We derive the relativistic Vlasov equation from quantum Hartree dynamics for fermions with relativistic dispersion in the mean-field scaling, which is naturally linked with an effective semiclassic limit. Similar results in the non-relativistic setting have been recently obtained in Benedikter et al. (Arch Rat Mech Anal 221(1): 273–334, 2016). The new challenge that we have to face here, in the relativistic setting, consists in controlling the difference between the quantum kinetic energy and the relativistic transport term appearing in the Vlasov equation.
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We would like to thank the two anonymous referees for their useful comments that allowed us to improve the presentation of our paper. B.S. gratefully acknowledges support from the NCCR SwissMAP and from the Swiss National Foundation of Science through the SNF Grant “Effective equations from quantum dynamics” and the SNF Grant “Dynamical and energetic properties of Bose–Einstein condensates”.
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Dietler, E., Rademacher, S. & Schlein, B. From Hartree Dynamics to the Relativistic Vlasov Equation. J Stat Phys 172, 398–433 (2018). https://doi.org/10.1007/s10955-018-1973-5
- Hartree theory
- Relativistic Vlasov equation
- Many-body quantum mechanics