*M. E. LINES and A. M. GLASS*

- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198507789
- eISBN:
- 9780191709944
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507789.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials

In order to acquire a simple physical picture of the dynamic mechanism of a phase transition it is necessary to use the simplest of many-body approximations. It is instructive, in particular, to ...
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In order to acquire a simple physical picture of the dynamic mechanism of a phase transition it is necessary to use the simplest of many-body approximations. It is instructive, in particular, to study the mean-field response of the model system to a time-dependent applied field. In this way, one can obtain considerable insight into the nature of the collective excitations and into the relationship between the static aspects of a phase transition and the occurrence of temperature-dependent (that is, soft) modes and of critical fluctuations. This chapter discusses the static aspects of mean-field theory and the nature of the static singularities which accompany second-order phase transitions. Mean-field dynamics are then described in terms of deviations from the equilibrium mean-field state. Correlated effective-field theory, the quasi-harmonic limit and self-consistent phonons, the deep double-well limit and the Ising model, and the pseudo-spin formalism and tunnel mode are also considered.Less

In order to acquire a simple physical picture of the dynamic mechanism of a phase transition it is necessary to use the simplest of many-body approximations. It is instructive, in particular, to study the mean-field response of the model system to a time-dependent applied field. In this way, one can obtain considerable insight into the nature of the collective excitations and into the relationship between the static aspects of a phase transition and the occurrence of temperature-dependent (that is, soft) modes and of critical fluctuations. This chapter discusses the static aspects of mean-field theory and the nature of the static singularities which accompany second-order phase transitions. Mean-field dynamics are then described in terms of deviations from the equilibrium mean-field state. Correlated effective-field theory, the quasi-harmonic limit and self-consistent phonons, the deep double-well limit and the Ising model, and the pseudo-spin formalism and tunnel mode are also considered.

*F. Iachello and R. D. Levine*

- Published in print:
- 1995
- Published Online:
- November 2020
- ISBN:
- 9780195080919
- eISBN:
- 9780197560419
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195080919.003.0005
- Subject:
- Chemistry, Physical Chemistry

Algebraic theory makes use of an algebraic structure. The structure appropriate to ordinary quantum mechanical problems is that of a Lie algebra. We begin this chapter with a brief review of the ...
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Algebraic theory makes use of an algebraic structure. The structure appropriate to ordinary quantum mechanical problems is that of a Lie algebra. We begin this chapter with a brief review of the essential concepts of Lie algebras. The binary operation (“multiplication”) in the Lie algebra is that of taking the commutator. As usual, we denote the commutator by square brackets, [A, B] = AB - BA. A set of operators {X} is a Lie algebra when it is closed under commutation.
Less

Algebraic theory makes use of an algebraic structure. The structure appropriate to ordinary quantum mechanical problems is that of a Lie algebra. We begin this chapter with a brief review of the essential concepts of Lie algebras. The binary operation (“multiplication”) in the Lie algebra is that of taking the commutator. As usual, we denote the commutator by square brackets, [A, B] = AB - BA. A set of operators {X} is a Lie algebra when it is closed under commutation.

*F. Iachello and R. D. Levine*

- Published in print:
- 1995
- Published Online:
- November 2020
- ISBN:
- 9780195080919
- eISBN:
- 9780197560419
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195080919.003.0010
- Subject:
- Chemistry, Physical Chemistry

Potential functions
In this chapter we return to the question of the geometrical interpretation of the algebraic approach. Specifically, we need to make contact with the concept of the potential ...
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Potential functions
In this chapter we return to the question of the geometrical interpretation of the algebraic approach. Specifically, we need to make contact with the concept of the potential function which is central to the geometrical point of view. For example, in three dimensions,...Less

Potential functions

In this chapter we return to the question of the geometrical interpretation of the algebraic approach. Specifically, we need to make contact with the concept of the potential function which is central to the geometrical point of view. For example, in three dimensions,...