Physics 582, Fall Semester 2008
Professor Eduardo Fradkin
Problem Set No. 1: Classical Field Theory
Due Date: September 12, 2008
1
The Landau Theory of Phase Transitions as a
Classical Field Theory
In the LandauGinzburg approach to the theory of phase transitions, the ther
modynamic properties of a onecomponent classical ferromagnet in thermal equi
librium are described by a
free energy functional
of an orderparameter field
φ
(
vectorx
)
( the local magnetization).
This functional contains, in addition to gradient
terms, contributions proprtional to various powers of the local order parame
ter. Under some circumstances the coefficient
λ
of the
φ
4
term of the energy
functional may become negative.
This is what happens if the local magnetic
moments have spin1 rather the spin
1
2
. In this case, we have to include, in the
energy functional, a term with a higher power of
φ
( such as
φ
6
) in order to
insure the thermodynamic stability of the system.
The (free) energy density
E
for this system has the form
E
=
1
2
parenleftBig
vector
▽
φ
(
vectorx
)
parenrightBig
2
+
U
(
φ
(
vectorx
))
where the potential
U
(
φ
(
vectorx
)) is
U
(
φ
(
vectorx
)) =
m
2
0
2
φ
2
(
vectorx
) +
λ
4
4!
φ
4
(
vectorx
) +
λ
6
6!
φ
6
(
vectorx
)
.
with
m
2
0
=
a
(
T
−
T
0
) and
λ
4
<
0,
λ
6
>
0.
1. Use a variational principle to derive the saddlepoint equations (
i.e.,
the
LandauGinzburg equations) for this system.
2. Plot the potential
U
(
φ
) for a constant field
¯
φ
for
λ
4
<
0 ( and fixed) at
several temperatures. Show that, as the temperature
T
is lowered, there
exists a temperature
T
∗
>T
0
at which the state with lowest energy has
(
φ
) negationslash
= 0 (for fixed
λ
4
<
0 and
λ
6
>
0). Plot the qualitative behavior of
(
φ
)
as a function of
T
. Is this a continuous function? Is this a first order
or a second order transition? Find the value of the energy of the system
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 Fall '08
 Leigh
 Physics, Equations, Quantum Field Theory, Fundamental physics concepts, Dirac equation

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