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Wave Turbulence pp 231–268Cite as

Bose-Einstein Condensation

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Part of the book series: Lecture Notes in Physics ((LNP,volume 825))

Abstract

In this Chapter, Wave Turbulence (WT) in Bose-Einstein condensates (BEC) and nonlinear optics is considered. Models considered include 2D and 3D Gross-Pitaevskii equations, Nordheim quantum kinetic equation, Boltzmann equation and 1D nonlinear equation for optical WT in nematic liquid crystals. Roles of the thermodynamic Rayleigh- Jeans, cascading Kolmogorov-Zakharov, mixed warm cascade and critical balance states are discussed. A picture of WT life cycle where a coherent uniform condensate component arises from an inverse cascade process. In this setting the WT starts as a four-wave process followed by breakdown of weakly nonlinear description and creation of a gas of strongly nonlinear vortices which undergo annihilations resulting in a vortex-free coherent condensate with random acoustic Bogoliubov waves engaged in mutual three-wave interactions

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Notes

  1. 1.

    Paper [7] filtered out high k’s at the data processing, but not at the computational stage. Obviously, this could not prevent the backscatter/bottleneck effect discussed here.

  2. 2.

    To be precise, this is a special case of the Boltzmann equation for rigid-sphere particles.

  3. 3.

    The opposite statement about the negative energy flux was made in [17] based on the spectrum \(n=({c}/{\omega}) \ln^{1/3} ({ \omega} )\) thus effectively assuming an IR (rather than an UV) cutoff, which is unnatural because one should not assume a zero spectrum at the forcing scale.

  4. 4.

    The normal component in Helium is representing, on a course-grained level, motion of the gas of phonons.

  5. 5.

    In 1D NLS the solitons pass each other almost freely,—without change of their amplitudes or shapes but with small shifts of the original trajectories.

  6. 6.

    Index x = 0 corresponds to a formal solution with particle equipartition in the momentum space. However, this solution is not normalizable.

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  1. Inst. Mathematics, University of Warwick, Gibbet Hill Road, CV4 7AL, Coventry, UK

    Sergey V. Nazarenko

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Nazarenko, S.V. (2011). Bose-Einstein Condensation. In: Wave Turbulence. Lecture Notes in Physics, vol 825. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15942-8_15

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  • DOI: https://doi.org/10.1007/978-3-642-15942-8_15

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