On Tue, Jul 25, 2006 at 06:46:12PM -0600, Robert Holmes wrote:
> >
> >
> >One can certainly start from the partition function. But the partition
> >function is something that is additional to the microscopic
> >description, hence emergent. Indeed, the partition function is
> >different depending on whether you are using microcanonical, canonical
> >or grand canonical ensembles, each of which is a thermodynamic, not
> >microscopic concept.
>
>
> I'm surprised that you consider the partition function as being "in
> addition" to the microscopic description. Is this the common view in
> statistical mechanics? Just to be specific, if I've got a system of
> distinguishable particles and the energy levels aren't degenerate, the
> single particle partition function Zsp is given by:
>
> Zsp = sum( exp( -ei/k.T ) )
> where ei is the energy of the energy level i, the sum is over all i (i.e.
> over all energy levels), k is the Boltzmann constant and T is the
> temperature.
>
> Now that seems about as microscopic description of a system as you can get.
> Could you explain why it's not please?
>
> Thanks for your patience!
>
> Robert
You have just written the canonical partition function. This assumes
that the universe is divided into two parts, the system, and its
environment, and that these are in thermal contact with each other.
If you further assume that particles can move between the system and
environment, then you get the grand canonical partition function:
Z=\sum_{N=0}^{\infty}\sum_{{n_i}}\prod_i exp(-n_i(E_i-\mu)/kT)
These assumptions are not microscopic in nature, but how we choose
to divide up physical reality. (The choice is needn't be arbitrary - in
most stat phys situations, there is a clear "best choice", and choosing
any other way of looking at the system is crazy, but you must
recognise that it is still a choice independent of microscopic dynamics).
Cheers
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