Z, , and C for Hagedorn Spectrum

Thread starter dark_matter_is_neat
Start date Apr 13, 2024
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Statisical mechanics

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SUMMARY

The discussion focuses on calculating the partition function Z for the Hagedorn spectrum using the integral ##\int \alpha E^{3} e^{(B_{0}-B)E} dE##. The derived formula for Z is ##Z = \frac{ \alpha E^{3} e^{\Delta B E}}{\Delta B} - \frac{3 \alpha E^{2} e^{\Delta B}}{\Delta B^{2}} + \frac{6 \alpha E e^{\Delta B E}}{\Delta B ^{3}} - \frac{6 \alpha e^{\Delta B E}}{\Delta B ^{4}}##. The internal energy is computed as ##- \frac{\partial ln(Z)}{\partial B}##, while the heat capacity C is determined by ##k \frac{\partial }{ \partial T}##. The participant expresses concern about the complexity of the calculations for and C, questioning whether there is a more efficient method or if errors occurred in calculating Z.

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Apr 13, 2024

dark_matter_is_neat

Homework Statement: The Hagedorn Spectrum with E>=0 is ##\frac{dn}{dE} = \alpha E^{3} e^{B_{0}E}## where ##B_{0} = 1/kT_{0}## where ##T_{0}## is the Hagedorn Temperature. For ##T<T_{0}## determine the partition function Z, the average energy and the specific heat C for the system.

Relevant Equations: ##\frac{dn}{dE} = \alpha E^{3} e^{B_{0}E}##

So to get the partition function I do the integral ##\int \alpha E^{3} e^{(B_{0}-B)E} dE##, which substituting in ##/Delta B = B_{0} - B## is ##Z = \frac{ \alpha E^{3} e^{\Delta B E}}{\Delta B} - \frac{3 \alpha E^{2} e^{\Delta B}}{\Delta B^{2}} + \frac{6 \alpha E e^{\Delta B E}}{\Delta B ^{3}} - \frac{6 \alpha e^{\Delta B E}}{\Delta B ^{4}}##. To get you just calculate ##- \frac{\partial ln(Z)}{\partial B}## and to get C you just calculate ##k \frac{\partial }{ \partial T}##. The math gets pretty rough calculating and gets especially rough calculating C and I'm wondering if there is a much less tedious way to get them or if I messed up calculating Z.