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

Your Name]In summary, the partition function for a composite system, '3', composed of systems '1' and '2' is the product of the partition functions of '1' and '2' independently, only when the temperatures of the two systems are the same. The composite temperature can be defined as the average temperature of the two systems, weighted by their respective energy levels. The partition function is a statistical quantity and is not necessarily dependent on the specific energy levels of the system.
  • #1
Elwin.Martin
207
0

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!
 
Last edited:
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  • #2

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.
 

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

1. What is the partition function of a composite system?

The partition function of a composite system is a mathematical concept used in statistical mechanics to calculate the probability of a system being in a certain energy state. It is a product of the partition functions of the individual components of the system.

2. How is the partition function of a composite system calculated?

The partition function of a composite system is calculated using the product rule, which states that the partition function of a composite system is equal to the product of the partition functions of its individual components. This is because the energy states of a composite system are determined by the energy states of its individual components.

3. What is the significance of the partition function in statistical mechanics?

The partition function is a fundamental concept in statistical mechanics as it allows us to calculate the thermodynamic properties of a system, such as its energy, entropy, and free energy. It also provides a link between the microscopic properties of a system and its macroscopic behavior.

4. How does temperature affect the partition function of a composite system?

The temperature of a composite system affects its partition function by determining the energy states that are accessible to the system. As the temperature increases, more energy states become accessible, leading to a higher partition function and a greater probability of the system being in a higher energy state.

5. Can the partition function of a composite system be calculated for systems with infinite degrees of freedom?

Yes, the partition function can be calculated for systems with infinite degrees of freedom, such as a gas. In these cases, the partition function is often expressed as an integral over all possible energy states, rather than a sum. The integral can be evaluated using mathematical techniques such as the Euler-Maclaurin formula.

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