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2080 results in Statistical Physics

Page 54 of 104

Statistical Physics of Liquids at Freezing and Beyond
  • Shankar Prasad Das
  • Published online:
    07 September 2011
    Print publication:
    16 June 2011
  • Exploring important theories for understanding freezing and the liquid-glass transition, this book is useful for graduate students and researchers in soft-condensed matter physics, chemical physics and materials science. It details recent ideas and key developments, providing an up-to-date view of current understanding. The standard tools of statistical physics for the dense liquid state are covered. The freezing transition is described from the classical density functional approach. Classical nucleation theory as well as applications of density functional methods for nucleation of crystals from the melt are discussed, and compared to results from computer simulation of simple systems. Discussions of supercooled liquids form a major part of the book. Theories of slow dynamics and the dynamical heterogeneities of the glassy state are presented, as well as nonequilibrium dynamics and thermodynamic phase transitions at deep supercooling. Mathematical treatments are given in full detail so readers can learn the basic techniques.

  • 4 - Constraint satisfaction problems

  • Lorenza Saitta, Università degli Studi del Piemonte Orientale Amedeo Avogadro, Attilio Giordana, Università degli Studi del Piemonte Orientale Amedeo Avogadro, Antoine Cornuéjols
  • Book:
    Phase Transitions in Machine Learning
    Published online:
    05 August 2012
    Print publication:
    16 June 2011, pp 70-91

  • Summary

    An important class of NP-complete problems is that of constraint satisfaction problems (CSPs), which have been widely investigated and where a phase transition has been found to occur (Williams and Hogg, 1994; Smith and Dyer, 1996; Prosser, 1996). Constraint satisfaction problems are the analogue of SAT problems in first-order logic; actually, any finite CSP instance can be transformed into a SAT problem in an automatic way, as will be described in Section 8.4.

    Formally, a finite CSP is a triple (X, R, D). Here X = {xi|1 ≤ i ≤ n} is a set of variables and R = {Rh, 1 ≤ h ≤ m} is a set of relations, each defining a constraint on a subset of variables in X; D = {Di|1 ≤ i ≤ n} is a set of variable domains Di such that every variable xi takes values only in the Di, whose cardinality |Di| equals di. The constraint satisfaction problem consists in finding an assignment in Di for each variable xi ∈ X that satisfies all relations in R.

    In principle a relation Rh may involve any proper or improper subset of X. Nevertheless, most authors restrict investigation to binary constraints, defined as relations over two variables only. This restriction does not affect the generality of the results that can be obtained because any relation of arity higher than two can always be transformed into an equivalent conjunction of binary relations.

Page 54 of 104