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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The discussion focuses on deriving the partition function Z for the Hagedorn spectrum using the integral involving energy E and the parameters B and B0. The resulting expression for Z is complex, involving multiple terms with derivatives of E and powers of ΔB. To compute the average energy , the derivative of the logarithm of Z with respect to B is used, while the specific heat C is derived from the temperature derivative of . The participants express concerns about the complexity of the calculations for and C and seek a more efficient method or verification of their calculations for Z. The conversation highlights the challenges in statistical mechanics when dealing with the Hagedorn spectrum.

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.

Thread 'Help needed with Elliptic Integrals'

I want to find the solution to the integral ##\theta = \int_0^{\theta}\frac{du}{\sqrt{(c-u^2 +2u^3)}}## I can see that ##\frac{d^2u}{d\theta^2} = A +Bu+Cu^2## is a Weierstrass elliptic function, which can be generated from ##\Large(\normalsize\frac{du}{d\theta}\Large)\normalsize^2 = c-u^2 +2u^3## (A = 0, B=-1, C=3) So does this make my integral an elliptic integral? I haven't been able to find a table of integrals anywhere which contains an integral of this form so I'm a bit stuck. TerryW

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