### R. Hilfer

Physical Review E **48**, 2466 (1993)

10.1103/PhysRevE.48.2466

submitted on

Monday, March 16, 1992

The paper introduces a classification of phase transitions in which each transition is characterized through its generalized order and a slowly varying function. This characterization is shown to be applicable in statistical mechanics as well as in thermodynamics albeit for different mathematical reasons. By introducing the block ensemble limit the statistical classification is based on the theory of stable laws from probability theory. The block ensemble limit combines scaling limit and thermodynamic limit. The thermodynamic classification on the other hand is based on generalizing Ehrenfest’s traditional classification scheme. Both schemes imply the validity of scaling at phase transitions without the need to invoke renormalizaton-group arguments. The statistical classification scheme allows derivation of a form of finite-size scaling for the distributions of statistical averages while the thermodynamic classification gives rise to multiscaling of thermodynamic potentials. The different nature of the two classification theories is also apparent from the fact that the generalized thermodynamic order is unbounded while the statistical order is restricted to values less than 2. This fact is found to be related to the breakdown of hyperscaling relations. Both classification theories predict the possible existence of phase transitions having orders less than unity. Such transitions are termed anequilibrium transitions. Systems near anequilibrium transitions cannot be described by conventional equilibrium thermodynamics or equilibrium statistical mechanics because of very strong fluctuations. Anequilibrium transitions are found to exist in statistical-mechanical model systems. The identification of the Lagrange parameter β in the canonical ensemble becomes invalid if a reservoir and a system of the same substance are in thermal contact and anequilibrium transitions are present. Based on the ergodic hypothesis and the theory of convolution semigroups it is shown that near anequilibrium transitions the equations of motion for macroscopic observables of infinite systems may involve modified time derivatives as generators of the macroscopic time evolution. The general solution to the modified equations of motion exhibits very slow dynamics as frequently observed in a nonequilibrium experiment.

For more information see