Multiplicity

Hyperphysics Article

Topics

• Microstate

$\Omega =\frac{1}{N!h^{3N}}{\int }_{\text{volume}}\prod _{i=1}^N{\mathrm{d}}^3{x}_i{\mathrm{d}}^3{p}_i$

• $\Omega$ is the number of microstates of a system. $g$ is another common synonymous symbol
• $N$ is the number of particles, which are all indistinguishable
• $h$ is Planck Constant

$\Omega ={\Omega }_1{\Omega }_1=e^{{\sigma }_1+{\sigma }_2}$

• For two different systems 1 and 2, their multiplicities multiply together by the multiplication principle of counting

$\Omega (U,V,N)=\frac{V^N}{N!h^{3N}}\frac{2{\pi }^{3N/2}}{\frac{3}{2}N!}(2mU{)}^{3N/2}$

• Ideal gas multiplicity function

$\frac{P(\epsilon )}{P(0)}=e^{-\epsilon /\tau }$

• Probability of a system having energy $\epsilon$ over probability of having 0 energy for temperature $\tau$
• Related to the partition function