Hardy–Ramanujan Formula

[Shustaire]

I've been searching for some details on the partition number of the Hardy–Ramanujan formula with regards to open and closed Strings.

 

Any help is appreciated

[Vmedvil]

I've been searching for some details on the partition number of the Hardy–Ramanujan formula with regards to open and closed Strings.

 

Any help is appreciated

 

https://en.wikipedia.org/wiki/Hardy%E2%80%93Ramanujan_theorem

 

 

[Shustaire]

Yeah that didn't help with the lack of end points of the closed Strings but I was looking at it wrong.

 

Its the harmonic oscillations of the fundamental frequency. This realization solved my dilemna with how to use it on the closed strings.

 

Thanks for the assist though. I wasn't sure how to apply it under string theory specifically.

 

The D brane application you posted is certainly related

 

the relation I missed is

 

[math]E_{\ell},n_\ell=n_\ell\hbar(n_\ell\omega)[/math]

 

the state occupation numbers is

[math]N|\hat{\psi}\rangle=N|\psi\rangle[/math]

 

so [math]N=\displaystyle\sum^{\infty}_{\ell=1}\ell N_\ell[/math]

 

ah got it the entropy as a function of energy is

 

[math]E=\frac{\pi^2}{6}\frac{1}{\hbar\omega_0}\frac{1}{\beta^2}=\frac{\pi^2}{6}(\frac{KT}{\hbar\omega_0})^2[/math]

 

entropy as a function of temp

[math]S=K\frac{\pi^2}{3}(\frac{KT}{\hbar\omega_0})[/math]

 

cool got what I need now to figure out Hagedorn energy

 

Currently studying "String theory a first course" textbook. Missed the first relation to understand the latter equations.

Edited December 31, 2017 by Shustaire