Landau theory in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
is a theory that
Lev Landau
Lev Davidovich Landau (russian: Лев Дави́дович Ланда́у; 22 January 1908 – 1 April 1968) was a Soviet- Azerbaijani physicist of Jewish descent who made fundamental contributions to many areas of theoretical physics.
His a ...
introduced in an attempt to formulate a general theory of continuous (i.e., second-order)
phase transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
s. It can also be adapted to systems under externally-applied fields, and used as a quantitative model for discontinuous (i.e., first-order) transitions. Although the theory has now been superseded by the
renormalization group
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the ...
and scaling theory formulations, it remains an exceptionally broad and powerful framework for phase transitions, and the associated concept of the
order parameter
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
as a descriptor of the essential character of the transition has proven transformative.
Mean-field formulation (no long-range correlation)
Landau was motivated to suggest that the free energy of any system should obey two conditions:
*Be analytic in the order parameter and its gradients.
*Obey the symmetry of the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
.
Given these two conditions, one can write down (in the vicinity of the critical temperature, ''T''
''c'') a phenomenological expression for the free energy as a
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
in the
order parameter
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
.
Second-order transitions
Consider a system that breaks some symmetry below a phase transition, which is characterized by an order parameter
. This order parameter is a measure of the order before and after a phase transition; the order parameter is often zero above some critical temperature and non-zero below the critical temperature. In a simple ferromagnetic system like the
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
, the order parameter is characterized by the net magnetization
, which becomes spontaneously non-zero below a critical temperature
. In Landau theory, one considers a free energy functional that is an analytic function of the order parameter. In many systems with certain symmetries, the free energy will only be a function of even powers of the order parameter, for which it can be expressed as the series expansion
:
In general, there are higher order terms present in the free energy, but it is a reasonable approximation to consider the series to fourth order in the order parameter, as long as the order parameter is small. For the system to be thermodynamically stable (that is, the system does not seek an infinite order parameter to minimize the energy), the coefficient of the highest even power of the order parameter must be positive, so
. For simplicity, one can assume that
, a constant, near the critical temperature. Furthermore, since
changes sign above and below the critical temperature, one can likewise expand
, where it is assumed that
for the high-temperature phase while
for the low-temperature phase, for a transition to occur. With these assumptions, minimizing the free energy with respect to the order parameter requires
:
The solution to the order parameter that satisfies this condition is either
, or
:
It is clear that this solution only exists for