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We consider solutions of the repulsive Vlasov–Poisson system which are a combination of a point charge and a small gas, i.e., measures of the form\delta_{(\mathcal{X}(t),\mathcal{V}(t))}+\mu^{2}d\mathbf{x}d\mathbf{v}for some(\mathcal{X}, \mathcal{V})\colon \mathbb{R}\to\mathbb{R}^{6}and a small gas distribution\mu\colon \mathbb{R}\to L^{2}_{\mathbf{x},\mathbf{v}}, and study asymptotic dynamics in the associated initial value problem. If initially suitable moments on\mu_{0}=\mu(t=0)are small, we obtain a global solution of the above form, and the electric field generated by the gas distribution \mudecays at an almost optimal rate. Assuming in addition boundedness of suitable derivatives of \mu_{0}, the electric field decays at an optimal rate, and we derive modified scattering dynamics for the motion of the point charge and the gas distribution. Our proof makes crucial use of the Hamiltonian structure. The linearized system is transport by the Kepler ODE, which we integrate exactly through an asymptotic action-angle transformation. Thanks to a precise understanding of the associated kinematics, moment and derivative control is achieved via a bootstrap analysis that relies on the decay of the electric field associated to\mu. The asymptotic behavior can then be deduced from the properties of Poisson brackets in asymptotic action coordinates.
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Phonon redshift and Hubble friction in an expanding BEC
We revisit the theoretical analysis of an expanding ring-shapedBose-Einstein condensate. Starting from the action and integrating overdimensions orthogonal to the phonon’s direction of travel, we derive aneffective one-dimensional wave equation for azimuthally-travellingphonons. This wave equation shows that expansion redshifts the phononfrequency at a rate determined by the effective azimuthal sound speed,and damps the amplitude of the phonons at a rate given by \dot{\mathcal{V}}/{\mathcal{V}} 𝒱 ̇ / 𝒱 ,where \mathcal{V} 𝒱 is the volume of the background condensate. This behavior is analogousto the redshifting and ``Hubble friction’’ for quantum fields in theexpanding universe and, given the scalings with radius determined by theshape of the ring potential, is consistent with recent experimental andtheoretical results. The action-based dimensional reduction methods usedhere should be applicable in a variety of settings, and are well suitedfor systematic perturbation expansions.
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- PAR ID:
- 10311211
- Date Published:
- Journal Name:
- SciPost Physics
- Volume:
- 10
- Issue:
- 3
- ISSN:
- 2542-4653
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.21468/SciPostPhys.10.3.064
