状態数

${\displaystyle W(X)=\sum _{\omega \in \varOmega (X)}1=\sum _{\omega \in \varOmega }\chi _{\varOmega (X)}(\omega )}$

${\displaystyle \chi _{A}(\omega )={\begin{cases}1&\omega \in A\\0&\omega \notin A\\\end{cases}}}$

${\displaystyle \varOmega (E,\delta E)=\{\omega \in \varOmega |E-\delta E<{\mathcal {E}}(\omega )\leq E\}}$

${\displaystyle \chi _{\varOmega (E,\delta E)}(\omega )=\eta (E-{\mathcal {E}}(\omega ))-\eta (E-\delta E-{\mathcal {E}}(\omega ))}$

${\displaystyle W(E,\delta E)=\sum _{\omega \in \varOmega }\{\eta (E-{\mathcal {E}}(\omega ))-\eta (E-\delta E-{\mathcal {E}}(\omega ))\}=N(E)-N(E-\delta E)}$

${\displaystyle N(E)=\sum _{\omega \in \varOmega }\eta (E-{\mathcal {E}}(\omega ))}$

${\displaystyle W(E,\delta E)\approx {\frac {dN}{dE}}\,\delta E=D(E)\,\delta E}$

${\displaystyle D(E)={\frac {dN}{dE}}}$

${\displaystyle D(E)=\sum _{\omega \in \varOmega }\delta (E-{\mathcal {E}}(\omega ))}$

${\displaystyle W(E,\delta E)=\int _{E-\delta E}^{E}D(E')\,dE'}$

${\displaystyle N(E)=\int _{-\infty }^{E}D(E')\,dE'}$

${\displaystyle W(E,\delta E)=\int _{-\infty }^{\infty }\{\eta (E-E')-\eta (E-\delta E-E')\}\,D(E')\,dE'}$

${\displaystyle W^{\text{g}}(E,\delta E)=\int _{-\infty }^{\infty }e^{-(E-E')^{2}/(\delta E)^{2}}\,D(E')\,dE'}$

${\displaystyle \sum _{\omega \in \Omega }\to {\frac {1}{h^{f}}}\int d^{f}p\,d^{f}q}$

${\displaystyle D(E)={\frac {1}{h^{f}}}\int \delta (E-{\mathcal {H}}(p,q))\,d^{f}p\,d^{f}q}$

${\displaystyle n=N(E_{\text{F}})=\int _{-\infty }^{E_{\text{F}}}D_{1}(E)\,dE}$

${\displaystyle A=\int _{-\infty }^{E_{\text{F}}}{\mathcal {A}}(E)D_{1}(E)\,dE}$

${\displaystyle A=\int _{-\infty }^{\infty }{\mathcal {A}}(E)\eta (E_{\text{F}}-E)D_{1}(E)\,dE}$

${\displaystyle A=\int _{-\infty }^{\infty }{\mathcal {A}}(E)f(E)D_{1}(E)\,dE}$