The investigation of the decay from a metastable state has been the subject of numerous studies over very many years, but the analogous problem of the decay from an unstable state and the selection of one of several isolated minima as the final state, has received comparatively little attention. However this situation is frequently encountered when complex systems driven far from equilibrium encounter instabilities. In this talk I will describe the problem of state selection for a stochastic system,  initially in an unstable stationary state, when multiple metastable states compete for occupation. The lack of adequate theoretical tools in a problem which involves coupled non-linear Langevin equations has been the major obstacle to progress, but I will show how path-integral methods are able to provide a systematic approach to the calculation of probabilities of various states being selected.

Vegetation in (semi-)arid zones tends to form clumps instead of being homogeneously distributed. Surprisingly, very little seems to be known, to date, about the fundamental mechanisms responsible for  this form of self-organization. I will present an attempt (work still in progress !) to formulate a minimal model of the biomass dynamics accounting for this type of pattern formation. In order to motivate this work further, I will also qualitatively compare our statistical physicists' approach to the typical one of ecologists.  Finally, I intend to highlight the surprising and very strong similarity between this minimal model and an existing one for the neurophysiology of the eye.

The dynamic response of a forest-fire model to the harmonic variation of an external parameter is studied by means of numerical simulations. Second-order phase transitions are found, driven by the harmonic input. By means of epidemic studies the relevant exponents can be determined, which allow us to place the transition in the directed-percolation universality class.

The time evolution of structure factors in the disordering process of an initially phase separated lattice depends crucially on the microscopic mechanism which drives the disordering, such as Kawasaki dynamics or vacancy-mediated disordering.  Monte Carlo simulations show unexpected "dips" in the structure factors, which mean-field theory completely fails to capture. The disordering via vacancies is slower by a surprisingly large constant factor compared to Kawasaki dynamics.  A phenomenological model is introduced in order to understand the dips, and an analytical solution of Kawasaki dynamics is derived, in excellent agreement with simulations.  An outline is given on how to extend the analytical solution to the more complicated case of a wandering vacancy.

We use Monte Carlo simulations of the 3D uniformly frustrated XY model, with uncorrelated random couplings, to model the effects of random point pins on the vortex line phases of an extreme type-II superconductor.  Mapping out the phase diagram as a function of temperature and disorder strength p, we find that the vortex line lattice melting transition remains 1st order as p increases.  At large enough p, the phase boundary turns parallel to the temperature axis, so that vortex lattice order is always destroyed above a critical disorder strength p_c. No evidence is found for a phase coherent vortex glass above p_c.

What happens if ordinary matter is compressed to a density that exceeds the density inside atomic nuclei? This question is of interest not only for our understanding of quantum chromodynamics (QCD), but it has practical consequences for the structure of neutron stars and the physics of relativistic heavy ion collisions. We will summarize recent progress in the theory of superdense matter. We will show that the phase structure of dense matter is very rich. Some of these phases exhibit chiral symmetry breaking and the formation of a mass gap in a regime where the coupling constant is weak and controlled calculations are possible.

For equilibrium systems, the qualitative shape of the phase diagram can often be obtained from simple energy-entropy arguments. As yet, we do not know how to extend such methods to systems far from equilibrium. It is therefore interesting to study models with simple equilibrium phase diagrams, and drive them out of equilibrium in a controlled fashion. As an example, I will discuss the phase diagram of a system with two layers of an Ising lattice gas at half filling, as a function of the inter-layer coupling J, keeping the usual intra-layer nearest neighbor attraction constant. In equilibrium, the phase diagram is symmetric under J to -J, but exhibits different ground states. When the system is subjected to a uniform drive, the phase diagram changes shape in a completely non-intuitive way. Both simulation and field theory results will be presented, analyzing the phase boundaries and the transitions across them.

This is a continuation of previous seminars on the dynamics of the Penna-Desai model. After briefly reviewing the model and previous results,  more recent studies will be covered. This will include an analytical description of two similar species competition without mutations, two species competitions with mutations including similar and dissimilar species, sticky phases, transition times, and scaling, and three species competitions both with and without mutations. Finally a plethora of future research directions will be discussed.

Phase transitions in thermodynamics are ordinarily defined for infinite, homogenous systems, making use of the singular behavior of e.g. the  specific heat at the phase tranistion temperature. For finite systems this singular behavior dwindles to a nonspectacular hump until for a single atom thermodynamics can no longer be applied. 
Modern theoretical physics concepts are presented for the quantum statistics and thermodynamics of finite systems. Phase transitions show up again as singular structures, but now in the complex temperature plane. The whole bouquette of unique other features of the thermodynamic of small systems is demonstrated and the respective tools such as path integral formalism, recursive partition functions, shell model theory, are revealed.

Publications

While structural phase transitions as a function of temperature have been studied extensively, relatively little work has been performed at high pressures. The application of pressure to such systems is expected to strongly renormalise the order parameter behaviour and to suppress any dynamic behaviour associated either with intermediate "flip mode" phases or the high-temperature para phases. In addition, pressure couples linearly with the strain in the expression of the free energy of the solid, through the PV term. Thus, unlike temperature, pressure can drive transitions directly through the strain, or by inducing changes in the elastic constants. Recent advances in the experimental techniques for single-crystal high-pressure X-ray diffraction now allow lattice parameters to be determined routinely to a precision of ~1 part in 40,000. The use of internal diffraction standards to measure pressure yields a precision in pressure measurement of better than 0.01 GPa over the pressure range of 0-10 GPa. These advances therefore open up the opportunity of characterising structural phase transitions at high pressures through the determination of the evolution of the spontaneous strain accompanying the transition. Because the coupling of the spontaneous strain to the primary order parameter of the phase transition is governed by symmetry rules, lattice parameter measurements can reveal the order parameter behaviour of the system. Examples of recent work on high-pressure phase transitions in titanites, melilites, clinopyroxenes and lead phosphate will be presented.

In this talk we consider in detail the connection between the problem of a polymer in a random medium and that of a quantum particle in a random potential. We are interested in a system of finite volume where the polymer is known to be localized inside a low minimum of the potential. We show how the end-to-end distance of a polymer which is free to move can be obtained from the density of states of the quantum particle using extreme value statistics. We review and give a physical interpretation to the recently discovered one-step replica-symmetry-breaking solution for the polymer (Y. Goldschmidt Phys. Rev. E 61, 1729 (2000)) in terms of the statistics of localized tail states. Numerical solutions of the variational equations for chains of different length are performed and compared with quenched averages computed directly by using the eigenfunctions and eigenenergies of the Schrodinger equation for a particle in a one-dimensional random potential. The glassiness of the system is explained and is estimated from the variance of the measured quantities.