Computational Physics - Department of Physics

426 13 Monte Carlo Methods in Statistical Physics

eigenvalues resulting in a partition function

ZN=λ 1 N+λ 2 N= 2 N

(

[cosh(βJ)]N+ [sinh(βJ)]N

)

.

In the limitN→∞the two partition functions with free ends and periodic boundary conditions
agree, see below for a demonstration.
In the development phase of an algorithm and its pertinent code it is always useful to test
the numerics against closed-form results. It is therefore instructive to compute properties
like the internal energy and the specific heat for these two cases and test the results against
those produced by our code. We can then calculate the mean energy with free ends from the
above formula for the partition function using

〈E〉=−

∂lnZ

∂ β

=−(N− 1 )Jtanh(βJ).

Helmholtz’s free energy is given by

F=−kBT lnZN=−NkBT ln( 2 cosh(βJ)).

If we take our simple system with just two spins in one-dimension, we see immediately that
the above expression for the partition function is correct.Using the definition of the partition
function we have

Z 2 =

2

∑

i= 1

e−βEi= 2 e−βJ+ 2 eβJ= 4 cosh(βJ)

If we take the limitT→ 0 (β→∞) and setN= 2 , we obtain

lim

β→∞

〈E〉=−Je

Jβ−e−Jβ

eJβ+e−Jβ

=−J,

which is the energy where all spins point in the same direction. At lowT, the system tends
towards a state with the highest possible degree of order.
The specific heat in one-dimension with free ends is

CV=

1

kT^2

∂^2

∂ β^2

lnZN= (N− 1 )k

(

βJ

cosh(βJ)

) 2

.

Note well that this expression for the specific heat from the one-dimensional Ising model does
not diverge or exhibit discontinuities, as can be seen from Fig. 13.3.

In one dimension we do not have a second order phase transition, although this is predicted
by mean field models [57].
We can repeat this exercise for the case with periodic boundary conditions as well.
Helmholtz’s free energy is in this case

F=−kBT ln(λ 1 N+λ 2 N) =−kBT

{

Nln(λ 1 )+ln

(

1 + (

λ 2

λ 1

)N

)}

,

which in the limitN→∞results inF=−kBT Nln(λ 1 )as in the case with free ends. Since
other thermodynamical quantities are related to derivatives of the free energy, all observables
become identical in the thermodynamic limit.
Hitherto we have limited ourselves to studies of systems with zero external magnetic field,
vizB= 0. We will mostly study systems which exhibit a spontaneous magnitization. It is
however instructive to extend the one-dimensional Ising model toB 6 = 0 , yielding a partition