Computational Physics

7.3 Phase transitions 185

T<Tt

T=T

T<T

1st order 2nd order

c

c

T>Tc

t

t

T=T

T>T

Figure 7.4. Typical behaviour of the free energy as a function of the order para-

meter and temperature. The left hand side corresponds to the first order case, with

transitions temperatureTf, and the right hand side to the continuous case, with

critical transition temperatureTc.

and for this minimum the order parameter assumes a particular value. It is possible
to define a free energy for any fixed value of the order parameter by calculating
the partition function for exclusively those configurations that have the prescribed
value of the order parameter. As an example, we can define the free energy,F(m),
for the Ising model with fixed magnetic field in terms of a partition function,Z(m),
defined as

Z(m)=

∑

{si}

e−βHδ

(∑

i

si−Ldm

)

(7.46a)

F(m)=−kBTlnZ(m), (7.46b)

wheredis the dimension of the system. Note the delta-function in the definition
ofZ(m)restricting the sum to configurations with a fixed magnetisationm.Itis
instructive to consider how this free energy as a function of the order parameter
changes with an external parameter (the temperature for example) across the trans-
ition for the two different types of phase transitions. Typical examples are shown
inFigure 7.4.
The equilibrium situation is characterised by the minimum of the free energy.
If we imagine the leftmost minimum in the first order case to correspond to the
liquid phase and the right hand one to the gas phase, we see that, away from the
transition temperature, one of the two phases is stable and the other one metastable.