Statistical mechanics - Partition function of a system of N particles

AI Thread Summary
The discussion focuses on finding the partition function for a system of N distinguishable particles, each capable of existing in two energy states: -ε and +ε. The user struggles with the correct formulation of the partition function, initially misapplying the summation over energy states. Clarification is provided that the partition function is a summation over the exponential of the energy states, not the energies themselves. The correct result for the partition function is identified as Z = (exp(βε) + exp(-βε))^N. Recommendations for further reading include "Thermal Physics" by Kittel and Kroemer for better understanding.
Jalo
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Homework Statement



Imagine a system with N distinguishable particles. Each particle may be in two states of energy: -ε and +ε.

Find the the partition function of the system

Homework Equations



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The Attempt at a Solution



I know that I have to find the partition function for a single function, Z, and my final result will be ZN. Now, I'll say that:

(Where it says ε it's meant to be ε(r) )

Z = Ʃr exp(-β(ε - ε) ) = Ʃr exp(-βε) * exp(βε) =
= Ʃr exp(-βε) * Ʃr exp(βε)



I'm sure this is incorrect. It doesn't make sense in my head.. E(r) is the energy associated with each microstate, therefore saying that E(r) = ε(r) - ε(r) can't make any sense! I know that the result is:

Z = ( exp(βε) + exp(-βε) )N

I have no idea how to get there tho. How did it became a sum? How do I get rid of the summatories?

Any help will be appreciated!
Thanks.
 
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The partition function is a summation over states. You simply are using the summation wrong. It is not a summation over the energy levels of within the exponent. It is a summation over e(-Es/T).
 
Look at any example problem in a thermo book for a 2-state system
 
8ikmAm I not summing over the expoent of the energy of each microstate?

klawlor419 said:
Look at any example problem in a thermo book for a 2-state system

Do you know any good statistical mechanics book you'd advise me reading?

EDIT:
Is it a summation over all the states of energy instead of the energies of each microstate? Because then the solution would make sense!
 
Last edited:
Jalo said:
Am I not summing over the expoent of the energy of each microstate?



Do you know any good statistical mechanics book you'd advise me reading?

No. You are summing over the exponential function raised to the -Es/T.

A good book is Thermal Physics by Kittel + Kroemer
 
Where Es is the energy of the s-th state
 

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