Chapter

Mean‐field theories

Hidetoshi Nishimori and Gerardo Ortiz

in Elements of Phase Transitions and Critical Phenomena

Published in print December 2010 | ISBN: 9780199577224
Published online January 2011 | e-ISBN: 9780191722943 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199577224.003.0002

Series: Oxford Graduate Texts

Mean‐field theories

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The present chapter explains the mean-field approximation, the Landau theory, the infinite-range model, and the Bethe approximation, and shows that all these (mean-field) theories are essentially equivalent to each other. The Landau theory is a phenomenological approach that uses the concept of symmetry and the order parameter, a measure of the breaking of that symmetry, as fundamental collective degrees of freedom. Also described are the Landau theory of tricritical behaviour, correlation functions, the limit of applicability of the mean-field theory, known as the Ginzburg criterion, and dynamic critical phenomena. Mean-field theories yield the exact critical exponents for dimensions larger than the upper critical dimension, and their solutions provide a reasonable starting point for more advanced methods including the renormalization group.

Keywords: Landau theory; order parameter; critical exponents; correlation functions; Ginzburg criterion; upper critical dimension; dynamic critical phenomena

Chapter.  17213 words.  Illustrated.

Subjects: Mathematical and Statistical Physics

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