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Friction in a model of Hamiltonian dynamics. (English) Zbl 1263.82033

This paper is a presentation of a motion study of a heavy tracer particle weakly coupled to a dense ideal Bose gas exhibiting Bose-Einstein condensation. The dynamics of this system approaches one determined by nonlinear Hamiltonian evolution equations describing a process of emission of Cerenkov radiation of sound waves into the Bose-Einstein condensate along the particle’s trajectory. The emission of Cerenkov radiation results in a friction force with memory acting on the tracer particle and causing it to decelerate until it comes to rest.
The problem of constructing the effective dynamics of a particle coupled to dispersive reservoirs has been studied in the literature. In this paper, the authors introduced a family of quantum mechanical models describing tracer particles moving through a Bose gas exhibiting Bose-Einstein condensation. They define a regime in which the dynamics of this system approaches one governed by classical Hamiltonian evolution equations. This problem is presented in successive sections of this paper. Section 2 shows the paper’s main results. In Section 3, a theoretical approach is given that allows for an equation containing a linear part and a higher-order nonlinear part. This linearized equation is carefully studied in Section 4. The proof of the main result is completed in Section 5. There are also five appendices that present other auxiliary technical results used in this paper.

MSC:

82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
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References:

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[5] Fröhlich, J., Sigal, I.M., Soffer, A., Gang, Z.: In preparation
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