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arXiv:cond-mat/9903161

Real space renormalisation for reaction-diffusion systems

Jef Hooyberghs, Carlo Vanderzande

Published Mar 10, 1999• 5 citations
arXivHTMLPDF

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 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 $d=1$). 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.

DOI
10.48550/arxiv.cond-mat/9903161

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Real-space renormalization for reaction-diffusion systems

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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.

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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 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.
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.
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.
Introduction The real space renormalisation group has a wide application in the study of classical critical phenomena. For quantum systems, several versions of the real space renormalisation group have been proposed (Jullien, 1981; Pfeuty et al ., 1982; dos Santos, 1982). The real space RG is the most direct way of implementing the scaling ideas presented in Chapter 1. The operation of elimination of degrees of freedom as the system is reduced to blocks of smaller sizes under a change of length scale is directly and intuitively represented as mathematical equations. In particular, the block method has been used to study different problems of strongly correlated systems (Pfeuty et al ., 1982) and spin chains (White, 1992, 1993) with different degrees of sophistication. We show here in detail a simple form of this technique applied to the case of the one-dimensional Ising model in a transverse field. The purpose is to illustrate the ideas introduced earlier in Chapter 1 with a concrete example where a quantum critical point is associated with an unstable fixed point. Also, we show how the stable attractors or fixed points can characterise the different phases and how the critical exponents can be obtained. Although the actual values for the quantum critical point and critical exponents that we obtain using a small block are not exact, a procedure is outlined which can be used to improve these results. In this sense, the block method presented here represents a controlled approximation, which can be systematically improved to yield in the limit of very large blocks essentially the correct results. When compared to the k -space version of the RG (to be presented in the next two chapters), the real space version turns out to be in general more difficult to control, in the sense of producing a systematic improvement of the results. On the other hand, this method yields without much effort the attractive fixed points of the different phases, allowing for their characterisation. This turns out to be useful since the nature of the phase transition, whether first or second order, can be determined by exponents calculated in the attractor of this phase (Nienhuis and Naunberg, 1975).
The real-space renormalisation group method can be applied to the Chalker–Coddington model of the quantum Hall transition to provide a convenient numerical estimation of the localisation critical exponent, ν . Previous such studies found ν ∼ 2 . 39 which falls considerably short of the current best estimates by transfer matrix ( ν = 2 . 593 + 0 . 005 − 0 . 006 ) and exact-diagonalisation studies ( ν = 2 . 58 ( 3 ) ). By increasing the amount of data 500 fold we can now measure closer to the critical point and find an improved estimate ν = 2 . 51 + 0 . 11 − 0 . 11 . This deviates only ∼ 3% from the previous two values and is already better than the ∼ 7% accuracy of the classical small-cell renormalisation approach from which our method is adapted. We also study a previously proposed mixing of the Chalker–Coddington model with a classical scattering model which is meant to provide a route to understanding why experimental estimates give a lower ν ∼ 2 . 3 . Upon implementing this mixing into our RG unit, we find only further increases to the value of ν .
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.