Computational Physics - Department of Physics

418 13 Monte Carlo Methods in Statistical Physics

arising extensive (depend on the size of the systems such as the number of particles) and
intensive variables (apply to all components of a system), in addition to associated potentials.

Table 13.1Overview of the most common ensembles and their variables. Here we have defineM- to be the
magnetization,D- the electric dipole moment,H- the magnetic field andE- to be the electric field. The last
two replace the pressure as an intensive variable, while themagnetisation and the dipole moment play the
same role as volume, viz they are extensive variables. The invers temperaturβregulates the mean energy
while the chemical potentialμregulates the mean number of particles.

MicrocanonicalCanonicalGrand canonicalPressure canonical

Exchange of heat no yes yes yes

with the environment

Exchange of particles no no yes no

with the environemt

Thermodynamical V,M,D V,M,D V,M,D P,H,E

parameters E T T T

N N μ N

Potential Entropy Helmholtz PV Gibbs

Energy Internal Internal Internal Enthalpy

13.2.1Microcanonical Ensemble

The microcanonical ensemble represents an hypotheticallyisolated system such as a nucleus
which does not exchange energy or particles via the environment. The thermodynamical
quantity of interest is the entropySwhich is related to the logarithm of the number of pos-
sible microscopic statesΩ(E)at a given energyEthat the system can access. The relation
is
S=kBlnΩ.

When the system is in its ground state the entropy is zero since there is only one possible
ground state. For excited states, we can have a higher degeneracy than one and thus an
entropy which is larger than zero. We may therefore loosely state that the entropy measures
the degree of order in a system. At low energies, we expect that we have only few states
which are accessible and that the system prefers a specific ordering. At higher energies, more
states become accessible and the entropy increases. The entropy can be used to compute
observables such as the temperature

1

kBT

=

(

∂logΩ

∂E

)

N,V

,

the pressure