4.3 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 and on , as follows:

(4.49) |

(4.50) |

From the grand partition function we can easily derive expressions for the various thermodynamic observables. For instance, putting and we find

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.

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.

2005-01-25