I present an analytic nonlinear-response approximation which yields estimates for the field, temperature, and orientation dependences of the velocity of an interface in two-dimensional kinetic Ising and Solid-On-Solid (SOS) models, driven by a  nonzero field at temperatures below the bulk critical temperature [1-3]. The interface mobility depends on the local interface structure, and the SOS approximation is used to estimate field-dependent mean spin-class populations, from which the mean interface velocity can be obtained for any specific single-spin-flip dynamic. In the low-temperature limit the standard polynuclear growth and single-step models are recovered for interfaces making small and large angles with the latice-symmetry directions, respectively. In the case of Glauber dynamics the analytic results are compared with Monte Carlo simulations. Very satisfactory agreement is found in a wide range of field, temperature, and interface orientation, both for the Ising and SOS models. For the latter model I also present results for the correlations between nearest-neighbor step heights, illustrating how the up-down symmetry of the interface is gradually destroyed as the field is increased. The field dependence of the interface velocity depends strongly on the form of the Monte Carlo transition probabilities. For dynamics in which the effects of the interaction energies and the field energy do not factorize (``hard'' dynamics), the velocity increases dramatically with the field, while there is no field dependence for dynamics in which the effects factorize (``soft'' dynamics) [2].

1. P. A. Rikvold and M. Kolesik, J. Stat. Phys. 100, 377 (2000).
2. P. A. Rikvold and M. Kolesik, J. Phys. A 35, L117 (2002).
3. P. A. Rikvold and M. Kolesik, Phys. Rev. E 66, 066116 (2002).

We conduct Monte Carlo renormalization group simulations of two-dimensional two-state driven lattice gases. The quantity we calculate is the measure entropy (the entropy density) from a converging sequence of conditional entropies of small shapes. We use this entropy in the renormalization procedure from which we calculate some of the critical exponents. We discuss the applicability of some new information-theoretic measures of spatial two-dimensional patterns in the study of systems out of equilibrium.

In a historical introduction I will review the properties of the classical two-dimensional XY (or O(2)) and Heisenberg (or O(3)) model. I will argue why it is interesting to investigate the O(3) model with a nonlinear nearest-neighbor interaction V(\vec{s},\vec{s}\,')=2K[(1+\vec{s}\cdot\vec{s}\,')/2 ]^p. The analogous nonlinear interaction for the XY model was studied by Domany, Schick, and Swendsen, who found that for large p the Kosterlitz-Thouless transition is preempted by a first-order transition. I will show that, whereas the standard (p=1) Heisenberg model has no phase transition, for large enough p a first-order transition appears. Both phases have only short range order, but with a correlation length that jumps at the transition.

The boundaries of clusters in 2d systems at a second-order phase transition may be thought of as certain kinds of dynamically generated random walks. The continuum limit of this process is described by an evolution equation called SLE, many of whose properties are analytically tractable. Using this approach, all the known results about 2d critical behaviour may be recovered, as well as some new ones.

Boltzmann's hypothesis is at the heart of equilibrium statistical mechanics. For classical systems it asserts that the probability distribution function in phase space is uniform on the constant energy surface for a N-particle system. It is natural to ask about the properties of the phase space distribution for non-equilibrium systems. Here we consider some stationary and non-stationary non-equilibrium distribution functions for simple chaotic systems. We will show that these non-equilibrium distribution functions have strong fractal properties, and that important macroscopic properties of the non-equilibrium systems are encoded in the mathematical structure of these fractal forms. Of particular interest is a formula relating a transport coefficient - in the case to be discussed, a diffusion coefficient - to the Lyapunov exponent characterizing the chaotic dynamics and the fractal dimension of a curve appearing naturally in the description of the non-equilibrium state of the system. We conclude with a brief discussion of the way non-equilibrium entropy production is connected to the fractal distribution functions.

Using Monte Carlo simulations and finite-size scaling methods we study ``wetting'' in Ising systems in a L x L x L_y pore with quadratic cross section. Antisymmetric surface fields H_s act on the free L x L_y surfaces of the opposing wedges, and periodic boundary conditions are applied along the y-direction. Our results represent the first simulational observation of fluctuation effects in three dimensional wetting phenomena and corroborate recent predictions on wedge filling. In the limit L -> oo, L_y / L^3 = const. the system exhibits a new type of phase transition, which is the analog of the ``filling transition'' that occurs in a single wedge. It is characterized by critical exponents alpha=3/4, beta=0, gamma =5/4 for the specific heat, order parameter, and susceptibility, respectively.

A model for the solar coronal magnetic field is proposed where multiple directed loops evolve in space and time. Loops injected at small scales are anchored by footpoints of opposite polarity moving randomly on a surface. Nearby footpoints of the same polarity aggregate, and loops can reconnect when they collide. This may trigger a cascade of further reconnection, representing a solar flare. Numerical simulations show that a power law distribution of flare energies emerges, associated with a scale-free network of loops, indicating self-organized criticality.

I will present recent results concerning the phase diagram of a lattice gas driven far from equilibrium. In our model two species of particles are driven in opposite directions around a periodic lattice. The particles interact via excluded volume and nearest-neighbor attractions, leading to a rich diagram of three distinct phases.

Pure and doped crystals of the palmierite-type like lead phosphate or lead arsenate undergo ferroelastic phase transitions from a rhombohedral paraphase to monoclinic low-temperature phases forming simple ferroelastic domain patterns. Dynamic precursor clusters lead to diffuse scattering in the paraphase. The critical behaviour is renormalised by impurities. In the ferrophase the intersection of ferroelastic W domain walls with the (100) surface and their internal gradientstructure has been studied by atomic force microscopy, inelastic light scattering and diffraction techniques using synchrotron radiation. Technically important might be the observation that ferroelastic domainwalls show impurity-controlled conductivity.

A general physical model for foam evolution is developed. The involved partial differential equations, i.e. the Navier Stokes and convection diffusion equation, are solved numerically with the help of the Lattice Boltzmann method (LBM). The LBM utilizes very simple rules on a microscopic level, which obey mass, momentum, and energy conservation. It will be outlined by a multiscale expansion that the microscopic properties of this system follow in a good approximation the desired continuum equations. Local free surface boundary conditions are constructed to model the gas-liquid interface of the foam system. Various simulation results of the evolution of the foam structure are presented.