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Journal of Physics A Mathematical and General

Real-space renormalization for reaction-diffusion systems

Jef Hooyberghs, Carlo Vanderzande

Published Feb 1, 2000• 27 citations
Publisher

Abstract

The stationary state of stochastic processes such as reaction-diffusion systems can be related to the ground state of a suitably defined quantum Hamiltonian. Using this analogy, we investigate the applicability of a real-space renormalization group approach, originally developed for quantum spin systems, to interacting particle systems. We apply the technique to an exactly solvable reaction-diffusion system and to the contact process (both in d = 1). In the former case, several exact results are recovered. For the contact process, surprisingly good estimates of critical parameters are obtained from a small-cell renormalization.

DOI
10.1088/0305-4470/33/5/307

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The stationary state of stochastic processes such as reaction-diffusion systems can be related to the ground state of a suitably defined quantum Hamiltonian. Using this analogy, we investigate the applicability of a real-space renormalization group approach, originally developed for quantum spin systems, to interacting particle systems. We apply the technique to an exactly solvable reaction-diffusion system and to the contact process (both in d = 1). In the former case, several exact results are recovered. For the contact process, surprisingly good estimates of critical parameters are obtained from a small-cell renormalization.
The stationary state of stochastic processes such as reaction-diffusion systems can be related to the ground state of a suitably defined quantum Hamiltonian. Using this analogy, we investigate the applicability of a real space renormalisation group approach, originally developped for quantum spin systems, to interacting particle systems. We apply the technique to an exactly solvable reaction-diffusion system and to the contact process (both in d=1d=1d=1). In the former case, several exact results are recovered. For the contact process, surprisingly good estimates of critical parameters are obtained from a small-cell renormalisation.
In the theory of stochastic many particle systems one models phenomena of non-equilibrium statistical mechanics by means of a Markov process. The master equation, describing the time evolution of such a process, can formally be interpreted as a Schrodinger equation in imaginary time. In this approach, the (non-Hermitian) generator of the process plays the role of the Hamiltonian. This mathematical equivalence permits the successful application to stochastic systems of a number of exact, approximative or numerical methods from quantum mechanics.A particularly interesting type of stochastic systems is formed by the processes having a phase transition between an absorbing state and an active state. Examples of these systems can be found in physics, chemistry and even biology. The best known process of this kind is the contact process. In this thesis real-space renormalisation group techniques, which were originally developed for quantum mechanical systems, are applied to study stochastic models on a one-dimensional lattice, more precisely the critical behaviour and the universality of absorbing state phase transitions. First, the standard renormalisation group (SRG) is adapted to a stochastic context resulting in one of the few analytical techniques capable of producing approximations for systems that can not be solved exactly. The SRG is successfully applied to the contact process for which it characterises the stationary critical behaviour with a competitive accuracy. Next a generalisation of the contact process to a model with several absorbing states is studied by means of the density matrix renormalisation group (DMRG) algorithm. The results for the cases with one and two absorbing states are consistent with what was known from simulations done by an other author. For the models with more than two absorbing states new results were found, which are supported by analytical considerations in a particular limit. Finally the experience of the previous two projects is used in the investigation of the quenched random contact process. In this model the local interactions on a lattice site are time-invariant stochastic variables, independently drawn from a given distribution with a certain variance, which gives a measure for the disorder. Very little was known about the stationary behaviour of this model since its extremely slow dynamics makes simulations inefficient. The renormalisation techniques in this thesis, suffer less from this problem. First, for the case of small disorder the SRG indicates that the critical behaviour is changed by the presence of disorder and the cross-over exponent is consistent with the Harris criterion. For intermediate disorder a DMRG study reveals that the critical exponents are continuously varying with the amount of disorder. Finally, an SRG calculation, suitably adapted to strong disorder, shows that in this case the phase transition is characterised by a fixed point at infinite randomness. In this limit the renormalisation is expected to be exact. It is the first time exact critical exponents are found for a phase transition out of an absorbing state. Moreover, the exponents are the same as those previously found in a class of disordered quantum chains, suggesting a new kind of universality for strongly disordered systems.
Abstract We develop a time-dependent real-space renormalization-group approach which can be applied to Hamiltonians with time-dependent random terms. To illustrate the renormalization-group analysis, we focus on the quantum Ising Hamiltonian with random site- and time-dependent (adiabatic) transverse-field and nearest-neighbour exchange couplings. We demonstrate how the method works in detail, by calculating the off-critical flows and recovering the ground state properties of the Hamiltonian such as magnetization and correlation functions. The adiabatic time allows us to traverse the parameter space, remaining near-to the ground state which is broadened if the rate of change of the Hamiltonian is finite. The quantum critical point, or points, depend on time through the time-dependence of the parameters of the Hamiltonian. We, furthermore, make connections with Kibble–Zurek dynamics and provide a scaling argument for the density of defects as we adiabatically pass through the critical point of the system.
An efficient Monte Carlo simulation method for bosonic reaction-diffusion systems which are mainly used in the renormalization group (RG) study is proposed. Using this method, one-dimensional bosonic single species annihilation model is studied and, in turn, the results are compared with RG calculations. The numerical data are consistent with RG predictions. As a second application, a bosonic variant of the pair contact process with diffusion (PCPD) is simulated and shown to share the critical behavior with the PCPD. The invariance under the Galilean transformation of this boson model is also checked and discussion about the invariance in conjunction with other models are in order.
We present a real-space renormalization-group scheme for both spin and gauge systems within a Hamiltonian formalism. The approximation, in particular, preserves gauge invariance at every step of the calculation. We apply this scheme to the (1 + 1)-dimensional Ising model in a transverse field and to the (2 + 1)-dimensional Ising gauge theory. We find reasonable results for the critical coupling and for those critical exponents which are related to energy gaps. We also obtain the correct qualitative behavior for order and disorder parameters and correlation functions. In particular, the calculation yields exponential decay for correlation functions in the disordered phase. However, the critical indices we find for spacelike quantities are not good. This defect of the approximation is related to the asymmetric scaling of space and time under the renormalization group.