This paper is a brief summary of a talk that was designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena – scale invariance and universality – can be useful in guiding research on a broad class of complex phenomena. We shall see that while scale invariance has been tested for many years, universality is relatively more rarely discussed. In particular, we shall develop a heuristic argument that serves to make more plausible the universality hypothesis in both thermal critical phenomena and percolation phenomena, and suggest that this argument could be developed into a possible coherent approach to understanding the ubiquity of scale invariance and universality in a wide range of complex systems.This paper is a brief summary of a talk that was designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena – scale invariance and universality – can be useful in guiding research on a broad class of complex phenomena. We shall see that while scale invariance has been tested for many years, universality is relatively more rarely discussed. In particular, we shall develop a heuristic argument that serves to make more plausible the universality hypothesis in both thermal critical phenomena and percolation phenomena, and suggest that this argument could be developed into a possible coherent approach to understanding the ubiquity of scale invariance and universality in a wide range of complex systems.

Physica A Statistical Mechanics and its ApplicationsJun 1, 2000122 citations

One—Scale-Factor Universality for Critical Phenomena

Donald D. Betts, Douglas S. Ritchie

An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of ${V}_{X}=\frac{{n}_{X}{q}_{X}{{K}_{C}}^{X}}{{g}_{X}{l}_{X}}$ , where ${q}_{X}$ is the coordination number, ${{K}_{C}}^{X}$ is the dimensionless inverse critical temperature, and ${n}_{X}$ , ${g}_{X}$ , and ${l}_{X}$ are the universality scale factors. ${V}_{X}$ is an exact invariant for the two-dimensional $S=\frac{1}{2}$ Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however, $\frac{{V}_{X}}{{l}_{X}}$ is an exact invariant.An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of ${V}_{X}=\frac{{n}_{X}{q}_{X}{{K}_{C}}^{X}}{{g}_{X}{l}_{X}}$, where ${q}_{X}$ is the coordination number, ${{K}_{C}}^{X}$ is the dimensionless inverse critical temperature, and ${n}_{X}$, ${g}_{X}$, and ${l}_{X}$ are the universality scale factors. ${V}_{X}$ is an exact invariant for the two-dimensional $S=\frac{1}{2}$ Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however, $\frac{{V}_{X}}{{l}_{X}}$ is an exact invariant.

Physical Review LettersMar 31, 19753 citations

Critical Phenomena

Javier Brey, Robert B. Jones

Springer eBooksJan 1, 19763 citations

Local scale invariance and its applications to strongly anisotropic critical phenomena

Malte Henkel, Alan Picone A, Michel Pleimling +1 more

The generalization of dynamical scaling to local scale invariance is reviewed. Starting from a recapitulation of the phenomenology of ageing phenomena, the generalization of dynamical scaling to local scale transformation for any given dynamical exponent $z$ is described and the two distinct types of local scale invariance are presented. The special case $z=2$ and the associated Ward identity of Schrödinger invariance is treated. Local scale invariance predicts the form of the two-point functions. Existing confirmations of these predictions for (I) the Lifshitz points in spin systems with competing interactions such as the ANNNI model and (II) non-equilibrium ageing phenomena as occur in the kinetic Ising model with Glauber dynamics are described.The generalization of dynamical scaling to local scale invariance is reviewed. Starting from a recapitulation of the phenomenology of ageing phenomena, the generalization of dynamical scaling to local scale transformation for any given dynamical exponent $z$ is described and the two distinct types of local scale invariance are presented. The special case $z=2$ and the associated Ward identity of Schrödinger invariance is treated. Local scale invariance predicts the form of the two-point functions. Existing confirmations of these predictions for (I) the Lifshitz points in spin systems with competing interactions such as the ANNNI model and (II) non-equilibrium ageing phenomena as occur in the kinetic Ising model with Glauber dynamics are described.

arXiv:cond-mat/0307649Jul 25, 20030 citations

An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of

{V}_{X}=\frac{{n}_{X}{q}_{X}{{K}_{C}}^{X}}{{g}_{X}{l}_{X}}

, where

{q}_{X}

is the coordination number,

{{K}_{C}}^{X}

is the dimensionless inverse critical temperature, and

{n}_{X}

{g}_{X}

, and

{l}_{X}

are the universality scale factors.

{V}_{X}

is an exact invariant for the two-dimensional

S=\frac{1}{2}

Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however,

\frac{{V}_{X}}{{l}_{X}}

is an exact invariant.

z

is described and the two distinct types of local scale invariance are presented. The special case

z=2

and the associated Ward identity of Schrödinger invariance is treated. Local scale invariance predicts the form of the two-point functions. Existing confirmations of these predictions for (I) the Lifshitz points in spin systems with competing interactions such as the ANNNI model and (II) non-equilibrium ageing phenomena as occur in the kinetic Ising model with Glauber dynamics are described.