Statistical and Non-linear Physics, Introduction to

Abstract

The field of statistical and nonlinear physics combines the venerable subject of statistical mechanics with the newer area of nonlinear science to create a highly interdisciplinary and exciting area of research. It has its roots in equilibrium statistical physics, but it has evolved to encompass and emphasize nonequilibrium and dynamical phenomena. It is a field in rapid evolution, with a constantly changing focus. It overlaps naturally with many other disciplines, including fluid dynamics, computational physics, biological physics, condensed matter physics, and polymer physics. Statistical physics aims to describe the large-scale collective behavior of systems composed of a large number of interacting units or degrees of freedom. Starting with a microscopic model, one performs various types of “coarse‐graining” to describe phenomena that occur on length and time scales large compared to microscopic ones, such as the size of the particles or the characteristic interaction times. At this large scale the system exhibits collective or emergent behavior that is far richer than that of the individual units. One of the hallmarks of collective behavior is phase transitions. An everyday example is the change from liquid to solid that occurs upon tuning a parameter such as temperature or pressure. Another familiar example arises in the study of the properties of magnetic materials and is highlighted in the entry “▶Complex Systems and Emergent Phenomena,” which is a good starting point for the reader of this section. When going from the deterministic Hamiltonian description of a system of many interacting units to the large-scale coarse‐grained description of the same system in terms of a continuum or hydrodynamic theory, an element of randomness is naturally introduced into the problem as one looses the invariance under time reversal that was present in the Hamiltonian description. At the same time randomness on the other hand is an intrinsic property of the dynamics of individual nonlinear systems that are known to often exhibit sensitive dependence to initial conditions and chaotic behavior. There is in fact a deep connection between the randomness of chaotic systems and the irreversible transport properties of extended systems (see “▶Chaotic Dynamics in Nonequilibrium Statistical Mechanics”). This connection highlights the unity of the two areas of statistical and nonlinear science. Rather than presenting an exhaustive review of the field of statistical and nonlinear physics, the articles in this section focus on nonequilibrium phenomena that are the subject of current research. Even then, only a fraction of the problems and physical systems that are studied using the tools and ideas of statistical and nonlinear physics are described here. Many of the others can be found in other entries throughout the rest of the encyclopedia. The topics highlighted in this entry may at a first glance seem disparate but are unified by the ideas that are used to study them. Principal among those are the notions of scaling and universality. The concept of universality, which has been around for some time, has its roots in the study of phase transitions and critical phenomena. Near a phase transition, the system is universal in that its behavior at large scales does not depend on the microscopic physics. The systems and physical