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

The effects of an attractive logarithmic potential $u_0\ln (r\mathrm{/}{r}_0)$ 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 $N\mathrm{/}{V}_D<\mathrm{\infty }$ 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, $N\mathrm{/}{V}_D\to 0$, 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 $V_D\to \mathrm{\infty }$, according to a complicated exponential relationship, as a consequence of the exponentially increasing density of states, specific of the logarithmic trap.