Classification of phase transitions in small systems
klick on the image to see an animated gif of the evolution of the distribution of zeros for small magnetic clusters with increasing magnetic field
(A publication list on this topic is available at the bottom of this page, klick here!)
Physicist, experimentalist as well as theoreticians, have paid continuous
interest to phase transition phenomena. About thirty years ago Wilson and
Kadanoff established a powerfull tool to classify phase transitions within
the frame of statistical mechanics. It is undoubted that this description
works amazingly well for all kinds of phase changes in macroscopic systems.
Small systems are somewhat different. Since the system is finite there are no observable singularities of thermodynamic functions, like the specific heat, the ground state occupation number for finite Bose-Einstein condensates, the multiplicity of fragments within the multifragmentation process of intermediate heavy-ion collisions or the susceptibility of small metal clusters, to name a few. Those properties show humps, i.e., to be more precise, smooth curves as a function of the order parameter, rather than sharp peaks or discontinuities.
Up to now phase transition phenomena in small systems are a widely discussed topic.
In order to give a proper description of phase change effects in small systems we have proposed a classification scheme for small systems which, for itself, has a true thermodynamic limit. Already in 1952 Yang and Lee have suggested that phase transitions within the grand-canonical ensemble could be descriped by the zeros of the partition function, simply due to the fact that most thermodynamic functions are derivatives of the logarithm of the partition function. Therefore they claimed that a distribution of zeros within the complex plane approaching the real axis infinitly close separates two phases and the point of intersection with the real axis is equal to the critical point. Few years later M.E. Fisher (1963) and again some years later Grossmann and coworkers (1967-69) have given a description of phase transitions within the canonical ensembel by the zeros of the canonical partition function in the inverse complex temperature plane. Grossmann and coworkers were able to classify phase transitons as first, second, and higher order in a distinct way. In the same period Abe and Suzuki have elaborated quite the same results as Grossmann and coworkers.
Thirty years later, in 1997, we have started working on the classification of phase transitions in small systems. The distribution of zeros of the canonical partition functions provides a powerful tool. We are not only able to detect whether there is a phase change or not but also can we make a non-ambiguous statement about the order of the transition in small systems. The first results we obtained were presented by P. Borrmann at the International Cluster Workshop (ICW) in Berlin (1997) and are written down in the diploma thesis of Oliver Mülken. Further studies, extensions and improvements of the previous studies were published in Physical Review Letters 84, 3511 (2000).
The most popular example of phase transitons in small systems might be the Bose-Einstein condensation in magnetically or optically trapped Alkali-atoms. There, the number of particles is fixed (from about 2000 atoms in the first experimentally measured condensated up to over a million atoms in recent experiments) and the theoretical description is relatively easy to handle if one considers the condensate as a bunch of non-interacting atoms. Therefore, this systems is the ideal candidate to study. We have given a recursive treatment of the canonical partition function first in 1993 (G. Franke and P. Borrmann, Journal of Chemical Physics 98, p. 3496 (1993)) which was enhanced in 1999 (P. Borrmann, J. Harting, O. Mülken, and E.R. Hilf, Physical Review A 60, 1519 (1999)). An extension of this recursion formulas to the complex temperature plane and the calculation of the ground-state occupation number for various particle numbers lead to enlightening pictures of the condensation transition
The analysis of those calculations showed that the order of the transition does not only depend on the dimension of the system but also for very small particle numbers on the particle number itself. We obtained - for a parabolically trapped ideal Bose-gas in three dimension - a second order transition for particle numbers greater than approximately 50. For smaller particle numbers the transition is of higher, i.e. third, order. A detailed study is available as preprint at arXiv: cond-mat/0006293.
Another systems which could be described theoretically in close relationship with Bose-Einstein condensates are heavy-ion collision at intermediate energies. The so-called multifragmentation process is observable. Calculations reveal that the multifragmentation phase transition is of first order, although the description is somewhat similar to the one for Bose-Einstein condensates. A description of the multifragmentation model and the calculations and analysis of the phase transition is also available as a preprint at arXiv: nucl-th/0009020 and is accepted for publication in Physical Review C.
Further studies of other small systems are on the way. Currently we are investigating the magnetic properties of small magnetic clusters. Those clusters exhibit phase transitions mainly in their structural arrangement, as they are detectable as different isomers but with almost same ground state energies but huge differences in their magnetic moment.
Last updated on 7.12.2000