Computational Physics - Department of Physics

13.2 Review of Statistical Physics 419

p

kBT

=

(

∂logΩ

∂V

)

N,E

,

or the chemical potential.
μ
kBT=−

(

∂logΩ

∂N

)

V,E

It is very difficult to compute the density of statesΩ(E)and thereby the partition function
in the microcanonical ensemble at a given energyE, since this requires the knowledge of all
possible microstates at a given energy. This means that calculations are seldomly done in the
microcanonical ensemble. In addition, since the microcanonical ensemble is an isolated sys-
tem, it is hard to give a physical meaning to a quantity like the microcanonical temperature.

13.2.2Canonical Ensemble

One of the most used ensembles is the canonical one, which is related to the microcanoni-
cal ensemble via a Legendre transformation. The temperature is an intensive variable in this
ensemble whereas the energy follows as an expectation value. In order to calculate expec-
tation values such as the mean energy〈E〉at a given temperature, we need a probability
distribution. It is given by the Boltzmann distribution

Pi(β) =

e−βEi

Z

withβ= 1 /kBTbeing the inverse temperature,kBis the Boltzmann constant,Eiis the energy
of a microstateiwhileZis the partition function for the canonical ensemble definedas

Z=

M

∑

i= 1

e−βEi,

where the sum extends over all microstatesM. The potential of interest in this case is
Helmholtz’ free energy. It relates the expectation value ofthe energy at a given temperatur
Tto the entropy at the same temperature via

F=−kBT lnZ=〈E〉−T S.

Helmholtz’ free energy expresses the struggle between two important principles in physics,
namely the strive towards an energy minimum and the drive towards higher entropy as the
temperature increases. A higher entropy may be interpretedas a larger degree of disorder.
When equilibrium is reached at a given temperature, we have abalance between these two
principles. The numerical expression is Helmholtz’ free energy. The creation of a macroscopic
magnetic field from a bunch of atom-sized mini-magnets, as shown in Fig. 13.1 results from
a careful balance between these two somewhat opposing principles in physics, order vs. dis-
order.
In the canonical ensemble the entropy is given by

S=kBlnZ+kBT

(

∂lnZ

∂T

)

N,V

,

and the pressure by

p=kBT

(

∂lnZ

∂V

)

N,T

.