Hans-Karl Janssen

New universal short-time scaling behaviour of critical relaxation processes

We study the critical relaxation properties of Model A (purely dissipative relaxation) starting from a macroscopically prepared initial state characterised by non-equilibrium values for order parameter and correlations. Using a renormalisation group approach we observe that even (macroscopically)early stages of the relaxation process display universal behaviour governed by a new, independent “initial slip” exponent. For large times, the system crosses over to the well-known long-time relaxation behaviour.The new exponent is calculated toO(ε2) in ε=4−d, whered is the spatial dimension of the system. The initial slip scaling form of general correlation and response functions as well as the order parameter is derived, exploiting a short-time operator expansion. The leading scaling behaviour is determined by initial states with sharp values of the order parameter. Non-vanishing correlations generate corrections to scaling.

On a Lagrangean for Classical Field Dynamics and Renormalization Group Calculations of Dynamical Critical Properties

AbstractFrom the path probability density for nonlinear stochastic processes a Lagrangean for classical field dynamics is derived. This formulation provides a convenient approach to the mode coupling equations and the renormalization group theory of critical dynamics. An application is given for the time-dependent isotropic Heisenberg ferromagnet. The dynamical exponent

Renormalized field theory of dynamical percolation

z = \frac{{d + 2 - \eta }}{2}

On the crossover to universal criticality in dilute Ising systems

is derived aboveTc for all dimensionsd>2.

Field theory of long time behaviour in driven diffusive systems

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 Critical Exponents and the Renormalization of the Coupling Constant in Growth Models with Surface Diffusion

The stochastic processes recently introduced by Grassberger and Cardy are shown to belong to the same universality class as dynamic percolation. To first order in ε=6−d, whered is the spatial dimensionality, known critical exponents of random percolation are rederived and the new dynamic exponent z=2−1/6ε+0(ε2) is calculated by use of the field-theoretic renormalization group method. The relation of the statistics of big clusters (animals) to the Yang-Lee edge singularity in random fields is presented.

Directed Percolation with Colors and Flavors

We present a field theoretic renormalisation group study for the critical behaviour of a diffusive system with a single conserved density subjected to an external driving force. The anisotropies induced by the external field require the introduction of two critical parameters associated with transverse and longitudinal order. The transition to transverse order is governed by a fixed point which is infrared stable below five dimensions. With the help of Ward-Takahashi identities based on Galilei invariance, we derive scaling forms for density correlation functions, critical exponents to all orders in ε=5−d, and the equation of state, taking care of a dangerous irrelevant composite operator. The transition is continuous and of mean-field type, with anomalous long-wavelength and long-time correlations in the longitudinal direction only. For the transition to longitudinal order, no infrared stable fixed point is found. An analysis of the mean-field equations indicates that the transition is discontinuous.

Non-equilibrium relaxation at a tricritical point

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.