Thermal Physics, Homework #1 problem #1

1. Jan 18, 2010

scikris

Consider a large number N of localized particles in an external magnetic field H. Each particle has spin 1/2.

Find the number of states, g(N,M), accessible to the system as a function of M=(Nup-Ndown), the magnetization.

Calculate the entropy per particle.

Determine the value of M for which the number of states is a maximum for a given N.

Equations that may help?
N=Nup+Ndown

M=Nup-Ndown

g(N,s)= N!/(Nup!Ndown!)

σ(N,U)=log(g(N,U))

This is my first thermal physics course and I am kinda confused (and overwhelmed) by this first homework assignment if anyone could explain what I am suposed to do, or set me in a direction, I would appreciate it.

Kris

2. Jan 18, 2010

scikris

I should note, this homework is due tomorrow morning :(

any help appreciated!

3. Jan 18, 2010

jdwood983

Are you familiar with Stirling's Approximation:

$$\ln[N!]\approx N\ln[N]-N$$

4. Jan 18, 2010

scikris

yes, the stirling's approximation just gives a means to calculate N! for very large N, but I still don't know how to apply that to entropy per particle...

5. Jan 18, 2010

jdwood983

Use the equations you are given:

$$\sigma=\ln\left[\frac{N!}{N_{up}!N_{down}!}\right]=\ln[N!]-\ln[N_{up}!]-\ln[N_{down}!]$$

Apply Stirling's approximation to the above, then use the fact that $N=N_{up}+N_{down}$ and $M=N_{up}-N_{down}$.

6. Jan 18, 2010

scikris

okay thank you!,

"Determine the value of M for which the number of states is a maximum for a given N."
The value of M should be 0 correct?, because the middle of the Gaussian distribution will be centered at the origin. M=Nup-Ndown

7. Jan 18, 2010

jdwood983

Correct, but I would say this as "the value of M will be zero because the maximum of the Gaussian distribution will be when $N_{up}=N_{down}=N/2\rightarrow M=N_{up}-N_{down}=N_{up}-N_{up}=0$" rather than how you have it.

8. Jan 18, 2010

scikris

okay cool, your equation didnt show up could you re-post it?

9. Jan 18, 2010

jdwood983

That is weird, I thought \rightarrow worked here.... It should read

$N_{up}=N_{down}=N/2$ --> $M=N_{up}-N_{down}=N_{up}-N_{up}=0$

10. Jan 18, 2010

scikris

thanks, I think I have got it now.