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

Loris Ferrari
Department of Physics and Astronomy (DIFA) of the University of Bologna
via Irnerio, 46 - 40126, Bologna,Italy

###### 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 _any_ temperature. If, instead, $N/V_{D}\rightarrow 0$, 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 $V_{D}\rightarrow\infty$, according to a complicated exponential
relationship, as a consequence of the exponentially increasing density of
states, specific of the logarithmic trap.

Key words: Logarithmic potential, Bose-Einstein Condensation, Fermi level.

*Corresponding author: e-mail<EMAIL_ADDRESS>
##  1 Introduction

In recent years, huge technological progresses made it possible to confine
neutral particles in a small region of space by magnetic traps [1, 2, 3, 4,
5], which results, in the most current cases, in a harmonic potential [6].
With the aid of laser cooling tecniques [7], the use of magnetic traps has led
to unprecedental results, like the direct observation of BEC [8, 9, 10] and
small gravimetric effects [11, 12, 13, 14]. In the mainstream of those
results, the theoretical literature on the harmonic traps and on the
gravitational field, is well represented [15, 16].

Due to the difficulty of realizing LT’s for neutral particles, the gas
thermodynamics in a logarithmic trap (LT) has less fields of concrete
application. The interest in LT’s dates back to ref [17], which studies the
orbits of charged particles in the logarithmic radial field produced by a
charged long wire, in the plane perpendicular to the wire itself. A
logarithmic baryon-baryon interaction was later suggested as a possible
interpretation of the composite hadrons’ mass formula [18, 19, 20]. Studies on
the gravitational field close to the Schwarzschild radius of black holes [21]
provide an example of logarithmic potential in a cosmological context. Besides
any possible concrete application, the present work primarly aims to stress
some surprising aspects of LT’s, quite ignored by the current literature, to
the author’s knowledge. In particular, it will be shown that LT’s challenge
some standard notions of Thermodynamics, like the thermodynamic limit (TL),
the size independence and the semi-classical limit at low gas density and/or
high temperatures.

The Hamiltonian:

$H=\frac{\mathbf{p}^{2}}{2m}+\begin{cases}u_{0}\ln(r/r_{0})&\quad(r<R)\\\ \\\
\infty&\quad(r>R)\end{cases}\quad,$ (1)

with $u_{0}>0$ and $r=|\mathbf{r}|$, describes an attractive TL, in a box of
radius $R$ and volume $V_{D}=\Omega_{D}R^{D}/D$ ($\Omega_{D}$ being the solid
angle in D-dimensions). In Section 2 the density of states (DOS) resulting
from the Hamiltonian (1) is calculated in 2 and 3 dimensions, showing the
separation between an increasing exponential shape
$\mathrm{e}^{D\epsilon/u_{0}}$, below a characteristic energy $\epsilon_{c}$,
and a free-particle shape $\epsilon^{D/2-1}$ above (Fig. 1). In Section 3, we
adopt the current definition of thermodynamic limit (TL):

$N,\>V_{D}\rightarrow\infty\quad;\quad\rho:=N/V_{D}<\infty\>,$

and study the possible relationships between $N$, $V_{D}$ and the LT’s
strength $u_{0}$. In particular, if $V_{D}$ diverges as follows:

(I)

$V_{D}\propto\mathrm{e}^{\epsilon_{c}D/u_{0}}u_{0}^{-D/2}\>;\quad
u_{0}\rightarrow 0$,

$\epsilon_{c}$ being a fixed parameter, independent from $u_{0}$, then
$\epsilon_{c}$ is shown to act like a gap between the ground state and the
free-particle DOS (Subsection 3.1). In this case, the chemical potential $\mu$
behaves in the current way, showing BEC at a finite temperature, for Bosons,
and a finite Fermi level, for Fermions.

If, instead:

(II)

$V_{D}\rightarrow\infty,\>u_{0}>0$,

one gets a really weird result (Section 3.2): the chemical potential of the
bosonic gas tends to the ground state energy $E_{0}$, and the Fermi level of a
fermionic gas tends to diverge, for all $T<\infty$. This means that the gas is
frozen in the ground state at _any_ temperature, independently from the
bosonic or fermionic nature of the particles. Since this forbids any thermal
activity of the log-trapped gas, one must assume the presence of a ‘normal’
LT-insensitive gas, which provides the thermal bath at $T>0$. If TL is
generalized to include the case of $N$ diverging less rapidly than $V_{D}$
(Subsection 3.3), a critical temperature $T_{c}$ can be defined, such that the
gas (bosonic or fermionic) is still frozen in the ground state for all $T\leq
T_{c}$, but expands itself for $T>T_{c}$, in a non equilibrium state of
borderless diffusion.

In short, the normal behavior of the gas follows from a fairly complicated
condition, like (I), involving an exponential divergence of the box size, with
the vanishing trap’s strength. Instead, what looks the most natural layout
(II) - a fixed trap in a divergingly large box - causes a quite unconventional
behavior, whose image can be clearly seen in Fig. 2 (Section 4), which shows
the _effective_ volume [5], i.e. the mean volume occupied by a single
particle, at a given temperature, _modulo_ the degeneracy effects and the
exclusion principle. EV is shown to increase abruptly from sizes comparable to
the volume occupied by the single particle’s ground state, to the whole box
volume, above the critical temperature $T_{D}=u_{0}/D$. It is clearly seen
that this is due to the competition between the exponential DOS
$\propto\mathrm{e}^{D\epsilon/u_{0}}$ and the Boltzmann factor
$\mathrm{e}^{-\beta\epsilon}$. In Section 5, we show that the physical origin
of the weird results described above is the special nature of the logarithmic
potential $u(r)=u_{0}\ln(r/r_{0})$, as a border case between a true trap
($u_{+}(r)=w_{0}(r/r_{0})^{\lambda}$) and a potential hole
($u_{-}(r)=-w_{0}(r/r_{0})^{-\lambda}$).

For the sake of brevity, the calculations are included in the Supplemental
Materials File, and quoted as references in the text.

##  2 The density of states (DOS)

###  2.1 The continuous limit (CL)

Let $\vec{\alpha}$ be a discrete set of parameters characterizing the excited
quantum levels $\epsilon_{\vec{\alpha}}$ and degeneracies $g_{\vec{\alpha}}$
of a generic spin-independent Hamiltonian $h(\mathbf{p},\>\mathbf{r})$. The
exact density of states (DOS) in the energy $\epsilon$, measured with respect
to the ground level $\epsilon_{0}=0$, can be written as:

$G_{ex}(\epsilon)=g_{0}\delta(\epsilon)+\overbrace{\sum_{\vec{\alpha}\neq
0}g_{\vec{\alpha}}\delta(\epsilon-\epsilon_{\vec{\alpha}})}^{g_{ex}(\epsilon)}$
(2)

where $\delta(\cdot)$ is the Dirac delta function. The continuous limit (CL)
aims to replace the excited-states DOS $g_{ex}(E)$ with a Riemann integrable
function:

$g(\epsilon)=\begin{cases}\frac{1}{\mathrm{h}^{D}}\frac{\mathrm{d}}{\mathrm{d}\epsilon}\int_{0<h(\mathbf{p},\>\mathbf{r})<\epsilon}\mathrm{d}\mathbf{p}\mathrm{d}\mathbf{r}&\quad(\epsilon>0)\\\
&\quad\quad\quad\quad\quad,\\\ 0&\quad(\epsilon<0)\end{cases}$ (3a) such that:

$G(\epsilon)=(2s+1)\left[g_{0}\delta(\epsilon)+g(\epsilon)\right]\>,$ (3b)

$s$ being the particle’s spin. Including the ground level as a separate term
$g_{0}\delta(\epsilon)$ is a convenient strategy, if the ground state itself
can be populated by an arbitrary large number of particles, as is the case of
a bosonic gas. When the energy spacings $\Delta_{\vec{\alpha}}$ between
nearest neighbor levels tend to vanish in the thermodynamic limit (TL),
equations (3) are rigorous, and one speaks about a rigorous continuous limit
(RCL). If the $\Delta_{\vec{\alpha}}$’s do not vanish in the TL, and CL
follows from a high-temperature approximation
$\beta\Delta\epsilon_{\vec{\alpha}}<<1$ for each $\vec{\alpha}$, we speak
about an _approximate_ continuous limit (ACL), in which the
$\Delta\epsilon_{\vec{\alpha}}$’s can be treated as ‘infinitesimal’
increments, with respect to the thermal energy $\kappa T$. In view of the
possible drawbacks inherent to CL, stressed in refs [22, 23], especially in 1
and 2D, and in view of applications to finite-sized gases, it makes sense
studying possible strategies to improve ACL, for instance, by applying the
continuous approximation only to the part of the spectrum whose energy
increments are actually smaller than $\kappa T$. If DOS is an increasing
function of $\epsilon$, this means re-writing Eq. (3b) as:

$G(\epsilon)=(2s+1)\left[g_{0}\delta(\epsilon)+\sum_{\vec{\alpha}\neq
0}^{\vec{\alpha}_{T}}g_{\vec{\alpha}}\delta(\epsilon-\epsilon_{\vec{\alpha}})+g_{T}(\epsilon)\right]\>,$
(4)

where $\epsilon_{\vec{\alpha}_{T}}$ is the maximum eigenvalue such that
$\beta\Delta_{\vec{\alpha}_{T}}>1$. The Riemann-integrable DOS

$g_{T}(\epsilon)=\begin{cases}\frac{1}{\mathrm{h}^{D}}\frac{\mathrm{d}}{\mathrm{d}\epsilon}\int_{0<h(p,r)-\epsilon_{\vec{\alpha}_{T}}<\epsilon}\mathrm{d}\mathbf{p}\mathrm{d}\mathbf{r}&\quad(\epsilon>0)\\\
&\\\ 0&\quad(\epsilon<0)\end{cases}\>,$ (5)

obtained by replacing $h(p,r)$ with $h(p,r)-\epsilon_{\vec{\alpha}_{T}}$ in
Eq. (3a), becomes thereby temperature-dependent. For the calculation of
$\epsilon_{\vec{\alpha}_{T}}$ in the continuous approximation, we refer to the
number of states

$N(\epsilon)=g_{0}+\int_{0}^{\epsilon}\mathrm{d}\epsilon^{\prime}\>g(\epsilon^{\prime})\>$

with energy smaller than $\epsilon$, and estimate the energy splitting between
consecutive levels as the energy difference

$\Delta\epsilon=\frac{1}{g(\epsilon)}\>,$

corresponding to the increment of a single state. Then the equation

$\kappa T\approx\frac{1}{g(\epsilon_{\vec{\alpha}_{T}})}$ (6)

provides an estimate for $\epsilon_{\vec{\alpha}_{T}}$. As we shall see in
what follows, improving ACL does not have a relevant influence on the results
obtained for LT.

###  2.2 DOS in a finite-sized LT

Let $\phi_{0}(r)$ be a s-wave, representing the ground state of the LT-
Hamiltonian Eq. (1). We assume that the localization length $\ell_{0}$ of
$\phi_{0}(r)$ is small, compared to $R$, i.e., that the rigid-wall confinement
has negligible effects on $\phi_{0}(r)$. Then the Schrödinger equation
$H\phi_{0}(r)=E_{0}\phi_{0}(r)$ can ignore the lower line of Eq. (1), and can
be written in the dimensionless form:

$[\underbrace{-\nabla_{\mathbf{x}}+\ln(x)}_{h_{red}}]f(x)=[\overbrace{\frac{E_{0}}{u_{o}}+\ln(x_{0})}^{A}]f(x)\>.$
(7)

by setting $\mathbf{x}:=\mathbf{r}\sqrt{2mu_{0}}/\hbar$, $\phi_{0}(r):=f(x)$
and $x_{0}:=r_{0}\sqrt{2mu_{0}}/\hbar$. The LT’s ground level is
$\epsilon_{m}=u_{0}A$. A simple variational calculation [24] yields: :

$A=\begin{cases}0.576\\\ \\\
1.076\end{cases}\quad;\quad\ell_{0}=\frac{\hbar}{\sqrt{4u_{0}m}}\times\begin{cases}1&\quad(D=2)\\\
&\quad\quad\quad\quad\quad\quad.\\\ 3&\quad(D=3)\end{cases}$ (8a) For the next
developments, it is useful to introduce the following definitions:

$r_{c}:=\frac{\hbar\>\mathrm{e}^{A}}{\sqrt{2mu_{0}}}\quad;\quad
v_{c}:=\frac{\Omega_{D}r_{c}^{D}}{D}\quad;\quad\widehat{V}:=\frac{V_{D}}{v_{c}}\quad;\quad\epsilon_{c}:=\frac{u_{0}}{D}\ln\widehat{V}\>.$
(8b)

Equation (7) shows that the length scale $r_{0}$ in Eq. (1) simply shifts the
energy origin by the quantity $u_{0}\ln(x_{0})$. Hence, it is convenient to
define the ground state energy as $E_{0}=\epsilon_{m}+u_{0}\ln x_{0}$, and the
Hamiltonian as:

$h(p,\>r):=H(p,\>r)-E_{0}=\frac{p^{2}}{2m}+\begin{cases}u_{0}\ln(r/r_{c})&\quad(r<R)\\\
\\\ \infty&\quad(r>R)\end{cases}\quad\quad\>,$ (9)

with ground level $\epsilon_{0}=0$. From Eq. (8a), the condition $\ell_{0}<<R$
yields:

$R\sqrt{u_{0}}>>\frac{\hbar}{\sqrt{4m}}\>.$ (10)

For brevity reasons, we take spinless Bosons and Fermions ($s=0$).

The Riemann-integrable part $g(\epsilon)$ of the DOS (Eq. (3a)) can be
calculated with the aid of ref [25], from Eq. (9), under the condition (10):

$g(\epsilon)=\frac{\widehat{V}\mathrm{e}^{2A}}{4u_{0}}\times\begin{cases}\mathrm{e}^{2(\epsilon-\epsilon_{c})/u_{0}}&\>\>\>\>(\epsilon\leq\epsilon_{c})\\\
\\\ 1&\>\>\>\>(\epsilon\geq\epsilon_{c})\end{cases}\quad\quad(D=2)$ (11a)
$\displaystyle g(\epsilon)$
$\displaystyle=\frac{\widehat{V}\mathrm{e}^{3A}}{u_{0}2\sqrt{27\pi}}\times$
$\displaystyle\times\begin{cases}\mathrm{e}^{3(\epsilon-\epsilon_{c})/u_{0}}&\>\>\>\>(\epsilon\leq\epsilon_{c})\\\
\\\
\mathrm{e}^{3(\epsilon-\epsilon_{c})/u_{0}}\mathrm{Erfc}(\sqrt{3(\epsilon-\epsilon_{c})/u_{0}})+\\\
\\\
+\frac{2}{\sqrt{\pi}}\sqrt{3(\epsilon-\epsilon_{c})/u_{0}}&\>\>\>\>(\epsilon\geq\epsilon_{c})\end{cases}\quad\quad(D=3)\>,$
(11b)

where use has been made of the definitions (8b). In Figure 1, the expressions
(11) are plotted as a function of $\epsilon/\epsilon_{c}$. It is seen that
above $\epsilon_{c}$, $g(\epsilon)$ tends to (in 3D), or even coincides with
(in 2D) the free-particle DOS, and decreases exponentially below. Hence,
$\epsilon_{c}$ determines the energy range above which the kinetic energy’s
loosing effect prevails over the attractive effect of LT.

Figure 1: Riemann-integrable DOS. Plot of $g(\epsilon)$ in 2 and 3 dimensions,
for different values of $\widehat{V}$, as a function of
$\epsilon/\epsilon_{c}$. In $D=2$ $g(\epsilon)$ coincides exactly with the
constant free-particle DOS, for $\epsilon>\epsilon_{c}$. The dashed curves in
$D=3$ show the approach to the free-particle shape $\epsilon^{1/2}$, starting
from $\epsilon_{c}$, for $\widehat{V}\rightarrow\infty$.

Treating _all_ the energy splittings $\Delta\epsilon_{\vec{\alpha}}$ as
‘infinitesimal’ increments in a LT, means taking $\kappa T>>u_{0}$, because
the levels are all proportional to $u_{0}$. While in case (I) $u_{0}$ vanishes
and RCL does apply, in case (II) $u_{0}$ is fixed and ACL is to be taken under
consideration. Since we are interested in the low energy states at low
temperature, we can refer to the case $\epsilon<\epsilon_{c}$, in Eq.s (11),
which involves a pure exponential DOS. A straightforward calculation then
yields, from Eq. (6):

$\epsilon_{\vec{\alpha}_{T}}\approx\frac{u_{0}}{D}\ln\left(\frac{u_{0}}{\lambda_{D}\kappa
T}\right)\quad;\quad\lambda_{D}=\mathrm{e}^{DA}\times\begin{cases}\frac{1}{4}&\>\>\>\>(D=2)\\\
\\\ \frac{1}{2\sqrt{27\pi}}&\>\>\>\>(D=3)\end{cases}\>,$ (12)

for the energy below which the discreteness of the spectrum must be recovered.
It is seen that $\epsilon_{\vec{\alpha}_{T}}$ diverges logarithmically with
$T^{-1}$ for vanishing temperature, and becomes _negative_ when $T$ exceeds
values comparable to

$T_{D}:=\frac{u_{0}}{D\kappa}\>$ (13)

(a special temperature that will come into play soon, in what follows). Since
the improved ACL applies to positive eigenvalues $\epsilon_{\vec{\alpha}}$
larger than $\epsilon_{\vec{\alpha}_{T}}$, equation (12) shows that the
improvement is practically uninfluent, for temperatures $T\gtrsim T_{D}$ and
leads to smoothly varying modifications for $T<T_{D}$.

###  2.3 The free-particle limit

The free-particle limit obviously implies the limit $u_{0}\rightarrow 0$. If
the trap is contained in a box, however, the box size must diverge in the same
limit, since the localization length of the trap’s ground state (and, _a
fortiori_ , that of the excited states) diverges with vanishing trap’s
strength (Eq. (10)). In addition, the special case of LT involves the border
energy $\epsilon_{c}(u_{0})$, which must vanish in turn. Hence the conditions
for the free-particle limit read:

$\lim_{u_{0}\rightarrow 0}R\sqrt{u_{0}}>0\text{
strictly}\quad,\quad\lim_{u_{0}\rightarrow
0}\epsilon_{c}=\lim_{u_{0}\rightarrow
0}u_{0}\ln\left(\frac{R\sqrt{2mu_{0}}}{\hbar\mathrm{e}^{A}}\right)=0\>.$ (14a)
In this case, one is left with the lowest lines of Eq.s (11), which yield:

$\lim_{u_{0}\rightarrow
0}g(\epsilon)=\frac{\Omega_{D}V_{D}}{2}\left(\frac{2m}{\mathrm{h}^{2}}\right)^{D/2}\epsilon^{D/2-1}=g_{free}(\epsilon)\>,$
(14b)

for $\epsilon>0$ (strictly), since $\widehat{V}/u_{0}^{D/2}$ is proportional
to $V_{D}$.

##  3 DOS and chemical potential in the thermodynamic limit (TL)

###  3.1 The case (I)

In contrast to the free-particle limit, in case (I) the border energy
$\epsilon_{c}$ is kept fixed and non zero, for $u_{0}\rightarrow 0$. In this
case, on dividing $g(\epsilon)$ by $V_{D}\propto\widehat{V}/u_{0}^{D/2}$, it
is not difficult to see that the first lines of $g(\epsilon)/V_{D}$ in Eq.s
(11) vanish for $u_{0}\rightarrow 0$.111In particular, the second line in Eq.
(11) vanishes because
$\mathrm{Erfc}(x)\rightarrow\mathrm{e}^{-x^{2}}/(x\sqrt{\pi})$ for
$x\rightarrow\infty$. On restoring $V_{D}$, the lowest lines lead to the
expression of the free-particle DOS for $\epsilon>\epsilon_{c}$, whence:

$\text{(I)}\quad\Rightarrow\quad g(\epsilon)\rightarrow
g_{I}(\epsilon)=\begin{cases}0&\quad\quad(\epsilon<\epsilon_{c})\\\ \\\
g_{free}(\epsilon-\epsilon_{c})&\quad\quad(\epsilon\geq\epsilon_{c})\>,\end{cases}$
(15)

and $G_{I}(\epsilon)=\delta(\epsilon)+g_{I}(\epsilon)$ is the total DOS (Eq.
(3b)). Figure 1 clearly confirms the content of Eq. (15), i.e. that
$g_{I}(\epsilon)$ is nothing but the limit of Eq.s (11) for
$\widehat{V}\rightarrow\infty$, at fixed $\epsilon_{c}$. Equation (15) shows
that LT creates a gap of width $\epsilon_{c}$, between the ground level
$\epsilon_{0}=0$ and the Riemann integrable DOS, which is simply the free
particle DOS, on redefining the energy as
$\epsilon^{\prime}=\epsilon-\epsilon_{c}$. Hence LT provides a rigorous
realization of an approximate model, sometimes used in elementary quantum
mechanics, i.e. a potential well, carrying a _single_ bound state at the
energy $\epsilon^{\prime}=-\epsilon_{c}$, whose effect on the unbound states
($\epsilon^{\prime}>0$) is assumed negligible. We shall return on this point
in Section 5.

Given a generic DOS $G(\epsilon)$, the equation for the chemical potential
$\mu$ is:

$1=\int_{0}^{\infty}\mathrm{d}\epsilon\frac{G(\epsilon)}{N(\mathrm{e}^{\beta(\epsilon-\mu)}\pm
1)}\>,$ (16)

where the double sign refers to Fermions ($+$) and Bosons ($-$). From Eq.
(15), equation (16) becomes, with the previously outlined substitution
$\epsilon-\epsilon_{c}\rightarrow\epsilon$:

$1=\overbrace{\frac{1}{N(\mathrm{e}^{-\beta\mu}\pm
1)}}^{n_{0}}+\underbrace{\int_{0}^{\infty}\mathrm{d}\epsilon\frac{\widehat{g}_{free}(\epsilon)}{\mathrm{e}^{\beta(\epsilon+\epsilon_{c}-\mu)}\pm
1}}_{n_{\pm}(T,\>\mu)}\quad;\quad\widehat{g}_{free}(\epsilon):=g_{free}(\epsilon)/N\>,$
(17)

where $n_{0}$ is the fractional population of the ground state, and
$\widehat{g}_{free}(\epsilon)$ is a size-independent DOS, proportional to
$\rho=V_{D}/N$ (Eq. (14b)). Of course, $n_{\pm}(T,\>\mu)$ is the fractional
population of the excited states. In the fermionic case ($+$), the quantity of
interest is the Fermi level $\epsilon_{F}=\lim_{T\rightarrow 0}\mu$, which can
be easily deduced from Eq. (17), since the occupation function
$[\mathrm{e}^{\beta(\epsilon-\epsilon_{c}-\mu)}+1]^{-1}$ becomes a step
function for $\beta\rightarrow\infty$ ($T\rightarrow 0$), jumping from 1 to 0
at $\epsilon=\epsilon_{F}-\epsilon_{c}$. Furthermore, in the TL
($N\rightarrow\infty$) $n_{0}$ vanishes for any value of $\mu$, whence the
equation for $\epsilon_{F}$ yields:

$1=\int_{0}^{\epsilon_{F}-\epsilon_{c}}\widehat{g}_{free}(\epsilon)\mathrm{d}\epsilon\quad\Rightarrow\quad\epsilon_{F}=\epsilon_{c}+\overbrace{\frac{\mathrm{h}^{2}}{2m}\left(\frac{2\rho
D}{\Omega_{D}}\right)^{2/D}}^{\epsilon_{F}^{free}}\>,$ (18)

which shows that $\epsilon_{F}$, in the limit (I), is just the Fermi level
$\epsilon_{F}^{free}$ of the free gas (Eq.(14b)), plus the border energy
$\epsilon_{c}$.

In the bosonic case ($-$), $\mu>0$ is forbidden by the condition $n_{0}\geq
0$. Until $\mu$ is strictly negative, $n_{0}$ vanishes in the TL. However, the
condition $n_{-}(T,\>0)<\infty$ for any $T>0$ is necessary and sufficient for
the existence of a critical temperature $T_{B}$ such that
$n_{-}(T_{B},\>0)=1$, which leads to (BEC) [26, 27]. From Eq. (17) it is
easily seen that $\epsilon_{c}>0$ forbids the vanishing of the denominator in
the integrand. Hence $n_{-}(T,\>\mu)$ is finite for any $T>0,\>\mu\geq 0$, and
BEC does actually occur at the temperature $T_{B}=1/(\kappa\beta_{B})$,
determined by the equation $1=n_{-}(T_{B},\>0)$, which yields:

$1=\int_{0}^{\infty}\mathrm{d}\epsilon\frac{\widehat{g}_{free}(\epsilon)}{\mathrm{e}^{\beta_{B}(\epsilon+\epsilon_{c})}-1}=\frac{\Omega_{D}\rho}{2}\left(\frac{2m}{\mathrm{h}^{2}\beta_{B}}\right)^{D/2}\overbrace{\int_{0}^{\infty}\mathrm{d}x\frac{x^{D/2-1}}{\mathrm{e}^{x+\beta_{B}\epsilon_{c}}-1}}^{\mathrm{PolyGamma}\left(\frac{D}{2},\>\mathrm{e}^{-\beta_{B}\epsilon_{c}}\right)}\>,$
(19)

where $\mathrm{PolyGamma}(\cdot\>,\>\cdot)$ is a special function [28]. The
resulting $T_{B}$ increases with $\epsilon_{c}$, and diverges for
$\epsilon_{c}\rightarrow\infty$. It should be noticed that under condition
(I), LT allows BEC even in 2D, which would be impossible, in the free gas case
$\epsilon_{c}=0$. Actually, for $D=2$, $\mathrm{PolyGamma}(1,1)=\infty$.

###  3.2 The case (II)

Condition (II) corresponds to what looks the simplest possible layout: a fixed
LT ($u_{0}>0$) in a diverging large box ($V_{D}\rightarrow\infty$). From Eq.s
(11), it is immediately seen that this leaves an exponential DOS at any
$\epsilon>0$:

$\text{(II)}\quad\Rightarrow\quad g(\epsilon)\rightarrow
g_{II}(\epsilon)=\mathrm{e}^{D\epsilon/u_{0}}\overbrace{\frac{\mathrm{e}^{DA}}{u_{0}}\times\begin{cases}\frac{1}{4}&\>\>\>\>(D=2)\\\
\\\ \frac{1}{2\sqrt{27\pi}}&\>\>\>\>(D=3)\end{cases}}^{g_{D}}\>.$ (20)

From Eq.s (16) and (20), the equation for $\mu$ now reads:

$1=\frac{1}{N(\mathrm{e}^{-\beta\mu}\pm
1)}+\frac{g_{D}}{N}\int_{0}^{\infty}\mathrm{d}\epsilon\frac{\mathrm{e}^{D\epsilon/u_{0}}}{\mathrm{e}^{\beta(\epsilon-\mu)}\pm
1}\>.$ (21)

The competition between the exponential DOS (20) and the Boltzmann factor
emerges by noticing that the integral in the r.h.s. of Eq. (21) converges for
$T<T_{D}$ and diverges for $T\geq T_{D}$ (recall Eq. (13)). This makes the
solution of Eq. (21) a non trivial task, in the limit $N\rightarrow\infty$. In
the Supplemental Material file [29, 30], a rigorous procedure, starting from
the finite-sized gas ($\widehat{V},\>N<\infty$) and performing TL after, shows
that the equation for the chemical potentials $\mu_{-}$ (Bosons) and $\mu_{+}$
(Fermions) can be put in the following form:

$\displaystyle\mathrm{e}^{-\beta\mu_{\pm}}$
$\displaystyle=\frac{1}{N(1\pm\mathrm{e}^{\beta\mu_{\pm}})}-\frac{\mathcal{G}T}{T_{D}N}\ln(1-\mathrm{e}^{\beta\mu_{-}})\frac{1\mp
1}{2}+$ (22a)
$\displaystyle+\begin{cases}\circ(1/N^{\delta_{\pm}})&(T<T_{D})\\\ \\\
\circ(\ln\widehat{V}/N)&(T=T_{D})\\\ \\\
\frac{\mathcal{G}}{N}\widehat{V}^{1-T_{D}/T}\mathcal{F}_{\pm}(T,\>\widehat{V})&(T>T_{D})\end{cases}$
(22b)

where $\mathcal{G}$ is a numerical factor, $\delta_{+}=T_{D}/T$,
$\delta_{-}=1$, and $\mathcal{F}_{\pm}(\cdot\>,\>\cdot)$ are _strictly
bounded_ functions of $\widehat{V}$.222This means that there exists
$M(T)<\infty$, such that $\mathcal{F}_{\pm}(\widehat{V},\>T)<M(T)$ for each
$\widehat{V}>0$, $T>T_{D}$. The factor $(1\mp 1)/2$ in Eq. (22) means that the
associated logarithmic singularity applies to Bosons only. It is intended that
$\circ(x)$ is a quantity proportional to $x$, to leading order. If
$N/\widehat{V}:=\widehat{\rho}$ is finite, it is immediately seen that the
expression (22b) vanishes in the TL (in particular, the lowest line vanishes
as $V_{D}^{-T_{D}/T}$), whence:

$\displaystyle\mu_{-}\rightarrow-\frac{\kappa
T}{N}\left[1+\frac{\mathcal{G}}{N}\ln(N)\right]\rightarrow 0$
$\displaystyle\text{\hskip 170.71652ptfor any }T>0\>.$
$\displaystyle\mu_{+}\rightarrow\infty$

In the bosonic case ($-$), this yields $\mu_{-}\rightarrow-\kappa T/N$, and
the ground state fractional population $n_{0}$ tends to 1 in the TL. All
Bosons are thereby ‘frozen’ in the ground state at _any_ temperature. Since
the excited states of a Fermi gas involve the occupation of levels lying
_above_ the chemical potential, the condition $\mu_{+}\rightarrow\infty$ means
that the fermionic gas is frozen in the ground state too, just like the
Bosons.

In conclusion, we get here to the result:

> (A)
>
> _Let $N$ Bosons or Fermions be trapped in a LT with fixed strength
> $u_{0}>0$, closed in box of volume $V_{D}$. Then the gas collapses in the
> ground state at _any_ temperature, if $V_{D}$ diverges, with finite gas
> density $\rho=N/V_{D}$ (which is the current form of TL)._

As mentioned in Section 1, a gas in the ground state is thermally inert.
Hence, what precedes underlies the presence of what we call a ‘normal’ gas,
i.e. a different _LT-insensitive_ gas, which provides the thermal bath at
$T>0$.

###  3.3 From freezing to borderless diffusion: an off-equilibrium transition

The competition between the exponential DOS and the Boltzmann factor emerges
again from the exponent $1-T_{D}/T$ in Eq. (22b), which makes it possible to
study an interesting generalization of TL. Let $N=N(\widehat{V})$ be simply a
non decreasing function, which warrants the existence of the non negative, but
otherwise arbitrary, limit:

$\gamma:=\lim_{\widehat{V}\rightarrow\infty}\frac{\ln
N(\widehat{V})}{\ln\widehat{V}}\>.$ (24)

The condition $\gamma\leq 1$ excludes any ‘overcrowded’ gas
($N/\widehat{V}\rightarrow\infty$), and makes it possible to define the
critical temperature:

$T_{c}:=\frac{T_{D}}{1-\gamma}>T_{D}\>.$ (25)

According to Eq. (24), $N/\widehat{V}$ tends to vanish as
$\widehat{V}^{\gamma-1}=\widehat{V}^{-T_{D}/T_{c}}$. Hence, equations (22)
yield:

$\displaystyle\mathrm{e}^{-\beta\mu_{\pm}}$
$\displaystyle=\frac{1}{N(1\pm\mathrm{e}^{\beta\mu_{\pm}})}-\frac{\mathcal{G}}{N}\ln(1-\mathrm{e}^{\beta\mu_{-}})\frac{T}{T_{D}}\left(\frac{1\mp
1}{2}\right)+$ (26a) $\displaystyle\text{\hskip 227.62204pt}(T>T_{D})$
$\displaystyle+\mathcal{G}\left[\widehat{V}^{T_{D}(1/T_{c}-1/T)}\mathcal{F}_{\pm}(\widehat{V},\>T)+\cdots\right]$
(26b)

showing that the gas is still frozen in the ground state for $T<T_{c}$, since
in this case the $\widehat{V}$-dependent expression in Eq. (26b) vanishes for
$\widehat{V}\rightarrow\infty$. In contrast, the same expression _diverges_
for $T>T_{c}$, which yields $\mu_{\pm}\rightarrow-\infty$. The negative
divergence of the chemical potential is an extreme non degeneracy condition
which eliminates any difference between the behavior of Fermions and Bosons.
In conclusion, the condition $\gamma<1$ and the limit
$\widehat{V}\rightarrow\infty$ lead to another weird result:

> (B)
>
> _Let $N$ Bosons or Fermions be trapped in a LT with fixed strength
> $u_{0}>0$, closed in a box of volume $V_{D}$. Let $V_{D}/N$ diverge in the
> TL. Then there exists a finite critical temperature $T_{c}$, marking the
> transition between a total freezing in the ground state below $T_{c}$, and a
> state of absolute non degeneracy, above $T_{c}$._

The reason can be understood as follows: if $T<T_{c}$, the confining effect of
the (weak) logarithmic potential is still able to overcome the thermal kinetic
energy even in the absence of the rigid-wall confinement. If $T>T_{c}$,
instead, the loosing effect of the kinetic energy prevails, and the lack of a
reservoir’s confinement makes the particles scatter in an unconfined space,
since the number density of particles tends to vanish, due to the condition
$\gamma<1$, under which $N$ diverges less rapidly than $\widehat{V}$. This
borderless diffusion through the normal LT-insensitive gas (which provides the
thermal bath at $T>0$), actually means that the log-trapped gas gets out of
thermal equilibrium above $T_{c}$.

It is important to stress that the results obtained in the limit
$\widehat{V}\rightarrow\infty$ are almost insensitive to the improvement of
ACL (Subsection 2.2), even for $T$ small compared to $T_{D}$. Actually, the
temperature-dependent DOS Eq. (5) is the same as Eq. (11), except for the
substitutions:

$r_{c}\rightarrow
r_{c}\mathrm{e}^{\epsilon_{\vec{\alpha}_{T}}/u_{0}}\quad;\quad\theta_{c}\rightarrow\theta_{c}-\frac{D\epsilon_{\vec{\alpha}_{T}}}{u_{0}}\quad;\quad\mu\rightarrow\mu-\epsilon_{\vec{\alpha}_{T}}\>.$

Hence, equation (21) becomes:

$\displaystyle 1$ $\displaystyle=\frac{1}{N(\mathrm{e}^{-\beta\mu}\pm
1)}+\frac{1}{N}\sum_{\vec{\alpha}\neq
0}^{\vec{\alpha}_{T}}\frac{1}{(\mathrm{e}^{-\beta(\mu-\epsilon_{\vec{\alpha}})}\pm
1)}+$ (27a)
$\displaystyle+\frac{g_{D}}{N}\int_{0}^{\infty}\mathrm{d}\epsilon\frac{\mathrm{e}^{D\epsilon/u_{0}}}{\mathrm{e}^{\beta(\epsilon-\mu+\epsilon_{\vec{\alpha}_{T}})}\pm
1}$ (27b)

The asymptotic behavior of the integral in Eq. (27b), for
$\widehat{V},\>N\rightarrow\infty$ remains the same as in Eq. (21). The only
significant change is that the single contribution of the ground level is
replaced by the finite sum in Eq. (27a), so that in the bosonic case the
particles’ condensation involves a set of low-energy levels
$0\leq\epsilon_{\vec{\alpha}}\leq\epsilon_{\vec{\alpha}_{T}}$, instead of the
ground level alone. In the fermionic case and for finite
$\epsilon_{\vec{\alpha}_{T}}$, the contribution of the sum simply vanishes in
the TL.

##  4 The Effective Volume

The peculiar features of LT can be clearly illustrated by the behavior of what
we call the _effective_ volume (EV) [5, 33], i.e. the mean volume occupied by
a particle at the temperature $T$, _modulo_ the degeneracy and the exclusion
principle’s effects. $V_{eff}$ represents the mean ideal volume available to a
particle if it were alone,333$V_{eff}$ can also be regarded as to the mean
volume occupied by a gas of _distinguishable_ bosons. and is useful to
illustrate the effects produced by the external fields on the confinement:

$V_{eff}=\int\mathrm{d}\mathbf{r}\>r_{1}\cdots r_{D}\>P(\mathbf{r},\>T)\>,$
(28a) where

$P(\mathbf{r},\>T)=\frac{|\phi_{0}(r)|^{2}+\frac{1}{\mathrm{h}^{D}}\int_{h(p,\>r)>0}\mathrm{d}\mathbf{p}\>\mathrm{e}^{-\beta
h(p,\>r)}}{1+\frac{1}{\mathrm{h}^{D}}\int_{h(p,\>r)>0}\mathrm{d}\mathbf{p}\>\mathrm{d}\mathbf{r}\>\mathrm{e}^{-\beta
h(p,\>r)}}\>,$ (28b)

is the probability distribution of the particle’s co-ordinate $\mathbf{r}$, of
Cartesian components $r_{j}$, in the non degeneration limit
$\mathrm{e}^{\beta\mu}<<1$. Following the same method as in Section 2, here we
introduce the probability density $|\phi_{0}(r)|^{2}$ of the ground state as a
contribution apart, with respect to the excited states contribution, expressed
by the integrals in the r.h.s. of Eq. (28b), according to the continuous
approximation. In view of TL, a further useful approximation follows from Fig.
1: as already outlined in Subsection 3.1, equation (15) can be used as a large
$\widehat{V}$ (or $V_{D}$) approximation, even for the finite-sized gas. In
particular, one can approximate the Hamiltonian as a potential well with a
single bound state $\phi_{0}(r)$ at the ground level $\epsilon_{0}=0$,
separated from the free-particle Hamiltonian by a gap of width $\epsilon_{c}$.
In this case the probability density (28b) reads simply:

$P(\mathbf{r},\>T)=\begin{cases}\frac{|\phi_{0}(r)|^{2}+\frac{\mathrm{e}^{-\beta\epsilon_{c}}}{\mathrm{h}^{D}}\int\mathrm{d}\vec{p}\>\mathrm{e}^{-\beta
p^{2}/(2m)}}{1+\frac{\mathrm{e}^{-\beta\epsilon_{c}}}{{\mathrm{h}^{D}}}V_{D}\int\mathrm{d}\vec{p}\>\mathrm{e}^{-\beta
p^{2}/(2m)}}&\quad\quad(r<R)\\\ &\text{\hskip 56.9055pt}.\\\
0&\quad\quad(r>R)\end{cases}$ (29)

On expressing the Gaussian integrals and the average volume $v_{0}$ of the
ground state [24] as:

$\int\mathrm{d}\vec{p}\>\mathrm{e}^{-\beta
p^{2}/(2m)}=\Omega_{D}(2m/\beta)^{D/2}\mathcal{K}_{D}\quad,\quad
v_{0}=\left(\frac{\hbar}{\sqrt{2mu_{0}}}\right)^{D}\mathcal{C}_{D}^{-1}$ (30a)
with:

$\mathcal{K}_{D}:=\begin{cases}1/2&\\\ \\\
\sqrt{\pi}/4&\end{cases},\quad\frac{1}{\mathcal{C}_{D}}:=\begin{cases}3/2\quad&(D=2)\\\
\\\ 15\sqrt{2}\quad&(D=3)\end{cases}\>,$ (30b)

equations (28a), (29), (30) yield:

$V_{eff}=\frac{v_{0}+\frac{\Omega_{D}^{2}\mathrm{e}^{-\beta\epsilon_{c}}}{2\mathrm{h}^{D}D}R^{2D}(2m/\beta)^{D/2}\mathcal{K}_{D}}{1+\frac{\Omega_{D}\mathrm{e}^{-\beta\epsilon_{c}}}{{\mathrm{h}^{D}}}V_{D}(2m/\beta)^{D/2}\mathcal{K}_{D}}\>.$

From the definitions (8b) one has $R=(\widehat{V}/v_{c})^{1/D}$,
$\exp(\beta\epsilon_{c})=\widehat{V}^{T_{D}/T}$ and
$v_{c}/v_{0}=\mathrm{e}^{DA}\mathcal{C}_{D}$, whence the ratio $V_{eff}/v_{0}$
can be expressed in terms of the dimensionless volume
$\widehat{V}=V_{D}/v_{c}$ and dimensionless temperature $T/T_{D}$:

$\frac{V_{eff}}{v_{0}}=\frac{1+\frac{\mathcal{K}_{D}\mathcal{C}_{D}D^{1-D/2}\mathrm{e}^{2DA}}{2(2\pi)^{D}}(T/T_{D})^{D/2}\widehat{V}^{2-T_{D}/T}}{1+\frac{\mathcal{K}_{D}\Omega_{D}D^{1-D/2}\mathrm{e}^{DA}}{(2\pi)^{D}}(T/T_{D})^{D/2}\widehat{V}^{1-T_{D}/T}}\>.$
(31)

Figure 2: Effective Volume. Semi-logarithmic plot of $V_{eff}/v_{0}$ as a
function of $T/T_{D}$ for different values of $\widehat{V}$, showing the
drastic increase of EV for $T>T_{D}/2$ up to extensive values proportional to
$\widehat{V}$ for $T>T_{D}$. In the logarithmic scale, the differences between
$D=2$ and $D=3$ are almost inappreciable.

Figure 2 shows a plot of expression (31) for different values of $\widehat{V}$
and $D=2,\>3$. For $T<T_{D}/2$, EV depends weakly on $T$ and is comparable in
magnitude with the minimum size $v_{0}$, corresponding to the mean volume
occupied by the ground state. For $T_{D}/2<T<T_{D}$, EV increases rapidly with
$T$ and attains an asymptotic value, above $T_{D}$, proportional to
$\widehat{V}$. Equation (31) shows that removing the rigid walls makes EV
‘explode’ from the mean volume $v_{0}$ occupied by the single particle in the
ground state, to a diverging value, at the critical temperature $T_{D}$.

##  5 Looking for the origin of the LT’s weirdness

The deep origin of the anomalous results expressed by statements (A) and (B)
can be found by comparing the DOS Eq.s (11) with the DOS’s resulting from the
Hamiltonians $H_{\pm}=p^{2}/(2m)\pm w_{0}r^{\pm\lambda}$ ($\lambda,w_{0}>0$),
with homogeneous power-law potentials [34, 35, 36]. $H_{+}$ yields a _true_
trap, which confines the gas even in the absence of any rigid-wall box
around.444A typical example is the harmonic trap with $\lambda=2$. Notice that
on setting $w_{0}=u_{0}/R^{\lambda}$, the rigid-wall box of radius $R$ is
realized by $\lambda\rightarrow\infty$. The spectrum is positive and discrete
up to arbitrary large energies, and TL is realized by the limit
$w_{0}\rightarrow 0$, together with CL. In the case $H_{-}$, instead, the very
notion of trap evaporates. Shifting the energy origin to the negative ground
level $\epsilon_{0}$, the spectrum has a discrete branch in the interval
$]0,|\epsilon_{0}|[$, and $|\epsilon_{0}|$ is an accumulation point of levels
$\epsilon_{n}$, with multiplicity $g_{n}$, corresponding to bound states with
localization length $\ell_{n}$, such that
$\epsilon_{n}\rightarrow|\epsilon_{0}|$, $\ell_{n}\rightarrow\infty$ for
$n\rightarrow\infty$.555A typical example is the Coulombic potential
$\lambda=1$. Above $|\epsilon_{0}|$, the spectrum is continuous and tends to
the free-particle DOS with diverging $\epsilon$. In this case the rigid-wall
box is necessary for two related reasons: first, to have the gas confined;
second, to have a finite partition function. Actually, the condition that the
localization length of the states must be smaller than the box radius
($\ell_{n}<R$), results in an upper limit $n^{*}$ for $n$, which makes the
diverging series $\sum_{n=0}^{\infty}g_{n}\mathrm{e}^{-\beta\epsilon_{n}}$
become a finite partition function
$\sum_{n=0}^{n^{*}}g_{n}\mathrm{e}^{-\beta\epsilon_{n}}$. In short, the
passage from $H_{+}$ to $H_{-}$ marks the difference between a true trap and
what can be loosely called a ‘potential hole’. LT is right the border case
between the preceding ones: it is a trap in that the spectrum, _without_ the
box, is positive and discrete up tu arbitrary large energies; it is a
potential hole in that there exists a finite energy $\epsilon_{c}$ which
separates the trapped localized states from those occupying the whole volume
_within_ the box. However, in contrast to a standard potential hole, in which
the border energy $|\epsilon_{0}|$ depends only on the attractive potential,
the border energy

$\epsilon_{c}=u_{0}\ln\left(\frac{R\sqrt{2mu_{0}}\mathrm{e}^{-A}}{\hbar}\right)\>,$
(32)

does depend on the box size too, which leads to the difference between the
approaches (I) and (II) to TL. As outlined in Subsection 3.1, the case (I)
realizes a perfect potential hole, with a single bound level separated from
the free DOS by a gap $\epsilon_{c}$, held fixed in the TL. The case (II),
instead, implements a perfect trap in an infinite space, confining the whole
gas in the ground level at any temperature. The origin of the LT’s weirdness,
is thereby the ambiguous nature of the logarithmic potential, bridging between
the true trap’s and the potential hole’s behavior, which leads to statement
(A). The competition between the exponential DOS, specific of LT, and the
Boltzmann factor, completes the weirdness, leading to statement (B).

##  6 Just a toy model?

As a mere toy model, a log-trapped gas is a workbench, aiming to submit some
current notions, like TL and size independence, to a sort of stress test, when
a logarithmic field is added to the usual rigid-wall confinement in a box. The
extreme consequences expressed by statements (A) and (B) show that approaching
TL as in case (II) ($u_{0}>0$, $V_{D}\rightarrow\infty$) cannot be used as an
ideal limit of concrete experimental layouts. However, one could wonder if
this would be possible in a finite sized gas, approaching TL as in case (I).
As a reference point, let a 3D bosonic gas, with a typical BEC temperature
$T_{B}^{0}$ of few Kelvin, in the free case, be exposed to the action of a LT,
in an appropriate box, such that the relationship (32) does yield
$\epsilon_{c}\approx 1\mathrm{eV}$. Since $\epsilon_{c}/\kappa
T_{B}^{0}\approx\mathrm{10}^{4}$, a numerical estimate of the solution of Eq.
(19) (see Fig. 3) yields $\kappa T_{B}\approx\epsilon_{c}/10$. LT would
thereby enhance the BEC temperature up to values comparable to
$10^{3}\,{}^{\circ}\mathrm{K}$.

Figure 3: BEC temperature $T_{B}$ as a function of $\epsilon_{c}$ in 3D.
$\epsilon_{c}/\kappa T_{B}$ is plotted against the ratio $\epsilon_{c}/\kappa
T_{B}^{0}$, $T_{B}^{0}$ being the BEC temperature of the free 3D gas. Dots are
calculated numerically from Eq. (19). The dashed straight line shows an
asympthotic linear interpolation, described by the equation
$\epsilon_{c}/\kappa T_{B}=3.1713\ln_{10}(\epsilon_{c}/\kappa
T_{B}^{0})-3.3395$.

As for Fermions, the Fermi level $\epsilon_{F}^{free}$ of an atomic free gas
in 3D, in standard laboratory conditions ($\rho\approx
10^{19}\mathrm{cm}^{-3}$, $m\approx 10^{-23}\mathrm{g}$), attains values
comparable to $10^{-2}\mathrm{eV}$. With $\epsilon_{c}\approx 1\mathrm{eV}$,
equation (18) in Subsection 3.1 would open the possibility of an _atomic_
Fermi level $\epsilon_{F}=\epsilon_{F}^{free}+\epsilon_{c}$ comparable in
magnitude with that of the electrons in solids (some $\mathrm{eV}$), which
would yield a fermionic ‘cousin’ of the Bose-Einstein condensate [31],
surviving extremely high temperatures.

What precedes would be astonishingly promising, if it were not for the
difficulty of realizing a LT for _neutral_ particles. To the author’s
knowledge, no attempt has been made with magnetic fields. As mentioned in
Section 1, an attractive logarithmic potential acts on _charged_ particles,
due to the electric field produced by a very long opposite charged wire, in
the plane normal to the wire itself. This, however, calls into play the
particle-particle interactions, which can be provisionally ignored (as we did
so far) in case of Lennard-Jones potentials between neutral particles, but
definitely cannot, in case of Coulombic interactions. So, the possibility of
promoting LT from a toy model to a realistic plan for promising laboratory
experiences, remains open to future investigations, which are in progress,
aiming to study the thermodynamics of _orbitrons_ [17, 32], whose core is an
electric field realizing a LT for charged particles.

##  7 Conclusions

A gas in a $D$-dimensional logarithmic trap (LT), confined in a box of volume
$V_{D}$, exhibits very unconventional aspects, which challenge the standard
notion of thermodynamic limit (TL), defined as $N,\>V_{D}\rightarrow\infty$,
with $0<N/V_{D}<\infty$. For a log-trapped gas to exhibit a conventional
behavior, characterized by BEC (Bosons), and by a finite Fermi level $E_{F}$
(Fermions), TL must be accompained by the condition that the energy
$\epsilon_{c}$ (Eq. (32)) is held fixed at a finite value (case (I)). This
implies an exponential divergence of the box radius $R$, with vanishing LT
strength $u_{0}$. If, instead, $u_{0}>0$ is the fixed parameter (case (II)),
the gas (fermionic or bosonic) collapses in the ground state at _any_
temperature (statement (A)). This implies the presence of a ‘normal’ gas
insensitive to LT, which provides the thermal bath. If the number of particles
$N$ diverges less rapidly than the confinement volume $V_{D}$, one can define
a non negative parameter $\gamma<1$ (Eq. (24)), and a critical temperature
$T_{c}=T_{D}/(1-\gamma)$, under which the gas is frozen in the ground state,
and gets out of thermal equilibrium above, by expanding through the normal gas
without limit (statement (B)). Figure 2 provides a clear picture of this off-
equilibrium transition, showing the ‘explosion’ of the mean effective volume
(EV) occupied by a single particle, _modulo_ the degeneration effects. The
discussion in Section 5 shows that the origin of the LT’s weirdness, is the
special nature of the logarithmic potential $u(r)=u_{0}\ln(r/r_{0})$, as a
border case, between a true trap ($u(r)=w_{0}r^{\lambda}$) and a potential
hole ($u(r)=-w_{0}r^{-\lambda}$).

Section 6 provides a brief outline of the astonishing consequences resulting
from LT’s, if those were realizable for _neutral_ particles, which looks a
very difficult technological task. Studies are in progress to include the
Coulombic interactions between the particles, and apply the results to the
_ionic_ LT’s, which are the core of _orbitrons_.

As a toy model, however, it is possible to use the extreme consequences (A)
and (B) to play a mind game, which we call the Logarithmic Universe Game,
bridging between the Big Bang [37] and the Steady State [38] theory. Ignore
any effect of general relativity666We stress that what follows has nothing to
do with the logarithmic potential of ref [21], resulting from general
relativity. and imagine that a LT is placed in the ‘centre’ of an infinite
space ($V_{D}=\infty$), populated by a _finite_ number $N<\infty$ of log-
trapped massive particles, at a temperature $T$ provided by a thermal bath of
LT-insensitive photons. In this case, one would have
$\gamma=\lim_{\widehat{V}\rightarrow\infty}\ln
N(\widehat{V})/\ln\widehat{V}=0$ and $T_{c}=u_{0}/(D\kappa)=T_{D}$. For
$T<T_{D}$, all the massive matter would be buried in the ground state. If,
however, the photonic gas temperature were raised above $T_{D}$ (Big Bang),
the log-trapped gas would begin to expand almost freely in a just-born LT-
universe (obviously non isotropic, due to the LT), according to statement (B).
The time scale of diffusion, the possibility of a ‘dark’ matter, still present
in the LT ground state, and the continuous creation of visible matter (Steady
State), as a result of a still lasting ‘evaporation’ of the dark matter
itself, could all be items for amusing speculations.

The authors report there are no competing interests to declare

## References

  * [1] D.E. Pritchard, _Cooling Neutral Atoms in a Magnetic Trap for Precision Spectroscopy_. Physical Review Letters. 51 (15) (1983) 1336 1339. Bibcode:1983PhRvL..51.1336P. doi:10.1103/PhysRevLett.51.1336.
  * [2] J.J. Tollett, C.C. Bradley, C.A. Sackett and R.G. Hulet, _Permanent magnet trap for cold atoms_ , Phys. Rev. A 51 (1995) R22.
  * [3] V. Vuletic, T. H nsch and C. Zimmermann, _Steep magnetic trap for ultra cold atoms_ , Europhys. Lett. 36 (1996) 349-354.
  * [4] J. Fort gh and C. Zimmermann, _Magnetic microtraps for ultra cold atoms_ , Rev. Mod. Phys. 79 (2007) 235-289.
  * [5] J. Pérez-Ríos and A.S. Sanz, _How does a magnetic trap work?_ , Am. J. of Phys. 81 (2013).
  * [6] See, for instance: V. Romero and S. Bagnato: _Thermodynamics of an Ideal Gas of Bosons Harmonically Trapped: Equation of State and Susceptibilities_ , Braz.J. Phys. 35 (2005) 607-613.
  * [7] W.D. Phillips, _Laser cooling and trapping of neutral atoms_ , Rev. of Mod. Phys. 70 (1988) 721-741.
  * [8] K.B. Davis, M-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Dufree, D.M. Kurm and W. Ketterle, _Bose-Einstein Condensation in a Gas of Sodium Atoms_ , Phys. Rev. Lett. 75 (1995) 3969-3973.
  * [9] C.C. Bradley, C.A. Sackett, J.J. Tollett and R.G. Hulet: _Evidence of Bose-Einstein Condensation in an Atomic Gas with Attractive Interactions_ , Phis. Rev. Lett. 75, 1687-1690 (1995).
  * [10] K. Frye, S. Abend, W. Barttosch _et al_ : _The Bose-Einstein Condensate and Cold Atom Laboratory_ , Eur. Phys. J. _Quantum Technology_ 8 (2021) DOI: 10.1140/epjqt/s40507-020-00090-8.
  * [11] H. Mütinga _et al_ : _Interferometry with Bose-Einstein Condensates in Microgravity_ , Phys. Rev. Lett. 110 (2013) 093602-093607.
  * [12] Olivier Carraz, Christian Siemes, Luca Massotti, Roger Hiagmans and Pierluigi Silvestrin: _A spaceborn gravity gradiometer concept based on cold atom interferometers for measuring Eath’s gravity field_ : arXiv: 1406.0765v2 $[\text{phys-atom-ph}]$ (2014).
  * [13] Matthias Meister, Albert Roura, Ernst M. Rasel and Wolfgang P. Schich: _The space atom laser: an isotropic souce for ultra-cold atoms in microgravity_ , New J. Phys. 21 (2019) 013039-013063.
  * [14] T. Bravo, D. Rätzel and I. Fuentes: _Phononic gravity gradiometry with Bose-Einstein Condensates_ , arXiv:2001.10104v2 $[\text{quant-phis}]$ (2020).
  * [15] V. Bagnato, D.F. Pritchard and D. Kleppner: _Bose-Einstein Condensation in an external potential_ , Phys. Rev. A 35 (1987) 4354-4358.
  * [16] R.M. Cavalcanti, P. Giacconi, G. Pupillo and R. Soldati: _Bose-Einstein condensation in the presence of a uniform field and a pointloke impurity_ , Phy. Rev. A 65 (2002) 0536056-0536063.
  * [17] R.H. Hooverman: _Charged particles orbits in a logarithmic potential_ , J. of Appl.Phys. 34 (1963) 3505 - 3508.
  * [18] Y. Muraki: _A new mass formula of elementary particles_ : Progr. Theor. Phys. 41 (1969) 473 - 478.
  * [19] Y. Muraki, K. Mori and N. Nakagawa: _Logarithmic mass formula for elementary particles and a new quantum number_ : N. Cim. 23 (1978) 27 - 31.
  * [20] K. Paasch: _The logarithmic potential and an exponential mass function for elementary particles_ : Progress in Physics 1 (2009) 36 - 39.
  * [21] N.I. Shakura and G.V. Lipunova: _Logarithmic potential for the gravitational field of Schwarzschild black holes_ : Monthly Notices of the Royal Astronomical Society 480 (2018) 4273-4277.
  * [22] R.K. Pathria: _An ideal quantum gas in a finite-sized container_ : Am. J. of Phys. 66 (1998) 1080 - 1085.
  * [23] S. Mogliacci, I. Kolb and W.A. Horowitz, _Geometrically confined thermal field theory: Finite size corrections and phase transitions_ , Phys.Rev. D 102 (2020) 116017-116052.
  * [24] Supplemental Materials: Section A.
  * [25] Supplemental Materials: Section B.
  * [26] I.F. Silveira: _Bose-Einstein Condensation_ , Am. J. of Phys. 65 (1999) 570 - 574. DOI: 10.119/1.18591.
  * [27] L. Ferrari: _Approaching Bose-Einstein Condensation_ , Europ. J. of Physics 32 (2011) 1 -11.
  * [28] Use, for example _Wolfram Mathematica 7.01.0_ or higher versions.
  * [29] Supplemental Materials: Section C.
  * [30] Supplemental Materials: Section D.
  * [31] B. de Marco and D. Jin, _Onset of Fermi degeneracy in a trapped atomic gas_ , Science 285 (1999) 1703-1706. DOI: 10.1126/science.285.543.1703.
  * [32] Sun Da-ming: _Ionizing characteristics of an orbitron-type pump_ , Jour. Phys. E: Sci. Instrum. 22 438 - 440 (1989).
  * [33] L. Ferrari and F. Pavan _Boson thermodynamics in the gravity field_ , arXiv:2206.12287v2[cond-mat.quant-gas] 1-20.
  * [34] H. Ciftci, E. Ateser and H. Koru, _The power-law and the logarithmic potentials_ , arXiv: math-ph/0212072v1 (2002) 1-16.
  * [35] A.K. Roy, _Calculation of the bound states of power-law and logarithmic potentials through a generalized pseudopotential method_ , arXiv: 1307.1525v1[quant-ph] (2013) 1-14.
  * [36] A.A. Abdel-Hardy, _Calculation of the power-law and logarithmic potential using the J-matrix method_ , Int. J. of App. Sc. IJBAS-IJENS 11 (2011) 53-59.
  * [37] P.J.E. Peebles, _Principles of physical cosmology_ , Princeton University Press (1993).
  * [38] H.C. Arp, G. Burbridge, F. Hoyle, J.V. Narlikar and N.C. Wickramasinghe, _The extragalactic universe: an alternative view_ , Nature 346 (1990) 807-812.