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PDE Models for Quantum Fluids

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Workshop
13th December 2010 to 17th December 2010

Organisers: Professor S Jin (Wisconsin) and Professor PA Markowich (Cambridge).

Workshop Theme

Quantum fluid modelling has attracted a lot of recent attention, mainly due to the exiting experimental discoveries in Bose-Einstein condensation of the last decade which may have possible applications in quantum computing. Typically, modelling is based on nonlinear Schrodinger equations, like the cubically nonlinear Gross-Pitaevskii equation describing the evolution of Bose-Einstein condensates. Topics of interest include vortex dynamics and vortex pattern, mixing of scales in random phase approximations (related to turbulence theory), lattice condensates and multi-component condensates modelled by coupled systems of nonlinear Schrodinger equations. Kinetic equations, relying mainly on the Wigner transformation, can also be applied to quantum fluid dynamics. Recently hybrid kinetic-quantum mechanical models, coupling the Gross-Pitaevskii equation to the boson Boltzmann equation also have become a topic of strong interest. Many deep and open mathematical problems arise in connection with the boson Boltzmann equation in the spatially inhomogeneous case, due to the weak growth of the entropy functional which may even permit the occurence of singularities in finite time. New efficient numerical methods for quantum Boltzmann equations, and their validity in the quantum hydrodynamic regimes, accurate simulations crossing the regimes from the Gross-Pitaevski equation to the quantum Boltzmann equations, efficient Bloch decomposition based numerical methods for quantum dynamics in periodic and random media, are also important research directions to be pursued in this programme.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons