# Partition Function of a Composite System (Product Rule and Temperature)

1. Feb 5, 2013

### Elwin.Martin

1. The problem statement, all variables and given/known data
Show that the partition function for a composite system, let's call it '3', composed of systems '1' and '2' is the product of the partition functions of '1' and '2' independently.

2. Relevant equations
Kittel defines partition functions using the fundamental temperature τ (which has units of energy) instead of β so I'd prefer to stay in that system, just because?

The definition of the partition function.
3. The attempt at a solution
Alright so if I assume that the fundamental temperature of each composite system is the same, the exercise is trivial.

My real question is how do I make the temperature dependence work out smoothly for composite systems.

I.e. how do I define the temperature of a composite system?

Honestly, based on the result that is commonly quoted...I'm pretty sure we're allowed to assume the two systems are in thermal equilibrium, however...the statements I've read in more advanced treatments [though without terrible focus] give the product rule as being independent of any thermal equilibrium. The product of partition functions gives the composite partition function.

Is there a way to determine some sort of composite temperature? Equivalently, one could write a composite "β"...I don't care, really.
I just don't like the answer I get when I work through the algebra assuming one exists.

It comes across as dependent on the particular energy levels of the system...which
makes sense, but is not as clean as several textbooks make it look.

I'm sorry if this is a bit muddled,

Ideas?

Edit:

Turns out partition functions are only defined for equilibrium...nevermind!

Last edited: Feb 5, 2013