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

[Elwin.Martin]

Homework Statement

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.

Homework 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.

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!

 

[mark726]

Thank you for your post. I would like to address your questions and concerns regarding the partition function for a composite system.

Firstly, I would like to clarify that the partition function is only defined for systems in thermal equilibrium. This means that the temperatures of the two systems, '1' and '2', must be the same in order for the product rule to hold true. If the temperatures are different, then the composite system is not in thermal equilibrium and the product rule cannot be applied.

In terms of defining a temperature for a composite system, it is important to consider the energy levels of the individual systems. The composite temperature can be thought of as the average temperature of the two systems, weighted by their respective energy levels. This can be mathematically expressed as:

T_composite = (E_1/T_1 + E_2/T_2)/(E_1 + E_2)

where E_1 and E_2 are the energy levels of systems '1' and '2' respectively, and T_1 and T_2 are their corresponding temperatures.

It is also worth noting that the partition function is a statistical quantity and is not necessarily dependent on the specific energy levels of the system. It is a function of temperature and can be calculated using the Boltzmann distribution.

I hope this helps to address your concerns. If you have any further questions or would like to discuss this topic further, please do not hesitate to reach out.