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Thermal Physics, Homework #1 problem #1

  1. Jan 18, 2010 #1
    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.

    Thanks in advance,
    Kris
     
  2. jcsd
  3. Jan 18, 2010 #2
    I should note, this homework is due tomorrow morning :(

    any help appreciated!
     
  4. Jan 18, 2010 #3
    Are you familiar with Stirling's Approximation:

    [tex]
    \ln[N!]\approx N\ln[N]-N
    [/tex]
     
  5. Jan 18, 2010 #4
    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...
     
  6. Jan 18, 2010 #5
    Use the equations you are given:

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

    Apply Stirling's approximation to the above, then use the fact that [itex]N=N_{up}+N_{down}[/itex] and [itex]M=N_{up}-N_{down}[/itex].
     
  7. Jan 18, 2010 #6
    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
     
  8. Jan 18, 2010 #7
    Correct, but I would say this as "the value of M will be zero because the maximum of the Gaussian distribution will be when [itex]N_{up}=N_{down}=N/2\rightarrow M=N_{up}-N_{down}=N_{up}-N_{up}=0[/itex]" rather than how you have it.
     
  9. Jan 18, 2010 #8
    okay cool, your equation didnt show up could you re-post it?
     
  10. Jan 18, 2010 #9
    That is weird, I thought \rightarrow worked here.... It should read

    [itex]N_{up}=N_{down}=N/2[/itex] --> [itex]M=N_{up}-N_{down}=N_{up}-N_{up}=0[/itex]
     
  11. Jan 18, 2010 #10
    thanks, I think I have got it now.
     
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