  The effects of an attractive logarithmic potential $u_0\ln(r/r_0)$ on a gas
of $N$ non interacting particles (Bosons or Fermions), in a box of volume
$V_D$, are studied in $D=2,3$ dimensions. The unconventional behavior of the
gas challenges the current notions of thermodynamic limit and size
independence. 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 critical
temperature $T_c$, such that the gas remains in the ground state at any
$T<T_c$, and "evaporates" above, in a non-equilibrium state of borderless
diffusion. 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, as
a consequence of the exponentially increasing density of states, specific of
the logarithmic trap.
