absorbing steady states is reduced to 2 and the model no longer shows DP critical behavior. The exponents θ=ν||/ν⊥ and β/ν⊥ are calculated numerically. The value of β/ν⊥ is close to the value of the parity conserving universality class, in spite of the absence of local conservation laws.

Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries

Journal of Physics A: Mathematical and General, 1996

We consider the one-dimensional partially asymmetric exclusion model with open boundaries. The model describes a system of hard-core particles that hop stochastically in both directions with different rates. At both boundaries particles are injected and extracted. By means of the method of Derrida, Evans, Hakim and Pasquier the stationary probability measure can be expressed as a matrix-product state involving two matrices forming a Fock-like representation of a general quadratic algebra. We obtain the representations of this algebra, which were unknown in the mathematical literature and use the two-dimensional one to derive exact expressions for the density profile and correlation functions. Using the correspondence between the stochastic model and a quantum spin chain, we obtain exact correlation functions for a spin-1 2 Heisenberg XXZ chain with non-diagonal boundary terms. Generalizations to other reaction-diffusion models are discussed.

Soluble two-species diffusion-limited models in arbitrary dimensions

Physical Review E, 2001

A class of two-species (three-states) bimolecular diffusion-limited models of classical particles with hard-core reacting and diffusing in a hypercubic lattice of arbitrary dimension is investigated. The manifolds on which the equations of motion of the correlation functions close, are determined explicitly. This property allows to solve for the density and the two-point (two-time) correlation functions in arbitrary dimension for both, a translation invariant class and another one where translation invariance is broken. Systems with correlated as well as uncorrelated, yet random initial states can also be treated exactly by this approach. We discuss the asymptotic behavior of density and correlation functions in the various cases. The dynamics studied is very rich. PACS number(s): 02.50.-r, 02.50.Ey, 05.50+q , 82.20.Db, 82.20.Mj

Bethe ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the Karder--Parisi--Zhang-type growth model

Physical Review E - PHYS REV E, 1995

A perturbative method is developed to calculate the finite size corrections of the low lying energies of the asymmetric XXZ hamiltonian near the stochastic line. The crossover from isotropic to anisotropic, Kardar-Parisi-Zhang (KPZ) scaling of the mass gaps is determined in terms of universal crossover scaling functions. At the stochastic line, the asymmetric XXZ hamiltonian describes the time evolution of the single-step or body-centered solid-on-solid growth model in one dimension. The mass gaps of the growth model are found as a function of the growth rate and the substrate slope. Higher order corrections to the growth model mass gaps are also calculated to obtain the first terms of the KPZ to Edward-Wilkinson crossover scaling function in the large argument expansion in the zero slope sector.

Mappings between reaction-diffusion and kinetically constrained systems: A+A <-> A and the FA model have upper critical dimension d_c=2

2006

We present an exact mapping between two simple spin models: the Fredrickson-Andersen (FA) model and a model of annihilating random walks with spontaneous creation from the vacuum, A+A <-> 0. We discuss the geometric structure of the mapping and its consequences for symmetries of the models. Hence we are able to show that the upper critical dimension of the FA model is two, and that critical exponents are known exactly in all dimensions. These conclusions also generalise to a mapping between A+A <-> 0 and the reaction-diffusion system in which the reactions are branching and coagulation, A+A <-> A. We discuss the relation of our analysis to earlier work, and explain why the models considered do not fall into the directed percolation universality class.

Critical Exponents in the One-Dimensional Hubbard Model

International Journal of Modern Physics B, 1994

Exact Bethe Ansatz results on the spectrum of large but finite Hubbard chains in conjunction with methods from conformal quantum field theory can be used to obtain exact results for the asymptotic behaviour of correlation functions. We review this method and discuss some interesting consequences of the results.

Critical behavior of a two-species reaction–diffusion problem in 2D

Physica A-statistical Mechanics and Its Applications, 2007

In this work we study the critical behavior of a model that simulates the propagation of an epidemic process over a population. We simulate the model on two-dimensional finite lattices to determine the critical density of the diffusive population. A finite size scaling analysis is employed to determine the order parameter and correlation length critical exponents. r