Statistical mechanics - Partition function of a system of N particles


by Jalo
Tags: function, mechanics, particles, partition, statistical
Jalo
Jalo is offline
#1
Jan2-13, 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 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.
Phys.Org News Partner Science news on Phys.org
SensaBubble: It's a bubble, but not as we know it (w/ video)
The hemihelix: Scientists discover a new shape using rubber bands (w/ video)
Microbes provide insights into evolution of human language
klawlor419
klawlor419 is offline
#2
Jan2-13, 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).
klawlor419
klawlor419 is offline
#3
Jan2-13, 11:03 PM
P: 112
Look at any example problem in a thermo book for a 2-state system

Jalo
Jalo is offline
#4
Jan3-13, 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?

Quote Quote by klawlor419 View Post
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!
klawlor419
klawlor419 is offline
#5
Jan3-13, 09:55 AM
P: 112
Quote Quote by Jalo View Post
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
klawlor419
klawlor419 is offline
#6
Jan3-13, 10:00 AM
P: 112
Where Es is the energy of the s-th state


Register to reply

Related Discussions
statistical thermodynamics: partition function Quantum Physics 0
Partition function in Statistical Physics.. General Physics 1
Statistical Physics: Partition function and fermions Advanced Physics Homework 0
classical statistical mechanics: dimensions of partition function Classical Physics 2