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

The effects of an attractive logarithmic potential u(r)=u0ln⁡(r/r0) (u0>0 being the trap strength and r0 an arbitrary length scale) on a gas of N non interacting particles (Bosons or Fermions), in a box of volume VD, are studied in D = 2, 3 dimensions. The unconventional behaviour of the gas challenges the current notions of thermodynamic limit and size independence. When VD and N diverge, with finite density N/VD<∞ and finite u0, the gas collapses in the ground state, independently from the bosonic/fermionic nature of the particles, at any temperature. If, instead, N/VD→0, there exists a critical temperature Tc, such that the gas remains in the ground state at any T<Tc, 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 u0 must vanish with VD→∞, according to a complicated exponential relationship, as a consequence of the exponentially increasing density of states, specific of the logarithmic trap.