10.4 Other ensembles 313
and annihilation of particles should be possible. Let us write down the probabil-
ity distribution of configurations in this ensemble. The grand canonical ensemble
average of a configurational physical quantityAis given by
〈A〉μVT=
∑∞
N= 01 /N!e
βμN− 3 N∫dRNA(RN)e−βU(RN)
ZG(μ,V,T)
; (10.35)
=h/
√
2 πmkBTin the case of a monatomic gas – it results from integrating out
the momentum degrees of freedom. We have attached a subscriptNto the positional
coordinateRbecause the number of particles is not fixed.
It is clear from(10.35)that the configurations are now defined by the number
of particlesN and by their positionsRN. The weight factor which replaces the
Boltzmann factor of the(NVT)ensemble now becomes
ρ(N,RN)=e−βU(RN)−^3 N/N!eβμN (10.36)
and the Metropolis algorithm can be applied directly, provided that in addition to
the usual particle moves, we allow for particle creations at random positions and
annihilations of randomly chosen particles. The algorithm for a Metropolis step
then becomes:
- Decide whether the next trial configuration is constructed via a creation, an
annihilation, or a particle move according to the probabilities for these
processes given by the matrixωXX′(note that the trial rates for creation and
annihilation should be equal in order to keepωXX′symmetric). This choice can
be made simply by dividing the interval[0, 1 ]up into three segments with sizes
equal to the respective probabilities, generating a random number uniformly
between 0 and 1, and then checking in which segment this number lies.
- If a creation is attempted, a random position in the system is selected and the
interactions between a new particle inserted at that position and the remaining
ones are added to yield a potential energy differenceU+=U(RN+ 1 )−
U(RN). As the probability that the new particle ends up in the volume element
d^3 rN+ 1 is given by d^3 rN+ 1 /V, we accept the creation with a probability
e−βU
+
−^3 V/(N+ 1 )eβμ. (10.37)
- If an annihilation is attempted, one of the existing particles is selected at
random, and its interaction with the remaining particles,U−=U(RN− 1 )−
U(RN), is calculated. The annihilation is then accepted with probability
e−βU
−
^3 N/Ve−βμ. (10.38)
- Particle moves are processed similarly to the canonical case. Only the potential
energy difference enters into the acceptance probability.
