Computational Physics - Department of Physics

420 13 Monte Carlo Methods in Statistical Physics

Similarly we can compute the chemical potential as

μ=−kBT

(

∂lnZ

∂N

)

V,T

.

For a system described by the canonical ensemble, the energyis an expectation value since
we allow energy to be exchanged with the surroundings (a heatbath with temperatureT).
This expectation value, the mean energy, can be calculated using

〈E〉=kBT^2

(

∂lnZ

∂T

)

V,N

or using the probability distributionPias

〈E〉=

M

∑

i= 1

EiPi(β) =

1

Z

M

∑

i= 1

Eie−βEi.

The energy is proportional to the first derivative of the potential, Helmholtz’ free energy. The
corresponding variance is defined as

σE^2 =〈E^2 〉−〈E〉^2 =^1

Z

M

∑

i= 1

Ei^2 e−βEi−

(

1

Z

M

∑

i= 1

Eie−βEi

) 2

.

If we divide the latter quantity withkT^2 we obtain the specific heat at constant volume

CV=

1

kBT^2

(

〈E^2 〉−〈E〉^2

)

,

which again can be related to the second derivative of Helmholtz’ free energy. Using the same
prescription, we can also evaluate the mean magnetization through

〈M〉=

M

∑

i

MiPi(β) =

1

Z

M

∑

i

Mie−βEi,

and the corresponding variance

σM^2 =〈M^2 〉−〈M〉^2 =

1

Z

M

∑

i= 1

Mi^2 e−βEi−

(

1

Z

M

∑

i= 1

Mie−βEi

) 2

.

This quantity defines also the susceptibilityχ

χ=

1

kBT

(

〈M^2 〉−〈M〉^2

)

13.2.3Grand Canonical and Pressure Canonical

Two other ensembles which are much used in statistical physics and thermodynamics are
the grand canonical and pressure canonical ensembles. In the first we allow the system (in
contact with a large heat bath) to exchange both heat and particles with the environment.
The potential is, with a partition functionΞ(V,T,μ)with variablesV,Tandμ,

pV=kBT lnΞ,