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

ABSTRACT

The effects of an attractive logarithmic potential $u\left(r\right)=u_0\ln \left(r/r_0\right)$ ($u_0>0$ being the trap strength and $r_0$ an arbitrary length scale) 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 behaviour of the gas challenges the current notions of thermodynamic limit and size independence. When $V_D$ and N diverge, with finite density $N/V_D<\mathrm{\infty }$ and finite $u_0$, the gas collapses in the ground state, independently from the bosonic/fermionic nature of the particles, at any temperature. If, instead, $N/V_D\to 0$, 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.

KEYWORDS:

• Logarithmic potential
• Bose–Einstein condensation
• Fermi level

Acknowledgments

The author is grateful to dr E. Aghion for crucial bibliographic suggestions and to one of the Philosophical Magazine’s referees for the pertinent comments and for the careful check of the calculations.

Disclosure statement

The author reports there are no competing interests to declare.

Notes

1 It is intended that $w_{\pm }>0$ and $\lambda >0$ are suitable parameters.

2 In particular, the second line in Equation (11b(11b) $\begin{array}{rl}g\left(ϵ\right)& =\frac{\hat{V}{\mathrm{e}}^{3A}}{u_{0}2\sqrt{27\pi }}\times \\ & \times \{\begin{array}{cc}{\mathrm{e}}^{3(ϵ-{ϵ}_c)/u_0}& (ϵ\le {ϵ}_c)\\ {\mathrm{e}}^{3(ϵ-{ϵ}_c)/u_0}\mathrm{\Psi }\left(\frac{ϵ-{ϵ}_c}{u_0}\right)+\frac{2}{\sqrt{\pi }}\sqrt{3(ϵ-{ϵ}_c)/u_0}& (ϵ\ge {ϵ}_c)\end{array}(D=3)\end{array}$(11b) ) vanishes because $\mathrm{E}\mathrm{r}\mathrm{f}\mathrm{c}\left(x\right)\to {\mathrm{e}}^{-x^2}/\left(x\sqrt{\pi }\right)$ for $x\to \mathrm{\infty }$.

3 This means that there exists $M\left(T\right)<\mathrm{\infty }$, such that ${\mathcal{F}}_{\pm }(\hat{V},T)<M\left(T\right)$ for each $\hat{V}>0$, $T>T_D$. The factor $(1\mp 1)/2$ in Equation (23a(23a) ${\mathrm{e}}^{-\text{β}{\mu }_{\pm }}=\frac{1}{N(1\pm {\mathrm{e}}^{\text{β}{\mu }_{\pm }})}-\frac{\mathcal{G}T}{T_{D}N}\ln (1-{\mathrm{e}}^{\text{β}{\mu }_{-}})\frac{1\mp 1}{2}+$(23a) ) means that the associated logarithmic singularity applies to Bosons only. It is intended that $\circ \left(x\right)$ is a quantity proportional to x, to the leading order.

4 A typical example is the harmonic trap with $\lambda =2$. Notice that on setting $w_{+}=u_0/R^{\lambda }$, the rigid-wall box of radius R is realised by $\lambda \to \mathrm{\infty }$.

5 A typical example is the Coulombic potential $\lambda =1$.