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## BoseGas with an Arbitrary Dispersion Law in a Trap

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### Ivan Franko National University of Lviv. L. Landau, E. Lifshitz, Statistical Physics. ... is studied using a simple technique based on the standard' ideal Bose ... – PowerPoint PPT presentation

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Title: BoseGas with an Arbitrary Dispersion Law in a Trap

1
Bose-Gas with an Arbitrary Dispersion Law in a
Trap
• Andrij Rovenchak
• Department for Theoretical Physics, Ivan Franko
National University of Lviv

2
Short Intro
• An ideal D-dimensional Bose-gas with an arbitrary
dispersion law
• is studied using a simple technique based on the
standard ideal Bose-gas theory 1.

(1)
We consider the system of N bosons with spin s
confined either in a box of volume V or in an
external potential. At low temperatures, the
occupation of the lowest energy state is
macroscopic, while the discreteness of the
excited states is neglected.
3
Particles in a Box
• Thermodynamic functions are calculated from the
energy E defined as follows

(2)
The chemical potential is defined from the
condition that the number of particles N is
fixed
(3)
4
Derivation Remark
It is easy to arrive at Eqs. (2), (3) integrating
the expression
(4)
The integral over the space variables gives the
volume V and it is convenient to pass to the
spherical variables in the integral over momenta.
Then, the substitution (1) leads to Eqs (2), (3).
5
Particles in an External Potential
• For the simplicity, we consider the potential
U(x) in the form of isotropic harmonic
oscillator. These like potentials are achieved in
the experiments on the Bose-Einstein condensation
(BEC).

(5)
Under an external potential the domain accesible
for a particle is not as trivial as V but is
defined by the energy of a particle. In the
semiclassical approach 2 the integration over
the space variables in (4) is made within the
classical turning points.
6
Some animations
No external field
Trap is turned on
7
Several cases of a special interest
• Power dispersion law 3,4
• Relativistic Bose-gas
• Elementary excitation spectrum in the
Bogoliubov's form

for the hard sphere model
for the Coulomb interaction
etc., cf. also 5.
8
Some results (D 2)
9
Conclusions
The described techniques allow to study the
interacting systems using the ideal system
formalism. Such an approach seems to be
utilizable for the description of diluted
weakly-interacting bosonic systems in traps. The
comparison of the predicted results with
experimental data is yet to be made.
10
References
• 1 L. Landau, E. Lifshitz, Statistical Physics,
Part 1.
• 2 V. Bagnato and D. Kleppner, Phys. Rev. A 44,
7439 (1991).
• 3 M. Li, L. Chen, and Ch. Chen, Phys. Rev. A
59, 3109 (1999).
• 4 Z. Yan, Phys. Rev. A 59, 4657 (1999).
• 5 A. A. Rovenchak, J. Low Temp. Phys. 138, 49
(2005).

11
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