Modified scattering for semiclassical Bose-Fermi mixtures
Abstract
The Vlasov-Hartree system models a mixture of bosons and fermions interacting via Coulomb forces in which the bosons are described quantum-mechanically, while the fermions are described classically. For small initial data, we prove that the system exhibits modified scattering, i.e. the bosonic and fermionic subsystems each converge to solutions of the free Schrödinger and Vlasov equations respectively, but with coupled logarithmic corrections owing to the long-range nature of Coulomb interactions. We use a mixed Lagrangian and Fourier-based approach to derive explicit descriptions of the asymptotic dynamics of the solution and associated fields and Lagrangian trajectories. Notably, due to our purely Lagrangian analysis on the Vlasov side, we require no derivatives on the initial fermion density, the asymptotic fermionic profile is as regular as the initial data, and convergence occurs in almost the same space as the initial data.
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BibTeX
@misc{cavanagh2026modified,
title = {{Modified scattering for semiclassical Bose-Fermi mixtures}},
author = {Kieran Cavanagh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.06143},
eprint = {2609.06143},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06143},
}
RIS
TY - UNPB
TI - Modified scattering for semiclassical Bose-Fermi mixtures
AU - Kieran Cavanagh
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.06143
UR - https://arxiv.org/abs/2609.06143
AB - The Vlasov-Hartree system models a mixture of bosons and fermions interacting via Coulomb forces in which the bosons are described quantum-mechanically, while the fermions are described classically. For small initial data, we prove that the system exhibits modified scattering, i.e. the bosonic and fermionic subsystems each converge to solutions of the free Schrödinger and Vlasov equations respectively, but with coupled logarithmic corrections owing to the long-range nature of Coulomb interactions. We use a mixed Lagrangian and Fourier-based approach to derive explicit descriptions of the asymptotic dynamics of the solution and associated fields and Lagrangian trajectories. Notably, due to our purely Lagrangian analysis on the Vlasov side, we require no derivatives on the initial fermion density, the asymptotic fermionic profile is as regular as the initial data, and convergence occurs in almost the same space as the initial data.
ER -