Research interests

The two central themes of my research have been the statistical physics of nonequilibrium systems and the equilibrium properties of surfaces. Brief summaries follow:

Statistical physics of nonequilibrium systems

     During my thesis work I have been concerned with the kinetic theory of dense gases and liquids. For hard spheres a formally exact theory has been formulated, which however, requires drastic simplification in practical applications. Generalizations have been made to square-well potentials (with G. Stell, J. Karkheck, J. Sengers and others) as well as to more general potentials (with J.A. Leegwater). Most used to date still is the most drastic simplification, now known as Revised Enskog Theory.     
     Over the past years I have been deeply interested in the connections between chaos theory and nonequilibrium statistical physics. With, among others, J.R. Dorfman, R. van Zon, H. Posch, A. de Wijn and O. Mülken I succeeded in calculating analytically several dynamical properties, such as Lyapunov exponents, Kolmogorov Sinai entropies and topological pressures, for dilute physical systems like the Lorentz gas and the hard-ball gas. For these calculations kinetic theory methods were crucial again. My present hope is to gain with these methods, a much better physical understanding of Sinai-Ruelle Bowen measures, which according to G. Gallavotti and E.G.D.Cohen are the proper generalizations of the famous Gibbs measures of equilibrium statistical physics to stationary nonequilibrium states.
     I have also worked extensively on the dynamics of stochastic systems, e.g. lattice gases undergoing hopping dynamics. For such systems, together with K. Kehr and R. Kutner, I have developed methods for calculating diffusion coefficients as well as time dependence of various correlation functions. I also gave simple rigorous derivations  of the so-called long-time-tails, occurring in such correlation functions. With L. Schulman I studied a simplifying limit in a class of driven diffusive systems, where the existence and properties of a phase transition in the stationary state could be demonstrated explicitly. With H. Spohn and R. Kutner I showed that one-dimensional driven diffusive systems exhibit quite anomalous collective diffusion, identified later as belonging to the KPZ universality class.
  Presently I am working, with G. Barkema and J. Kuipers, on nucleation phenomena in lattice gases with stochastic dynamics. We found that the classical Becker-Döring equations for the nucleation dynamics work very well, at least for two-dimensional systems with non-conserving dynamics. However, for a good quantitative agreement between theory and simulation results one has to refine the expressions for the free energy of a growing nucleus and for the jump rates between nuclei of different sizes. We are now considering systems with conserving dynamics and three-dimensional systems and in the near future we also plan to study continuum systems. Another point of great interest is the dynamics for very short times, which is especially important in systems where the first nucleation has a dramatic effect, like an explosion. For systems prepared by a quench into a metastable state, the first nucleation event is strongly delayed, because initially there are only very small nuclei present. These require a minimal time for reaching the critical size, beyond which growth proceeds unimpeded. We found the Becker-Döring equations are successful in describing this phenomenon too.

Equilibrium properties of surfaces

       My interest in this subject was raised already during my thesis work in Nijmegen. There G. Gallavotti reported on a conjecture by R. Dobrushin: he had noticed that at low temperatures the formation of large mounds or pits on a flat interface in a three-dimensional Ising model should be virtually impossible for energetic reasons. With increasing importance of entropy, this occurrence might well become feasible at higher temperatures, still below the critical one. This then ought to give rise to a phase transition, which later has been baptized the roughening transition. At about the same time I heard P. Bennema, in  a talk on crystal growth, explain the theory of Burton, Cabrera and Frank. They argued that on crystal facets there ought to be a phase transition, where the growth mode in the normal direction changes from a slow, nucleation driven, layer-by-layer growth to a fast, continuous accumulation of matter. I immediately was convinced these two observations were closely related and the subject kept drifting in my mind.
     As a first result I showed, by very simple means that, in the three-dimensional Ising model, the critical temperature of the corresponding two-dimensional model provides a rigorous lower bound for the roughening temperature, the temperature where the roughening transition occurs. Later, it was found numerically that this is only 10% below the actual roughening temperature for this system.
     A few years later I discovered that the six-vertex model, which had been solved exactly by Lieb, extended by Sutherland, Yang and Yang, may be reinterpreted as a model for a crystal surface. As a consequence of this, the phase transitions found by Lieb could be reinterpreted as roughening transitions in the corresponding surface models. This was the first rigorous proof that roughening transitions indeed may exist. In addition it followed that the transition is not at all of the Ising-type, as anticipated by Burton, Cabrera and Frank, but rather it is a Kosterlitz-Thouless transition. Chui and Weeks, on the basis of an approximate mapping of another surface model to a two-dimensional Coulomb gas, in fact, had anticipated this a little earlier.
     By now the roughening transition has also been found in several experimental systems. Most convincingly by S. Lipson et al. and by S. Balibar et al. in solid helium, but also in various metals, e.g. lead and indium, by the groups of J. C. Heyraud  and J.J. Métois and of A. Pavlovska and E. Bauer, and already before that (but with smaller accuracy) in plastic crystals by A. Pavlovska and Ch. Nenov.
     More recently, with E. Luijten, G. Mazzeo and E. Carlon, I have been looking at staggered six-vertex models, for which quite remarkable phase diagrams do emerge. We found inverse roughening transitions in certain variants of the model, where the surface is rough below and smooth above the roughening temperature, and phase diagrams with a rough phase confined between two roughening temperatures plus an Ising-type transition at still lower temperature. For certain combinations of coupling constants mean-field methods are very accurate. These predict very complex equilibrium surface structures, with several phase boundaries and exotic features like conical points.
     The theory of equilibrium states of crystal surfaces by now has reached some stage of maturity, but the dynamics of these surfaces is still very much under investigation.