# Assumptions of the Landau theory

Landau theory is based on some assumptions, which we now introduce:

• There exists an order parameter${\displaystyle \eta }$ for the system, such that:

${\displaystyle {\begin{cases}\eta =0&T>{\overline {T}}\\\eta \neq 0&T<{\overline {T}}\end{cases}}}$

• There exists a function ${\displaystyle {\mathcal {L}}}$ called Landau free energy[1], which is an analytic function of the coupling constants ${\displaystyle \lbrace K_{i}\rbrace }$ of the system and of the order parameter ${\displaystyle \eta }$.

Therefore, near ${\displaystyle {\overline {T}}}$ we can expand ${\displaystyle {\mathcal {L}}}$ for small ${\displaystyle \eta }$:

${\displaystyle {\mathcal {L}}=\sum _{n=0}^{\infty }a_{n}\eta ^{n}=a_{0}+a_{1}\eta +a_{2}\eta ^{2}+a_{3}\eta ^{3}+a_{4}\eta ^{4}+O(\eta ^{5})}$

• ${\displaystyle {\mathcal {L}}}$ has to be consistent with the symmetries of the system

• The equilibrium states of the system are the global minima of ${\displaystyle {\mathcal {L}}}$ with respect to ${\displaystyle \eta }$

We also assume that the thermodynamic properties of the system can be obtained by differentiating ${\displaystyle {\mathcal {L}}}$, just like we can do with thermodynamic potentials.[2]

From these assumption we can explicitly construct ${\displaystyle {\mathcal {L}}}$, depending of course on the system considered.

Note also that the general formulation of the Landau theory does not depend on the dimensionality of the system (although we will see that once a system has been chosen some details can depend on it).

1. To be more precise, ${\displaystyle {\mathcal {L}}}$ is the Landau free energy density; the "real" Landau free energy should be ${\displaystyle L=V{\mathcal {L}}}$.
2. Strictly speaking, Landau free energy is not really a thermodynamic potential: the correct interpretation of ${\displaystyle {\mathcal {L}}}$ (see , 5.6.1) is that it is a coarse grained free energy (not the exact one); see Functional partition function and coarse graining for coarse graining procedures.