Statistical mechanics  Partition function of a system of N particlesby Jalo Tags: function, mechanics, particles, partition, statistical 

#1
Jan213, 10:35 PM

P: 112

1. The problem statement, all variables and given/known data
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 2. Relevant equations 3. 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 Z^{N}. Now, I'll say that: (Where it says ε it's meant to be ε(r) ) Z = Ʃ_{r} exp(β(ε  ε) ) = Ʃ_{r} exp(βε) * exp(βε) = = Ʃ_{r} exp(βε) * Ʃ_{r} 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. 



#2
Jan213, 11:01 PM

P: 112

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).




#3
Jan213, 11:03 PM

P: 112

Look at any example problem in a thermo book for a 2state system




#4
Jan313, 09:39 AM

P: 112

Statistical mechanics  Partition function of a system of N particles
8ikmAm I not summing over the expoent of the energy of each microstate?
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! 



#5
Jan313, 09:55 AM

P: 112

A good book is Thermal Physics by Kittel + Kroemer 



#6
Jan313, 10:00 AM

P: 112

Where Es is the energy of the sth state



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