Notes on Mean Field Theory

y expanding the full equation s = tanh(mB + Jνs)/τ assuming that both
s and B are small. The result is
M = Nms = Nm2
τ − τ c
B (8)
The coefficient of B is a quantity called the magnetic susceptibility. Its
divergence at the critical temperature is a sign that the system is on the
verge of being ordered: a very small magnetic field will produce a large
magnetization.
Energy and heat capacity. The energy U of the system is best obtained
directly from Eq. (1), since the standard method of obtaining it from
Z mf would overcount the interaction terms. If the number of spins is N,
the number of nearest neighbor pairs is Nν/2. Therefore, Eq. (1) gives
The heat capacity is
U = −NmBs − 1 2 NνJs 2 . (9)
C = ∂U
∂τ
= −NmB
ds
dτ −1 2 NνJ ds2
dτ . (10)
Since the derivatives of s and s 2 with respect to τ are both negative, as
indicated in Figure (), the heat capacity is positive. When the external field
B is absent, C is zero above the critical temperature where s = 0. Eq. (10)
shows that C is discontinuous at τ c , since ds/dτ is negative and finite at
this point (see Eq. (7). The discontinuity of C at the critical temperature is
characteristic of this type of phase transition and shows that the two phases
have different physical properties.
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