Thermal Physics, Homework #1 problem #1

[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=(N_up-N_down), 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=N_up+N_down

M=N_up-N_down

g(N,s)= N!/(N_up!N_down!)

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.

Thanks in advance,
Kris

 

[scikris]

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

any help appreciated!

 

[jdwood983]

Are you familiar with Stirling's Approximation:

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

 

[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...

 

[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 $M=N_{up}-N_{down}$.

 

[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=N_up-N_down

 

[jdwood983]

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=N_up-N_down

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.

 

[scikris]

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

 

[jdwood983]

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

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

 

[scikris]

thanks, I think I have got it now.