A renormalisation group study of one-dimensional contact processes.

摘要

In principle all phenomena in nature can be described by fundamental physical laws. For realistic macroscopic systems the number of degrees of freedom is often too high. Instead, statistical approximative techniques are used. The best known case is represented by models for thermal equilibrium described by the Gibbs ensemble.; Most systems in nature are however not in thermal equilibrium . In this thesis one such type of systems is considered: reaction-diffusion systems. The main concept of reaction diffusion systems is that particles diffuse and/or react in a medium. In order to model the stochastic dynamics of these systems, a Markov process is used on a discrete configuration space. The master equation, describing the time evolution of such a process, can formally be interpreted as a Schrödinger 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.; In this thesis real-space renormalisation group techniques are applied to study stochastic models on a one-dimensional lattice. The critical behaviour and the universality of absorbing state phase transitions are studied.; 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. 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. Finally the experience of the previous two projects is used in the investigation of the quenched random contact process. In the limit of strong disorder exact results are derived for this model. 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.