PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 3- S C Chapman
Landau Theory II
This is the summary given in the lecture – see handout for full details.
Now relax assumption (2) of symmetric F ( M ) . Now F3 ! 0 .
Simplest case: let F3 = !
! > 0 constant, (eg: ferromagnet with applied B field).
Now we have
F = ! ( T ! Tc ) M 2 + " M 3 + # M 4
!F
= 2!( T " Tc ) M + 3" M 2 + 4# M 3
!M
= M [ 2! ( T ! Tc ) + 3" M + 4# M 2 ]
M =0
extrema are
M =
- again, and
!3! ± 9! 2 ! 4.2." ( T ! Tc ) 4#
2.4#
We will use notation as follows: write this as
!3! ± 3 ! 2 ! ! c2
M =
8"
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! c2 = "# ( T ! Tc )
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The real extrema are:
M = 0, and
M ! 0 when ! 2 > ! c2 ! 2 roots
when ! 2 = ! c2 ! 1 root
appear at
Importantly - M ! 0 extrema are at T > Tc
different to F3 = 0 case
- now look for max/min.
!2 F
= 2!( T " Tc ) + 6" M + 12# M 2
2
!M
For M = 0
min T > Tc
max T < Tc
as in F3 = 0 case
For M ! 0
We have
2! ( T ! Tc ) + 3" M + 4# M 2 = 0
# !F
&
%%
from →
= 0, M " 0 (((
%$ !M
'
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PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 3- S C Chapman
So,
!2 F
= 3! M + 8" M 2
2
!M
! 3"
$
= 8! M ## + M &&&
#" 8!
%
for the case where M ! 0
!3!
8"
- so one real M ! 0 root.
passes through zero at M =
this is when ! 2 = ! c2
otherwise ! ! ! c2 :
!2 F
= M #% ±3 ! 2 " ! c2 &(
2
$
'
!M
- need to look at signs (handout) to classify max/min
- but see that as we go from large T → small T the root goes Im→ Re
To summarise - max/min behaviour
M =0
M !0
min
max
real
real
2 min
x x x x
imaginary
1 max, 1 min
(+ root) (- root)
T = Tc
Tc ! Tc
! =!
2
2
c
! c2 > ! 2
!c = 0
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PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 3- S C Chapman
M
min
+ ve root
!3!
8"
- ve root
min
0
! 2 = ! c2
T ! Tc
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PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 3- S C Chapman
Dynamics with T
F
T > Tc
! c2 > ! 2
M
T > Tc , ! c2 < ! 2
T = Tc
!c = 0
T < Tc
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PX391 Nonlinearity, Chaos, Complexity SUMMARY Lecture Notes 3- S C Chapman
M
hysteresis
!
- Hysteresis
- Fluctuations don't matter
- 1st order phase transition
- discussed ferromagnets but anything can do this if satisfies F ( M ) , etc. – universality.
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