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# 4.3 Grand canonical ensemble

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.

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 and on , as follows:
 (4.49)

Summing this density over all possible values of and integrating - at each - over all we obtain the grand partition function
 (4.50)

Its value is just the total statistical weight'' of all possible states of system . Above all, it serves as the source function of thermodynamics.

THERMODYNAMICS IN THE GRAND CANONICAL ENSEMBLE
From the grand partition function we can easily derive expressions for the various thermodynamic observables. For instance, putting and we find
 (4.51) (4.52) (4.53)

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 statistical mechanics.

[To do: applet with MD simulation, averages taken only over particles in a partial volume -> same results!]

Example: Let us visit the ideal gas again. For the grand partition function we have

 (4.54)

Therefore

 (4.55)

Using the formulae for internal energy and pressure we find

 (4.56)

Consequently, or

 (4.57)

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 of microstates 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 thermodynamic observable.

Next: 4.4 Problems for Chapter Up: 4. Statistical Thermodynamics Previous: 4.2 Canonical ensemble
Franz Vesely
2005-01-25