Partition function next up previous
Next: Ideal gas Up: The grand canonical ensemble Previous: Thermodynamics

Partition function

Consider two canonical systems, 1 and 2, with particle numbers tex2html_wrap_inline523 and tex2html_wrap_inline525 , volumes tex2html_wrap_inline527 and tex2html_wrap_inline529 and at temperature T. The systems are in chemical contact, meaning that they can exchange particles. Furthermore, we assume that tex2html_wrap_inline533 and tex2html_wrap_inline535 so that system 2 is a particle reservoir. The total particle number and volume are

eqnarray101

The total Hamiltonian tex2html_wrap_inline537 is

displaymath103

If the systems could not exchange particles, then the canonical partition function for the whole system would be

eqnarray106

where

eqnarray111

However, tex2html_wrap_inline523 and tex2html_wrap_inline525 are not fixed, therefore, in order to sum over all microstates, we need to sum over all values that tex2html_wrap_inline523 can take on subject to the constraint tex2html_wrap_inline545 . Thus, we can write the canonical partition function for the whole system as

displaymath118

where tex2html_wrap_inline547 is a function that weights each value of tex2html_wrap_inline523 for a given N. Thus,

f(0,N) is the number of configurations with 0 particles in tex2html_wrap_inline527 and N particles in tex2html_wrap_inline529 .
f(1,N) is the number of configurations with 1 particles in tex2html_wrap_inline527 and N-1 particles in tex2html_wrap_inline529 .
etc.

Determining the values of tex2html_wrap_inline569 amounts to a problem of counting the number of ways we can put N identical objects into 2 baskets. Thus,
f(0,N)=1
f(1,N)=N=N!/1!(N-1)!
f(2,N)=N(N-1)/2=N!/2!(N-2)!
etc.

or generally,

displaymath126

which is clearly a classical degeneracy factor. If we were doing a purely classical treatment of the grand canonical ensemble, then this factor would appear in the sum for Q(N,V,T), however, we always include the ad hoc quantum correction 1/N! in the expression for the canonical partition function, and we see that these quantum factors will exactly cancel the classical degeneracy factor, leading to the following expression:

displaymath131

which expresses the fact that, in reality, the various configurations are not distinguishable from each other, and so each one should count with equal weighting. Now, the distribution function tex2html_wrap_inline583 is given by

displaymath134

which is chosen so that

displaymath138

However, recognizing that tex2html_wrap_inline585 , we can obtain the distribution for tex2html_wrap_inline587 immediately, by integrating over the phase space of system 2:

displaymath140

where the tex2html_wrap_inline589 prefactor has been introduced so that

displaymath148

and amounts to the usual ad hoc quantum correction factor that must be multiplied by the distribution function for each ensemble to account for the identical nature of the particles. Thus, we see that the distribution function becomes

displaymath152

Recall that the Hemlholtz free energy is given by

displaymath157

Thus,

eqnarray160

or

displaymath165

But since tex2html_wrap_inline591 and tex2html_wrap_inline593 , we may expand:

eqnarray169

Therefore the distribution function becomes

eqnarray173

Dropping the ``1'' subscript, we have

displaymath183

We require that tex2html_wrap_inline595 be normalized:

eqnarray189

Now, we define the grand canonical partition function

displaymath199

Then, the normalization condition clearly requires that

eqnarray207

Therefore PV is the free energy of the grand canonical ensemble, and the entropy tex2html_wrap_inline599 is given by

displaymath213

We now introduce the fugacity tex2html_wrap_inline601 defined to be

displaymath222

Then, the grand canonical partition function can be written as

displaymath225

which allows us to view the grand canonical partition function as a function of the thermodynamic variables tex2html_wrap_inline601 , V, and T.

Other thermodynamic quantities follow straightforwardly:

Energy:

eqnarray237

Average particle number:

displaymath248

This can also be expressed using the fugacity by noting that

displaymath252

Thus,

displaymath258


next up previous
Next: Ideal gas Up: The grand canonical ensemble Previous: Thermodynamics

Mark Tuckerman
Tue Feb 1 14:50:00 EST 2000