Once again we put a small system () in contact with a large one ().
However, this time we do not only permit the exchange of energy but also
the crossing over of particles from one subsystem to the other.
System in contact with an energy and particle reservoir:
grand canonical ensemble
And as before we can write down the probability density in the phase space of
the smaller system; it depends now both on the number of particles
, as follows:
Summing this density over all possible values of and integrating
- at each - over all
we obtain the
grand partition function
Its value is just the ``total statistical weight'' of all possible
states of system . Above all, it serves as the source function of
THERMODYNAMICS IN THE GRAND CANONICAL ENSEMBLE
From the grand partition function we can easily derive expressions for
the various thermodynamic observables. For instance, putting
As a rule the - permitted - fluctuations of the number of particles remain
small; in particular we have
Thus the grand ensemble is again equivalent to others ensembles of
[To do: applet with MD simulation, averages taken only over particles
in a partial volume -> same results!]
Let us visit the ideal gas again. For the grand partition function we have
Using the formulae for internal energy and pressure we find
in keeping with the phenomenological ideal gas equation.
SIMULATION IN THE GRAND CANONICAL ENSEMBLE: GCMC
The states within the grand ensemble may again be sampled in a random manner.
Just as in the canonical Monte Carlo procedure we produce a sequence
with the appropriate relative
frequencies. The only change is that we now vary also the number of
particles by occasionally adding or removing a particle.
The stochastic rule for these insertions and removals is such that it agrees
with the thermodynamic probability of such processes.
By averaging some quantity over the ``Markov chain'' of configurations we again obtain an estimate of the respective
Next:4.4 Problems for ChapterUp:4. Statistical Thermodynamics Previous:4.2 Canonical ensembleFranz Vesely