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Recently i am studying Statistical mechanics and reading about the Boltzman hypothesis about entropy $$S = k \,ln\,\Omega(E)$$ where it says $\Omega(E)$ is total no. of microstates , the available volume in phase space . Now if i consider an ideal gas confined in a volume V then the Total no. of microstate is $$ \Omega\,=\,\frac{V}{h^3}\int dp_1\,dp_2\,dp_3...dp_{3N}$$

And as for free particles $E\,=\,\frac{p^2}{2m}$ so if i write the integration in momentum space in spherical polar coordinate then Radius of the sphere is $p\,=\,\sqrt{2mE}$

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Now the total no. of microstates $\Omega(E)$ is no. of microstates from energy 0 to E. And we use this to calculate Entropy S. Now my question is in microcanonical ensemble if i say the system is in Energy E to E+$\Delta$E then the total no. of available microstate is $$ \Delta\Omega\,=\,\Omega(E+\Delta E)-\Omega(E)$$ the microstates which are in the hyper shell, but while writing entropy we use total no. of microstate $\Omega(E)$ which is actually total no. of microstate from energy o to E not from E to E+$\Delta$E. So apparently we will get wrong answer , we should calculate $$ S\,=\,k\,ln(\Delta \Omega)\,=\, k\,ln\,(\Omega(E+\Delta E)-\Omega(E))$$

So why we use total no. of microstate from energy 0 to E to calculate Entropy although the system can have energy only from E to E+$\Delta$E ?

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  • $\begingroup$ Where is your fixed energy E in your integral for omega? $\endgroup$ – anna v May 9 at 19:24
  • $\begingroup$ Here at formula 18 theory.physics.manchester.ac.uk/~xian/thermal/chap3.pdf the E enters in delta function. as far as I can tell the limits of intergration for dp are from -infinity to +infinity, it is the delta function that restrains the values. $\endgroup$ – anna v May 10 at 5:31

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