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arXiv:cond-mat/0010062 · Journal of Physics A Mathematical and General

Exact solution of a reaction-diffusion process with three-site interactions

Malte Henkel, Haye Hinrichsen B

Published Feb 19, 2001• 48 citations
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Abstract

The one-dimensional reaction diffusion process AA → A and A ∅ A → AAA is exactly solvable through the empty interval method if the diffusion rate equals the coagulation rate. Independently of the particle production rate, the model is always in the universality class of diffusion-annihilation. This allows us to check analytically the universality of finite-size scaling in a non-equilibrium critical point.

DOI
10.1088/0305-4470/34/8/303

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The one-dimensional reaction diffusion process AA → A and A ∅ A → AAA is exactly solvable through the empty interval method if the diffusion rate equals the coagulation rate. Independently of the particle production rate, the model is always in the universality class of diffusion-annihilation. This allows us to check analytically the universality of finite-size scaling in a non-equilibrium critical point.
We generalize nonperturbative renormalization group methods to nonequilibrium critical phenomena. Within this formalism, reaction-diffusion processes are described by a scale-dependent effective action, the flow of which is derived. We investigate branching and annihilating random walks with an odd number of offspring. Along with recovering their universal physics (described by the directed percolation universality class), we determine their phase diagrams and predict that a transition occurs even in three dimensions, contrarily to what perturbation theory suggests.
The steady-state phase diagram of the one-dimensional reaction-diffusion model $2\stackrel{\ensuremath{\rightarrow}}{A}3A,$ $2\stackrel{\ensuremath{\rightarrow}}{A}0$ 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 $\ensuremath{\theta}={\ensuremath{\nu}}_{\ensuremath{\Vert}}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ and $\ensuremath{\beta}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ are calculated numerically. The value of $\ensuremath{\beta}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ is close to the value of the parity conserving universality class, in spite of the absence of local conservation laws.
Many non-equilibrium systems display dynamic phase transitions from active to absorbing states, where fluctuations cease entirely. Based on a field theory representation of the master equation, the critical behavior can be analyzed by means of the renormalization group. The resulting universality classes for single-species systems are reviewed here. Generically, the critical exponents are those of directed percolation (Reggeon field theory), with critical dimension d_c = 4. Yet local particle number parity conservation in even-offspring branching and annihilating random walks implies an inactive phase (emerging below d_c' = 4/3) that is characterized by the power laws of the pair annihilation reaction, and leads to different critical exponents at the transition. For local processes without memory, the pair contact process with diffusion represents the only other non-trivial universality class. The consistent treatment of restricted site occupations and quenched random reaction rates are important open issues.
We present a Monte Carlo study in dimension d=1d=1d=1 of the two-species reaction-diffusion process $A+\stackrel{\ensuremath{\rightarrow}}{B}2B$ and $\stackrel{\ensuremath{\rightarrow}}{B}A.$ Below a critical value ${\ensuremath{\rho}}_{c}$ of the conserved total density $\ensuremath{\rho}$ the system falls into an absorbing state without B particles. Above ${\ensuremath{\rho}}_{c}$ the steady state B particle density ${\ensuremath{\rho}}_{B}^{\mathrm{st}}$ is the order parameter. This system is related to directed percolation but in a different universality class identified by Kree et al. [Phys. Rev. A 39, 2214 (1989)]. We present an algorithm that enables us to simulate simultaneously the full range of densities $\ensuremath{\rho}$ between zero and some maximum density. From finite-size scaling we obtain the steady state exponents $\ensuremath{\beta}=0.435(10),\ensuremath{\nu}=2.21(5),$ and $\ensuremath{\eta}=\ensuremath{-}0.606(4)$ for the order parameter, the correlation length, and the critical correlation function, respectively. Independent simulation indicates that the critical initial increase exponent takes the value ${\ensuremath{\theta}}^{\ensuremath{'}}=0.30(2),$ in agreement with the theoretical relation ${\ensuremath{\theta}}^{\ensuremath{'}}=\ensuremath{-}\ensuremath{\eta}/2$ due to Van Wijland et al. [Physica A 251, 179 (1998)].
We study in further detail particle models displaying a boundary-induced absorbing state phase transition [Deloubrière and van Wijland Phys. Rev. E 65, 046104 (2002) and Barato and Hinrichsen, Phys. Rev. Lett. 100, 165701 (2008)]. These are one-dimensional systems consisting of a single site (the boundary) where creation and annihilation of particles occur, and a bulk where particles move diffusively. We study different versions of these models and confirm that, except for one exactly solvable bosonic variant exhibiting a discontinuous transition and trivial exponents, all the others display nontrivial behavior, with critical exponents differing from their mean-field values, representing a universality class. Finally, the relation of these systems with a (0+1)-dimensional non-Markovian process is discussed.
The pair-contact process 2A->3A, 2A->0 with diffusion of individual particles is a simple branching-annihilation processes which exhibits a phase transition from an active into an absorbing phase with an unusual type of critical behaviour which had not been seen before. Although the model has attracted considerable interest during the past few years it is not yet clear how its critical behaviour can be characterized and to what extent the diffusive pair-contact process represents an independent universality class. Recent research is reviewed and some standing open questions are outlined.