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{\Large \bf BEC in Nonextensive Statistical Mechanics}
\vskip 0.8cm
Luca Salasnich \\
\vskip 0.5cm
Istituto Nazionale per la Fisica della Materia, Unit\`a di Milano,\\
Dipartimento di Fisica, Universit\`a di Milano, \\
Via Celoria 16, 20133 Milano, Italy
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\vskip 1.5cm
\begin{center}
{\bf Abstract}
\end{center}
We discuss the Bose-Einstein condensation (BEC) for an ideal gas
of bosons in the framework of Tsallis's nonextensive
statistical mechanics.
We study the corrections to the standard BEC formulas
due to a weak nonextensivity of the system.
In particular, we consider three cases
in the D-dimensional space:
the homogeneous gas, the gas in a harmonic trap and
the relativistic homogenous gas.
The results show that small deviations from
the extensive Bose statistics produce remarkably large
changes in the BEC transition temperature.
\vskip 0.5cm
PACS numbers: 05.30-d; 03.75.Fi
\newpage
A decade ago, Tsallis introduced a nonextensive statistical mechanics
(NSM) to describe systems for which the additivity property of entropy
does not hold.$^{1}$ The NSM can describe systems for which
long-range microscopic memory, fractal space-time constraints
or long-range interactions affect the thermalization process.$^{2}$
The NSM is characterized by a parameter $q$ such that
$(q-1)$ is a measure of the lack of extensivity: in the limit
$q\to 1$ one recovers the familiar statistical mechanics but for
$q\ne 1$ one obtains generalized
Boltzmann, Fermi and Bose distributions.$^{3}$
In the last few years the NSM has been applied in different contexts
like solar neutrinos,$^{4}$ high energy nuclear collisions$^{5}$ and
the cosmic microwave background radiation.$^{6}$
In such cases it has been found that a small deviation
from standard statistics is sufficient
for eliminating the discrepancy between
theoretical calculations and experimental data.
\par
Recently, there has been a renewed theoretical interest on Bose-Einstein
condensation (BEC) (for a review see Ref. 7),
motivated by the experimental achievement of BEC
with trapped weakly-interacting alkali-metal atoms.$^{8}$
In this paper we analyze the consequences of weak nonextensivity
on BEC for an ideal Bose gas. From the generalized Bose-Einstein
distribution we derive the BEC transition temperature,
the condensed fraction and the energy per particle
in three different cases: the homogeneous gas, the gas in a harmonic trap and
the relativistic homogenous gas. All the calculations are performed
by assuming a D-dimensional space.
\par
For a quantum gas of identical bosons in the grand canonical ensemble,
the NSM predicts that the average number of particles
with energy $\epsilon$ is given by
\beq
\langle n(\epsilon )\rangle_q = {1\over \left[1+
\beta (q-1) (\epsilon - \mu)\right]^{1/(q-1)} - 1} \; ,
\eeq
where $\mu$ is the chemical potential and $\beta=1/(kT)$
with $k$ the Boltzmann constant and $T$ the temperature.$^{2}$
This generalized distribution follows from the minimization of the
Tsallis's generalized entropy under
the dilute gas assumption, namely the different single-particle
states of the systems are regarded as independent.
Thus, this is not an exact formula
but it has been shown to be extremely accurate,
in particular near $q=1$.$^{9}$
When $q<1$ the generalized distribution has an upper cut-off:
$(\epsilon - \mu ) \le kT/(1-q)$. In the limit $q\to 1$
the generalized distribution becomes the standard Bose-Einstein
distribution. For $q>1$ there is no cut-off and the (power-law)
decay is slower than exponential.
Because of the unphysical cut-off for $q<1$, in this paper
we discuss only the case $q\ge 1$.
\par
We want study the effects of weak nonextensivity on the BEC
properties. We assume that $(q-1)<1$ and
by performing a Taylor expansion of the generalized Bose distributions
in the parameter $(q-1)$, at first order we obtain
\beq
\langle n(\epsilon )\rangle_q =
{1\over e^{\beta(\epsilon -\mu)}-1}
+ {1\over 2} (q-1) {\beta^2 (\epsilon - \mu)^2 \over
(e^{\beta(\epsilon -\mu)}-1)^2 } \; .
\eeq
This is the weak nonextensivity correction to the standard
Bose-Einstein distribution and the starting point for our calculations.
\par
The total number of particle for our system of non-interacting
bosons reads
\beq
N=\int_0^{\infty}
d\epsilon \; \rho(\epsilon ) \; \langle n(\epsilon )\rangle_q \; ,
\eeq
where $\rho (\epsilon )$ is the density of states.
It can be obtained from the formula
\beq
\rho(\epsilon ) = \int {d^D{\bf q} d^D{\bf p}\over (2\pi\hbar)^D}
\delta (\epsilon - H({\bf p},{\bf q})) \; ,
\eeq
where $H({\bf p},{\bf q})$ is the classical single-particle
Hamiltonian of the system in a D-dimensional space.
It is easy to show that for a homogenous gas
the density of states in a D-dimensional box of volume $V$ is given by
\beq
\rho(\epsilon ) = {V\over \Gamma(D/2)}
\left({m\over 2\pi \hbar^2}\right)^{D/2} \epsilon^{(D-2)/2} \; ,
\eeq
where $m$ is the mass of the particle.
Instead, for a gas in a harmonic trap one finds
\beq
\rho(\epsilon ) = {\epsilon^{D-1}\over
(\hbar \bar{\omega})^D \Gamma(D)} \; ,
\eeq
where $\bar{\omega}$ is the geometric average of the trap frequencies.
$\Gamma(x)$ is the factorial function.
\par
At the BEC transition temperature $T_q$,
the chemical potential $\mu$ is zero and
at $\mu=0$ the number of particles $N$
can be analytically determined from Eq. (2) and (3).
By inverting the function $N=N(T_q)$ one finds
the transition temperature. It is given by
\beq
kT_q = \left({2\pi \hbar^2 \over m}\right)
{(N/V)^{2/D}\over \zeta (D/2)^{2/D}}
\left[1 + {1\over 2}(q-1)
{\Gamma(D/2+2) \zeta(D/2+1)\over \Gamma(D/2) \zeta(D/2) }
\right]^{-2/D} \;
\eeq
for the homogenous gas, and by
\beq
kT_q ={\hbar \bar{\omega}\over \zeta (D)^{1/D}} N^{1/D}
\left[1 + {1\over 2}(q-1){\Gamma(D+2)\zeta(D+1)\over \Gamma(D)\zeta(D)}
\right]^{-1/D} \;
\eeq
for a gas in a harmonic trap.
$\zeta(x)$ is the Riemann $\zeta$-function.
Obviously, for $q=1$ one recovers
standard BEC formulas. Moreover one observes that for $D=2$
there is no BEC in the homogenous gas because $\zeta(1)=\infty$.
Instead, BEC is possible with $D=2$ in a harmonic trap.
Note that the inclusion of an attractive interaction can modify
the stability of the Bose condensate.
A discussion of the the role of dimensionality
in the stability of a weakly-interacting condensate
can be found in Ref. 10.
\par
An inspection of Eq. (7) and (8) shows that the critical temperature
$T_q$ grows by increasing the nonextensive parameter $q$ and
the space dimension $D$. It is important to stress that such effect is
quite strong. For example, with $q=1.1$ and $D=3$ we have that
the relative difference $(T_q-T_1)/T_1$ is $6.32\%$
for the homogenous gas and and $15.48\%$ for the gas
in a harmonic trap.
\par
Below $T_q$, a macroscopic number $N_0$ of particle occupies
the single-particle ground-state of the system. It follows
that Eq. (3) gives the number $N-N_0$ of non-condensed particles and
the condensed fraction is $N_0/N=1-(T/T_q)^{D/2}$ for the homogenous
gas and $N_0/N=1-(T/T_q)^D$ for the gas in harmonic trap.
For the sake of completeness, we calculate also the energy
\beq
E= \int_0^{\infty} d\epsilon \;
\epsilon \; \rho(\epsilon ) \; \langle n(\epsilon )\rangle_q \; .
\eeq
>From the energy one can easily obtain the specific heat and the other
thermodynamical quantities. We find
\beq
{E\over KT} = V \left({kT\over 2\pi\hbar^2}\right)^{D/2} {D\over 2}
\zeta(D/2+1)
\left[1 + {1\over 2}(q-1)
{\Gamma(D/2+3) \zeta(D/2+2)\over \Gamma(D/2+1) \zeta(D/2+1) }
\right] \;
\eeq
for the homogenous gas, and by
\beq
{E\over KT} = \left({kT\over \hbar \bar{\omega}}\right)^D D \zeta(D+1)
\left[1 + {1\over 2}(q-1)
{\Gamma(D+3) \zeta(D+2)\over \Gamma(D+1) \zeta(D+1) }
\right] \;
\eeq
for a gas in a harmonic trap. Note that our formulas of the energy can be
easily generalized above the critical temperature $T_q$ by substituting
the Riemann function $\zeta(D)$ with the polylogarithm
function $Li_{D}(z)=\sum_{k=1}^{\infty}z^k/k^D$, that depends
on the fugacity $z=e^{\beta \mu}$.
\par
In the case of a relativistic gas, the total number of particles is not
conserved because of the production
of antiparticles, which becomes relevant when $kT$ is comparable
with $mc^2$. The conserved quantity is the difference
between the number $N$ of particles and the number $\bar{N}$ of
antiparticles, i.e. the net conserved {\it charge}
\beq
Q=N-\bar{N}=
\int d\epsilon \; \rho(\epsilon )
\left[ \langle n(\epsilon )\rangle_q -
\langle \bar{n}(\epsilon )\rangle_q \right] \; ,
\eeq
where $\langle \bar{n}(\epsilon )\rangle_q$ is obtained
from $\langle n(\epsilon )\rangle_q$ with the substitution
$\mu \to - \mu$. Thus the chemical potential $\mu$
describes both bosons and antibosons: the sign of $\mu$
indicates whether particles outnumber antiparticles or vice.
Moreover, because both $\langle n(\epsilon )\rangle_q$ and
$\langle \bar{n}(\epsilon )\rangle_q$ must be positive definite,
it follows that $|\mu |\le mc^2$.$^{11}$
\par
As well known, the classical
single-particle Hamiltonian of a relativistic ideal gas is
$H=\sqrt{p^2c^2 + m^2c^4}$ and the density of states reads
\beq
\rho(\epsilon )={V 2\pi^{D/2} \over (2\pi \hbar c)^D \Gamma(D/2)}
\epsilon (\epsilon^2-m^2 c^4)^{(D-2)/2} \; .
\eeq
It is interesting to observe that
in the ultrarelativistic limit, the density of states is
$\rho(\epsilon )=(V 2\pi^{D/2})/((2\pi \hbar c)^D \Gamma(D/2))
\epsilon^{(D-1)}$ and it has the same power law of the
non-relativistic gas in a harmonic potential.
The critical temperature $T_q$ at which BEC occurs corresponds
to $|\mu| = mc^2$. In the ultrarelativistic region $kT \gg mc^2$
one can expand $Q$ at first order in $\mu$ and then obtains
$$
kT_q = \left( { (2\pi \hbar c)^D \Gamma(D/2)
\over 4\pi^{D/2} \Gamma(D) \zeta(D-1)}
{|Q|/V \over mc^2 } \right)^{1/(D-1)} \times
$$
\beq
\times \left[1 + {1\over 2}(q-1)
{(D-1)\Gamma(D+1)\zeta(D)
\over \Gamma(D)\zeta(D-1)} \right]^{-1/(D-1)} \; .
\eeq
Note that, as in the non-relativistic case, for a homogenous gas
there is BEC only for $D>2$. Also for the relativistic gas the
critical temperature $T_q$ is a growing function of the nonextensive
parameter $q$ (for $q\ge 1$) and of the space dimension $D$.
By using the previously introduced
values $q=1.1$ and $D=3$ we find $(T_q-T_1)/T_1=6.83\%$.
Finally, we obtain that below $T_q$
the condensed fraction reads $Q_0/Q=1-(T/T_q)^{(D-1)}$.
\par
In conclusion, we have analyzed the consequences of
Tsallis's nonextensive statistical mechanics on BEC.
We have studied three non-interacting systems with a generic
spatial dimension: the homogeneous gas, the gas in a harmonic trap and
the relativistic homogenous gas.
The calculations show that a very small deviation from
the extensive Bose statistics produces remarkable changes
in the BEC transition temperature.
This result may have important consequences, for instance
in the formation of Quark-Gluon Plasma$^{12}$
and in the thermodynamics of the Higgs field in the early Universe.$^{13}$
We observe that the inter-particle interaction can strongly modify
the BEC transition temperature and the condensate properties:
one of our future projects will be the study of nonextensive
statistical mechanics for interacting systems.
\newpage
\section*{References}
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\end{description}
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