## Abstract

In this report we are dealing with the thermodynamic theory of second-order phase transitions

or continuous transitions of unary systems. The first classification of these phase transitions is

due to Ehrenfest (1933), based on chemical potentials. First-order transitions are changes in

which the derivatives of the chemical potentials, with respect to temperature or pressure are

discontinuous. Second-order phase transitions are continuous in the first derivatives of the

chemical potentials, but exibits discontinuity for at least one second derivative. However, the

modern classification is based on systems, with or without latent heat, during the transitions.

Landau (1937) formulated the first theory, which is based on a power series of the Gibbs free

energy, including a so-called scalar order parameter. It’s a simple and powerful theory, but

fails for temperatures in the vicinity of the transition line. This is because there exists longrange

fluctuations in the systems. The order parameter is zero for the high temperature, high

symmetric, disordered phase, whereas it is non-zero for the low temperature, low symmetric,

ordered phase. The classical Landau theory can be outlined with statistical thermodynamics

and with symmetry considerations in crystals, which is in fact the group/representation theory.

There are two Landau symmetry rules: the space group of the low symmetric phase should be

a subgroup of the high symmetric phase. The second is that the continuous transitions leads to

crystal changes which corresponds to a single irreducible representation of the initial space

group. The last subject contains some remarks about the study of critical phenomena and

exponents and the renormalization group theory (temperatures near the transition). However a

thorough discussion of these subjects are beyond the scope of this report.

or continuous transitions of unary systems. The first classification of these phase transitions is

due to Ehrenfest (1933), based on chemical potentials. First-order transitions are changes in

which the derivatives of the chemical potentials, with respect to temperature or pressure are

discontinuous. Second-order phase transitions are continuous in the first derivatives of the

chemical potentials, but exibits discontinuity for at least one second derivative. However, the

modern classification is based on systems, with or without latent heat, during the transitions.

Landau (1937) formulated the first theory, which is based on a power series of the Gibbs free

energy, including a so-called scalar order parameter. It’s a simple and powerful theory, but

fails for temperatures in the vicinity of the transition line. This is because there exists longrange

fluctuations in the systems. The order parameter is zero for the high temperature, high

symmetric, disordered phase, whereas it is non-zero for the low temperature, low symmetric,

ordered phase. The classical Landau theory can be outlined with statistical thermodynamics

and with symmetry considerations in crystals, which is in fact the group/representation theory.

There are two Landau symmetry rules: the space group of the low symmetric phase should be

a subgroup of the high symmetric phase. The second is that the continuous transitions leads to

crystal changes which corresponds to a single irreducible representation of the initial space

group. The last subject contains some remarks about the study of critical phenomena and

exponents and the renormalization group theory (temperatures near the transition). However a

thorough discussion of these subjects are beyond the scope of this report.

Original language | English |
---|---|

Number of pages | 23 |

Publication status | Published - Sept 2009 |

## Keywords

- Ehrenfest classification
- Landau theory
- (Complex) order parameter
- Statistical thermodynamics
- Group theory
- Symmetry
- Irreducible representations
- Critical exponents