# Microstates in phase space

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 $$p\,=\,\sqrt{2mE}$$

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 ?

• Where is your fixed energy E in your integral for omega? – anna v May 9 at 19:24
• 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. – anna v May 10 at 5:31