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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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