Order, Disorder and Criticality

The following sections are included:

• Preface
• Contents

In the 1960's, four famous scaling relations were developed which relate the six standard critical exponents describing continuous phase transitions in the thermodynamic limit of statistical physics models. They are well understood at a fundamental level through the renormalization group. They have been verified in multitudes of theoretical, computational and experimental studies and are firmly established and profoundly important for our understanding of critical phenomena. One of the scaling relations, hyperscaling, fails above the upper critical dimension. There, critical phenomena are governed by Gaussian fixed points in the renormalization-group formalism. Dangerous irrelevant variables are required to deliver the mean-field and Landau values of the critical exponents, which are deemed valid by the Ginzburg criterion. Also above the upper critical dimension, the standard picture is that, unlike for low-dimensional systems, finite-size scaling is non-universal, at least at the critical point. Here we report on new developments which indicate that the current paradigm is flawed and incomplete. In particular, the introduction of a new exponent characterising the finite-size correlation length allows one to extend hyperscaling beyond the upper critical dimension. Moreover, finite-size scaling is shown to be universal provided the correct scaling window is chosen. These recent developments above the upper critical dimension also lead to the introduction of a new scaling relation analogous to one introduced by Fisher 50 years ago and deliver a statistical physics explanation for the emergence of effective four dimensionality as characteristic of generic field theories.

In the vicinity of the second order phase transition point long-range critical fluctuations of the order parameter appear. The second order phase transition in a critical binary mixture in the vicinity of the demixing point belongs to the universality class of the Ising model. The superfluid transition in liquid He belongs to the universality class of the XY model. The confinement of long-range fluctuations causes critical Casimir forces acting on confining surfaces or particles immersed in the critical substance. Last decade critical Casimir forces in binary mixtures and liquid helium were studied experimentally. The critical Casimir force in a film of a given thickness scales as a universal scaling function of the ratio of the film thickness to the bulk correlation length divided over the cube of the film thickness. Using Monte Carlo simulations we can compute critical Casimir forces and their scaling functions for lattice Ising and XY models which correspond to experimental results for the binary mixture and liquid helium, respectively. This chapter provides the description of numerical methods for computation of critical Casimir interactions for lattice models for plane-plane, plane-particle, and particle-particle geometries.

Anomalous diffusion, the deviation from classical Brownian motion with its linear-in-time mean squared displacement, is a widespread phenomenon in a large variety of complex systems, ranging from amorphous semiconductors to biological cells. Anomalous diffusion processes are no longer confined by the central limit theorem, that enforces the convergence of Brownian motion to the Gaussian shape of the probability density function. Instead, anomalous processes have a rich variety of physical origins and non-traditional mathematical properties. We here provide a summary of various anomalous diffusion models, paying special attention to their non-ergodic and ageing behaviour. Whether a process is ergodic or not is of vital importance when measurements are evaluated in terms of time averaged observables such as the mean squared displacement. For non-ergodic processes these can no longer be interpreted by comparison to the typically calculated ensemble averaged observables. Breaking of ergodicity occurs in many anomalous diffusion processes. We also consider their ageing properties, the explicit dependence of observables on the time span between the original system initiation and the start of the measurement.

The phenomenon of pattern formation is universal in nature and involves complex non-equilibrium processes that are highly important for both fundamental research and applied materials systems as well as for theoretical biology. Last decades, computational modelling provided an important tool for understanding pattern formation dynamics and self-organisation in many systems. In this chapter the brief overview of different methods of modelling of pattern formation phenomena has been done. The linking between mesoscopic and atomistic approaches is established. It is shown that some general conclusions about pattern formation can be done from the modelling of microstructure evolution in alloys at these two scales.

We review a model of sociophysics which deals with democratic voting in bottom up hierarchical systems. The connection to the original physical model and technics are outlined underlining both the similarities and the differences. Emphasis is put on the numerous novel and counterintuitive results obtained with respect to the associated social and political framework. Using this model a real political event was successfully predicted with the victory of the French extreme right party in the 2000 first round of French presidential elections. The perspectives and the challenges to make sociophysics a predictive solid field of science are discussed.

The following sections are included:

• Subject Index