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Physical Review Letters

Multiscaling and the classification of continuous phase transitions

Rudolf Hilfer A

Published Jan 13, 1992• 35 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.68.190

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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.
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.
The continuous phase transition, indicated by the macroscopic order parameter and the occurrence of the spontaneous symmetry breaking, is well illustrated based on the Ginzburg-Landau's paradigm. In systems described by one order parameter, the phase diagram is only composed of the normal phase and the ordered phase. However, in systems with two or more order parameters, much richer phase diagrams and critical phenomena may emerge. According to coupling terms of the free energy, we classify the systems with two order parameters into several categories, featuring different patterns of phase diagrams and critical scalings respectively. Our work propose the new avenue to studying and evaluating the complex systems only based on the coupling terms in expressions of free energy.
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 \ensuremath{\beta} 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.