The Thomas–Fermi approach to galaxy structure determines self-consistently and non-linearly the gravitational potential of the fermionic warm dark matter (WDM) particles given their quantum distribution function f(E). This semiclassical framework accounts for the quantum nature and high number of DM particles, properly describing gravitational bounded and quantum macroscopic systems as neutron stars, white dwarfs and WDM galaxies. We express the main galaxy magnitudes as the halo radius rh\documentclass[12pt]{minimal}
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\begin{document}$$ r_h $$\end{document}, mass Mh\documentclass[12pt]{minimal}
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\begin{document}$$ M_h $$\end{document}, velocity dispersion and phase space density in terms of the surface density which is important to confront to observations. From these expressions we derive the general equation of state for galaxies, i.e., the relation between pressure and density, and provide its analytic expression. Two regimes clearly show up: (1) Large diluted galaxies for Mh≳2.3×106M⊙\documentclass[12pt]{minimal}
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\begin{document}$$ M_h \gtrsim 2.3 \times 10^6 \; M_\odot $$\end{document} and effective temperatures T0>0.017\documentclass[12pt]{minimal}
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\begin{document}$$ T_0 > 0.017 $$\end{document} K described by the classical self-gravitating WDM Boltzman gas with a space-dependent perfect gas equation of state, and (2) Compact dwarf galaxies for 1.6×106M⊙≳Mh≳Mh,min≃3.10×104(2keV/m)165M⊙,T0<0.011\documentclass[12pt]{minimal}
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\begin{document}$$ 1.6 \times 10^6 \; M_\odot \gtrsim M_h \gtrsim M_{h,\mathrm{min}} \simeq 3.10 \times 10^4 \; (2 \, {\mathrm{keV}}/m)^{\! \! \frac{16}{5}} \; M_\odot , \; T_0 < 0.011 $$\end{document} K described by the quantum fermionic WDM regime with a steeper equation of state close to the degenerate state. In particular, the T0=0\documentclass[12pt]{minimal}
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\begin{document}$$ T_0 = 0 $$\end{document} degenerate or extreme quantum limit yields the most compact and smallest galaxy. In the diluted regime, the halo radius rh\documentclass[12pt]{minimal}
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\begin{document}$$ r_h $$\end{document}, the squared velocity v2(rh)\documentclass[12pt]{minimal}
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\begin{document}$$ v^2(r_h) $$\end{document} and the temperature T0\documentclass[12pt]{minimal}
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\begin{document}$$ T_0 $$\end{document} turn to exhibit square-root of Mh\documentclass[12pt]{minimal}
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\begin{document}$$ M_h $$\end{document}scaling laws. The normalized density profiles ρ(r)/ρ(0)\documentclass[12pt]{minimal}
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\begin{document}$$ \rho (r)/\rho (0) $$\end{document} and the normalized velocity profiles v2(r)/v2(0)\documentclass[12pt]{minimal}
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\begin{document}$$ v^2(r)/ v^2(0) $$\end{document} are universal functions of r/rh\documentclass[12pt]{minimal}
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\begin{document}$$ r/r_h $$\end{document} reflecting the WDM perfect gas behavior in this regime. These theoretical results contrasted to robust and independent sets of galaxy data remarkably reproduce the observations. For the small galaxies, 106≳Mh≥Mh,min\documentclass[12pt]{minimal}
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\begin{document}$$ 10^6 \gtrsim M_h \ge M_{h,\mathrm{min}} $$\end{document}, the equation of state is galaxy mass dependent and the density and velocity profiles are not anymore universal, accounting to the quantum physics of the self-gravitating WDM fermions in the compact regime (near, but not at, the degenerate state). It would be extremely interesting to dispose of dwarf galaxy observations which could check these quantum effects.
