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The effects of an attractive logarithmic potential $u_0\ln(r/r_0)$ on a gasof $N$ non interacting particles (Bosons or Fermions), in a box of volume$V_D$, are studied in $D=2,3$ dimensions. The unconventional behavior of thegas challenges the current notions of thermodynamic limit and sizeindependence. When $V_D$ and $N$ diverge, with finite density $N/V_D<\infty$and finite trap strength $u_0>0$, the gas collapses in the ground state,independently from the bosonic/fermionic nature of the particles, at \emph{any}temperature. If, instead, $N/V_D\rightarrow0$, there exists a criticaltemperature $T_c$, such that the gas remains in the ground state at any$T<T_c$, and "evaporates" above, in a non-equilibrium state of borderlessdiffusion. For the gas to exhibit a conventional Bose-Einstein condensation(BEC) or a finite Fermi level, the strength $u_0$ must vanish with$V_D\rightarrow\infty$, according to a complicated exponential relationship, asa consequence of the exponentially increasing density of states, specific ofthe logarithmic trap.

Gas in external fields: the weird case of the logarithmic trap