The energy of the Ising model is given by:

Statistical Physics 3 October 13th, 2010

Series 2

The energy of the Ising model is given by:

H = − ∑

hi, ji

Js _i s _j − h

N i=1 ∑

s _i . (1)

We have also seen that the free energy per site, in the mean field approach, can be written:

f = F

N = − 1

β ln 2 − 1

β ln[cosh(β Jmz + β h)] + Jm ² z/2 (2) where z is the number of nearest neighbours of every site and β = _k ¹

B

T .

We want to show that the mean field approach is not valid in the unidimensional case in the thermodynamic limit. In order to do so, compare the free energy of a perfectly ordered system with the free energy of a partially disordered case that introduces the less energetic frustrations possible.

∆F = ∆U − T ∆S

Fix the spin of one of the extremity to avoid the issue due to the global symmetry of the system.

1