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The Thomas-Fermi approach to galaxy structure determines selfconsistently andnonlinearly the gravitational potential of the fermionic WDM particles giventheir quantum distribution function f(E). Galaxy magnitudes as the halo radiusr_h, mass M_h, velocity dispersion and phase space density are obtained. Wederive the general equation of state for galaxies (relation between thepressure and the density), and provide an analytic expression. This clearlyexhibits two regimes: (i) Large diluted galaxies for M_h > 2.3 10^6 Msuncorresponding to temperatures T_0 > 0.017 K, described by the classical selfgravitating WDM Boltzman regime and (ii) Compact dwarf galaxies for 1.6 10^6Msun > M_h>M_{h,min}=30000 (2keV/m)^{16/5} Msun, T_0<0.011 K described by thequantum fermionic WDM regime. The T_0=0 degenerate quantum limit predicts themost compact and smallest galaxy (minimal radius and mass M_{h,min}). Allmagnitudes in the diluted regime exhibit square root of M_h scaling laws andare universal functions of r/r_h when normalized to their values at the originor at r_h. We find that universality in galaxies (for M_h > 10^6 Msun) reflectsthe WDM perfect gas behaviour. These theoretical results contrasted to robustand independent sets of galaxy data remarkably reproduce the observations. Forthe small galaxies, 10^6>M_h>M_{h,min} corresponding to effective temperaturesT_0 < 0.017 K, the equation of state is galaxy dependent and the profiles areno more universal. These non-universal properties in small galaxies account tothe quantum physics of the WDM fermions in the compact regime. Our results areindependent of any WDM particle physics model, they only follow from thegravitational interaction of the WDM particles and their fermionic quantumnature.

Equation of state, universal profiles, scaling and macroscopic quantum effects in Warm Dark Matter galaxies