I am trying to get the relation between developing a Ginzburg-Landau theory, let's say for a ferroma



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I am trying to get the relation between developing a Ginzburg-Landau theory, let's say for a ferromagnet with magnetization field m = m ( r ), and the formal expansion of the free energy density F = F ( m ) in terms of a Taylor series.
Considering an isotropic ferromagnet, the lowest-order terms in our Ginzburg-Landau theory should be given by
F = r 2 m 2 + U 4 ( m 2 ) 2 + J 2 [ ( x m ) 2 + ( y m ) 2 + ( z m ) 2 ]
with r<0 and U,J>0.
However, when I think of a Taylor expansion of F ( m ) around the origin
F = F 0 + m T D F + 1 2 m T D 2 F m +
this is giving me terms of all the individual powers in m , which are either zero or identified with the r- and U-term, but no gradient terms for the J-term? How to motivate these through a Taylor expansion?

Answer & Explanation

Cortez Hughes

Cortez Hughes

Beginner2022-05-19Added 23 answers

Note that this is rather an opinion than a fully rigorous statement:
For vanishing gradients, i.e. for uniform systems (the case originally considered by Landau), the expansion of the free energy is indeed a Taylor expansion (in even powers of m) near the transition. However the addition of gradient terms in the case of non-uniform systems removes this interpretation as is rather a phenomenological modification. However, you could still see this an expression for a classical field theory.
Raphael Mccullough

Raphael Mccullough

Beginner2022-05-20Added 1 answers

It looks to me like you are Taylor expanding F as a function of m without considering that m is a function of space. In particular, your D and D 2 involve things like m x but not x . You are implicitly assuming that F ( x ) only depends on m ( x ) and not also for example m ( x + d x ) for small d x

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