In the Landau theory of phase transitions one considers an effective potential U whose symmetry group G and degree d depend on the system under consideration; generally speaking, U is the most general G-invariant polynomial of degree d. When such a U turns out to be too complicate for a direct analysis, it is essential to be able to drop unessential terms, i.e., to apply a simplifying criterion. Criteria based on singularity theory exist and have a rigorous foundation, but are often very difficult to apply in practice. Here we consider a simplifying criterion (as stated by Gufan) and rigorously justify it on the basis of classical Lie-Poincaré theory as far as one deals with fixed values of the control parameter(s) in the Landau potential; when one considers a range of values, in particular near a phase transition, the criterion has to be accordingly partially modified, as we discuss. We consider some specific cases of group G as examples, and study in detail the application to the Sergienko–Gufan–Urazhdin model for highly piezoelectric perovskites.

Lie-Poincare' transformations and a reduction criterion in Landau theory / G. Gaeta. - In: ANNALS OF PHYSICS. - ISSN 0003-4916. - 312:2(2004 Aug), pp. 511-540.

Lie-Poincare' transformations and a reduction criterion in Landau theory

G. Gaeta
Primo
2004

Abstract

In the Landau theory of phase transitions one considers an effective potential U whose symmetry group G and degree d depend on the system under consideration; generally speaking, U is the most general G-invariant polynomial of degree d. When such a U turns out to be too complicate for a direct analysis, it is essential to be able to drop unessential terms, i.e., to apply a simplifying criterion. Criteria based on singularity theory exist and have a rigorous foundation, but are often very difficult to apply in practice. Here we consider a simplifying criterion (as stated by Gufan) and rigorously justify it on the basis of classical Lie-Poincaré theory as far as one deals with fixed values of the control parameter(s) in the Landau potential; when one considers a range of values, in particular near a phase transition, the criterion has to be accordingly partially modified, as we discuss. We consider some specific cases of group G as examples, and study in detail the application to the Sergienko–Gufan–Urazhdin model for highly piezoelectric perovskites.
Landau theory; Normal forms; Phase transitions; Singularity theory
Settore MAT/07 - Fisica Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/68902
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