functional method and statistical mechanics

a.k.a thermodynamics.

\(\int_{\Omega} \frac{p^2}{2m} \, dP(\omega) = \left\langle \frac{p^2}{2m} \right\rangle = \int_{\Omega} \frac{p^2}{2m} \, \frac{e^{-\frac{E}{kT}}}{\int e^{-\frac{E}{kT}} \, d\mu} \, d\mu = 3 \times \frac{1}{2} kT\)

\(\quad \text{where } d\mu \text{ is dimensionless, and}\)

\(d\mu = \frac{d^3x \, d^3p}{h^3}\)

\(d^3x : \mathrm{m}^3\)

\(d^3p : (\mathrm{kg \cdot m/s})^3 = \mathrm{kg^3 \, m^3 \, s^{-3}}\)

\(h : \mathrm{kg \cdot m^2 / s}\)

\(h^3 : \mathrm{kg^3 \, m^6 \, s^{-3}}\)

\(\frac{\mathrm{m}^3 \cdot \mathrm{kg^3 \, m^3 \, s^{-3}}}{\mathrm{kg^3 \, m^6 \, s^{-3}}} = 1\)