Statistical Mechanics - Physics at Oregon State University

196 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
〈V〉λ = −〈
(σi − µ)(σj − µ)〉λ = − 1
2 Nq〈(σi − µ)(σj − µ)〉λ
We now introduce the average magnetization mλ and write
(9.30)
〈V〉λ = − 1
2 Nq〈(σi − mλ)(σj − mλ)〉λ − 1
2 Nq〈2(mλ − µ)σi〉λ + 1
2 Nq(m2λ − µ 2 )
(9.31)
Next we approximate the correlations between the fluctuations around the average
by zero and obtain
or
or
This gives
〈V〉λ ≈ − 1
2 Nq〈2(mλ − µ)σi〉λ + 1
2 Nq(m2 λ − µ 2 ) (9.32)
〈V〉λ ≈ −Nq(mλ − µ)mλ + 1
2 Nq(m2 λ − µ 2 ) (9.33)
〈V〉λ ≈ − 1
2 Nq(mλ − µ) 2
∂G
= −
∂λ T,h,N
1
2 Nq(mλ − µ) 2
The magnetic moment is related to the Gibbs like free energy via
∂G
Nmλ = −
∂h
and hence
T,N
∂m
2 ∂m
∂ G
N
= −
= Nq(m − µ)
∂λ h,T ∂h∂λ T,N
∂h
λ,T
−
∂µ
∂h T
(9.34)
(9.35)
(9.36)
(9.37)
where we have m(h, λ, T ) and µ(h, T ). This equation shows that if mλ = µ the
magnetization will not change as a function of λ. Also, because for large fields
the magnetization approaches one, we see that if m > µ one needs dm dµ
dh < dh and
hence the right hand side of the partial differential equation is zero. This means
that for large values of λ the solutions for m will approach µ. Now suppose that
we use the mean field solution for the magnetization with λ = J for µ. Then
we conclude that the true magnetization of the system will be smaller than the
mean field value, because we integrate to J only and not to infinity. Of course,
this uses an approximation for the correlation function, so the conclusion does
not have to be general.