Multiplicity
#Physics
Hyperphysics Article
Topics
$\displaystyle \Omega=\frac{1}{N!h^{3N}}\int {\text{volume}}\prod{i=1}^{N} , \mathrm{d}^{3}x_{i}\mathrm{d}^{3}p_{i}$
- $\displaystyle \Omega$ is the number of microstates of a system. $\displaystyle g$ is another common synonymous symbol
- $\displaystyle N$ is the number of particles, which are all indistinguishable
- $\displaystyle h$ is Planck Constant
$\displaystyle \Omega=\Omega_{1}\Omega_{1}=e^{{\sigma}{1}+{\sigma}{2}}$
- For two different systems 1 and 2, their multiplicities multiply together by the multiplication principle of counting
$\displaystyle \Omega(U,V,N)=\frac{V^{N}}{N!h^{3N}}\frac{2\pi^{3N/2}}{\frac{3}{2}N!}(2mU)^{3N/2}$
- Ideal gas multiplicity function
$\displaystyle \frac{P(\varepsilon)}{P(0)}=e^{-\varepsilon/\tau}$
- Probability of a system having energy $\displaystyle \varepsilon$ over probability of having 0 energy for temperature $\displaystyle \tau$
- Related to the partition function