K. Oerding

WILSON RENORMALIZATION OF A REACTION-DIFFUSION PROCESS

Healthy and sick individuals (A and B particles) diffuse independently with diffusion constants DA and DB. Sick individuals upon encounter infect healthy ones (at rate k), but may also spontaneously recover (at rate 1/τ). The propagation of the epidemic therefore couples to the fluctuations in the total population density. Global extinction occurs below a critical value ρc of the spatially averaged total density. The epidemic evolves as the diffusion–reaction–decay process A+B→2B,B→A, for which we write down the field theory. The stationary-state properties of this theory when DA=DB were obtained by Kree et al. The critical behavior for DA DB remains unsolved.

Lévy-flight spreading of epidemic processes leading to percolating clusters

Abstract:We consider two stochastic processes, the Gribov process and the general epidemic process, that describe the spreading of an infectious disease. In contrast to the usually assumed case of short-range infections that lead, at the critical point, to directed and isotropic percolation respectively, we consider long-range infections with a probability distribution decaying in d dimensions with the distance as . By means of Wilsons momentum shell renormalization-group recursion relations, the critical exponents characterizing the growing fractal clusters are calculated to first order in an -expansion. It is shown that the long-range critical behavior changes continuously to its short-range counterpart for a decay exponent of the infection .

On the crossover to universal criticality in dilute Ising systems

Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.

Fluctuation-Induced First-Order Transition in a Nonequilibrium Steady State

We present the first example of a phase transition in a nonequilibrium steady state that can be argued analytically to be first order. The system of interest is a two-species reaction-diffusion problem whose control parameter is the total density ρ. Mean-field theory predicts a second-order transition between two stationary states at a critical density ρ=ρc. We develop a phenomenological picture that instead predicts a first-order transition below the upper critical dimension dc=4. This picture is confirmed by hysteresis found in numerical simulations, and by the study of a renormalization-group improved equation of state. The latter approach is inspired by the Coleman–Weinberg mechanism in QED.

Non-equilibrium relaxation at a tricritical point

We study the purely relaxational tricritical dynamics of a non-conserved order parameter (model A) in the upper critical dimension dc=3 and in 3- epsilon dimensions. We are especially interested in the relaxation, starting from a macroscopically prepared initial state with a small correlation length. Using the methods of renormalized field theory we obtain the scaling behaviour of the correlation and response functions and study the nonlinear relaxation of the order parameter M(t). In three dimensions M(t) displays a crossover from the purely logarithmic short-time behaviour M(t) approximately (In(t/t0))-a to a t- 14 / power law with logarithmic corrections. For dimensions d<3 we obtain the exponents which govern the tricritical relaxation at lowest non-trivial order in epsilon =3-d. The dynamic scaling exponent z is calculated at second order in epsilon .

Nonequilibrium critical relaxation with coupling to a conserved density

The authors study nonequilibrium critical relaxation properties of model C (purely dissipative relaxation of an order parameter coupled to a conserved density) starting from a macroscopically prepared initial state with short-range correlations. Using a field-theoretic renormalization group approach they show that all the stages of growth of the correlation length display universal behaviour governed by a new critical exponent theta . This exponent is calculated to second order in in =4-d where d is the spatial dimension of the system.

Global persistence in directed percolation

We consider a directed percolation process at its critical point. The probability that the deviation of the global order parameter with respect to its average has not changed its sign between 0 and t decays with t as a power law. In space dimensions the global persistence exponent that characterizes this decay is while for d<4 its value is increased to first order in . Combining a method developed by Majumdar and Sire with renormalization group techniques we compute the correction to to first order in . The global persistence exponent is found to be a new and independent exponent. Finally we compare our results with existing simulations.

Fermionic field theory for directed percolation in (1 + 1)-dimensions

We formulate directed percolation in (1 + 1) dimensions in the language of a reaction-diffusion process with exclusion taking place in one space dimension. We map the master equation that describes the dynamics of the system onto a quantum spin chain problem. From there we build an interacting fermionic field theory of a new type. We study the resulting theory using renormalization group techniques. This yields numerical estimates for the critical exponents and provides a new alternative analytic systematic procedure to study low-dimensional directed percolation.

Non-equilibrium critical relaxation with reversible mode coupling

The non-equilibrium critical relaxation of systems in which the order parameter couples reversibly to conserved densities is investigated. The initial state which may be macroscopically prepared by a quench from a temperature T >> Tc to the critical temperature Tc is characterized by short-range correlations. In the case where the order parameter itself is conserved the relaxation shows universal scaling. If, on the other hand, the order parameter is not conserved the scaling is governed by a new exponent which depends on the width of the initial distribution of the conserved fields. However, the response function of the conserved fields still shows universal scaling.

Equation of state for directed percolation

Using field-theoretic renormalization group methods we calculate the equation of state for non-equilibrium systems belonging to the universality class of directed percolation to second order in . By introducing a parametric representation the result can be written to this order in a very simple form. We use our result to obtain a universal amplitude ratio to second order in .