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Brian-san
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Homework Statement
a) Derive an asymptotic expression for the number of ways in which a given energy E can be distributed among a set of N, one-dimensional harmonic oscillators, the energy eigenvalues of the oscillators being [itex](N+\frac{1}{2})\hbar\omega, n=0, 1, 2, ...[/itex].
b)Find the corresponding expression for the "volume" of the relevant region of the phase space of the system. Establish the correspondence between the two results, showing that the conversion factor [itex]\omega_0[/itex] is precisely [itex]h^N[/itex].
c) In addition, derive the exact number of ways in which the energy can be divided among the N oscillators (before taking the asymptotic limit). Calculate the entropy for this system in the thermodynamic limit. Express the internal energy E in terms of the temperature T and oscillator number N, and plot this function versus T. Also, plot the heat capacity versus temperature. Find the asymptotic limits of the energy for large and small temperature.
Homework Equations
[itex]S=k_Bln\Omega[/itex] (Entropy)
[itex]lnN!\approx NlnN-N[/itex] (Stirling's Approximation)
The Attempt at a Solution
So for a given energy E, [itex]\frac{E}{\hbar\omega}=k[/itex], where k is a positive integer. Let [itex]m_i[/itex] denote the number of oscillators whose energy (per unit [itex]\hbar\omega[/itex]) is equal to i. Obviously then we know
[tex]\sum_{i=1}^k m_i=N[/tex]
Now the number of possible arrangements is the total number of ways we can divide N oscillators into k groups of sizes [itex]m_1, m_2, ..., m_k[/itex]. Then the number of ways to get a group of size [itex]m_1[/itex] is just
[tex]\binom{N}{m_1}[/tex]
Similarly, we want [itex]m_2[/itex] oscillators of the remaining [itex]N-m_1[/itex] oscillators, which is just
[tex]\binom{N-m_1}{m_2}[/tex]
Continuing this line of thought, the total number of arrangements should be
[tex]\Omega=\binom{N}{m_1}\binom{N-m_1}{m_2}...\binom{N-m_1-m_2-...-m_{k-1}}{m_K}=\frac{N!}{m_1!m_2!...m_k!}[/tex]
This would then be the exact expression that is wanted in part c, right? I can get the entropy from this by using Stirling's approximation, so
[tex]S=k_B(NlnN-N-\sum_{i=1}^k(m_ilnm_i-m_i))=k_B(NlnN-\sum_{i=1}^k(m_ilnm_i))[/tex]
Intuitively, I think the m's should go like
[tex]m_i=e^{-\frac{E_i}{k_BT}}[/tex]
So when we sum over all i we get
[tex]N=\sum_{i=1}^ke^{-\frac{E_i}{k_BT}}[/tex]
That's where I get stuck, I have a relation between N, T and [itex]E_i[/itex], but I can't figure out how to turn it into E=f(N,T). Also, I did the problem backwards finding the exact expression for the number of arrangements first (assuming it's right). Can I simply turn the exact expression into an asymptotic form, or is there a way to derive it without knowing the exact form?