Second-order phase transitions of pure substances

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.