10.4 Other ensembles 315
subsystem (V 2 ) minus the interactions it felt in its previous subsystem (V 1 ). These
moves are actually executed in the Gibbs ensemble MC method, alongside the
usual particle moves within the two subsystems and relative changes of the two
subsystem volumes. In order to be able to vary the subsystem volumes, we scale
the particle positionsRK=LKSKwhere the indexK=1, 2 labels the subsystem
andLK =VK^1 /^3. The weight of a configuration with subsystem volumesV 1 and
V 2 =V−V 1 , subsystem particle numbersN 1 andN 2 and subsystem configurations
S 1 andS 2 is given by
ρ(V 1 ,N 1 ,S 1 ,V 2 ,N 2 ,S 2 )=
V 1 N^1
N 1!
V 2 N^2
N 2!
e−βU(L^1 S^1 )e−βU(L^2 S^2 ). (10.39)
This expression follows directly from the weight factors for the ensembles con-
sidered in the two previous subsections (the pressure and the chemical potential
occur withV 1 +V 2 andN 1 +N 2 respectively, which are constant).
Equation (10.39) determines the detailed balance transition probabilities in a
Metropolis algorithm:
- The matrixωXX′is a probability for volume changes, particle transfers from
subsystem 1 to 2 or vice versa, and for particle moves in either 1 or 2
respectively. The matrix elements for particle moves are chosen such as to allow
for single particle moves only, with equal probability for each particle.
- Particle moves in each subsystem are processed with the probabilities based on
the factor
exp{−β[U(LSnewK )−U(LSKold)]}. (10.40)
- For subsystem volume changes we find for the acceptance rate (see also
Section 10.4.1):
∏
K=1,2
exp{−β[U(LnewK SK)−U(LoldK SK)]}
(
VKnew
VKold
)NK
. (10.41)
- Acceptance rates for particle transfers from subsytem 1 to subsystem 2 involve
the energy differenceUK±for removing (−) or adding (+) a particle to the
subsystem analogous to the grand canonical MC method:
exp[−β(U 2 +−U 1 −)]
N 1
N 2 + 1
V 2
V 1
(10.42)
and similarly for transfers from subsystem 2 to subsystem 1.
These transition rates define the Gibbs ensemble Monte Carlo method. It should
be noted that this method is still susceptible to the kind of problems described in
connection with the grand canonical Monte Carlo method: moving a particle from one
subsystem to another may have a prohibitively low acceptance rate at high densities,
because of the large increase of configurational energy in most such attempts.
