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&lt;0 and U,J&gt;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?

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I am trying to get the relation between developing a Ginzburg-Landau theory, let&#039;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&amp;lt;0 and U,J&amp;gt;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?

Cortez Hughes · Accepted Answer

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 · Accepted Answer

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      →

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

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