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

TL;DR

This paper investigates how a logarithmic external potential affects the thermodynamics of non-interacting quantum gases, revealing unconventional behaviors like collapse and non-standard phase transitions due to the trap's unique density of states.

Contribution

It introduces a detailed analysis of quantum gases in a logarithmic trap, showing how the potential alters thermodynamic limits and phase transition phenomena.

Findings

Gas collapses in the ground state at any temperature when volume and particle number diverge.

Existence of a critical temperature for evaporation when particle density tends to zero.

Conventional BEC or Fermi levels require the trap strength to vanish with volume.

Abstract

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.