Hypercontractivity estimate

$${\int }_{{ℝ}^n}{\left|u(x)\right|}^2\ln \left|u(x)\right|d{\mu }_n(x)\le \frac{1}{2}{\Vert \nabla u\Vert }_2^2+{\Vert u\Vert }_2^2\ln {\Vert u\Vert }_2$$

Releativistic interaction free gas

$$Z\; =\frac{{Z_1}^{\mathrm{N}}}{N!}$$ This implies for the free energy $$F\; =\; -k\; T\; ln(Z)$$