Classification of Phase Transitions

Lucien Gilles Benguigui, Lawrence S. Schulman

Wolfhard Janke E

Rudolf Hilfer A

A refined classification theory for phase transitions in thermodynamics and statistical mechanics in terms of their orders is introduced and analyzed. The refined thermodynamic classification is based on two independent generalizations of Ehrenfests traditional classification scheme. The statistical mechanical classification theory is based on generalized limit theorems for sums of random variables from probability theory and the newly defined block ensemble limit. The block ensemble limit combines thermodynamic and scaling limits and is similar to the finite size scaling limit. The statistical classification scheme allows for the first time a derivation of finite size scaling without renormalization group methods. The classification distinguishes two fundamentally different types of phase transitions. Phase transitions of order λ>1 correspond to well known equilibrium phase transitions, while phase transitions with order λ<1 represent a new class of transitions termed anequilibrium transitions. The generalized order λ varies inversely with the strength of fluctuations. First order and second order transitions play a special role in both classification schemes. First order transitions represent a limiting case separating equilibrium and anequilibrium transitions. The special role or second order transitions is shown to be related to the breakdown of hyperscaling. For anequilibrium transitions the nature of the heat bath in the canonical ensemble becomes important.

Peter Borrmann, Oliver Mülken, Jens Harting

We present a classification scheme for phase transitions in finite systems like atomic and molecular clusters based on the Lee-Yang zeros in the complex temperature plane. In the limit of infinite particle numbers the scheme reduces to the Ehrenfest definition of phase transitions and gives the right critical indices. We apply this classification scheme to Bose-Einstein condensates in a harmonic trap as an example of a higher order phase transition in a finite system and to small Ar clusters.

James G. Glimm, Arthur M. Jaffe

Rudolf Hilfer A

Multiscaling of the free energy is obtained by generalizing the classification of phase transitions proposed by Ehrenfest. The free energy is found to obey a new generalized scaling form which contains as special cases standard and multiscaling forms. The results are obtained by analytic continuation from the classification scheme of Ehrenfest.