Mean-field limit and Vlasov equation for a Gaussian-thermostatted Lorentz gas
Abstract
We study N particles moving in a dispersing billiard under an electric field and coupled through a Gaussian thermostat. The corresponding mean-field equation is a Vlasov-type equation with specular reflection. We reformulate the equation as a fixed point problem for the time-integrated thermostat coefficient. This yields existence of weak solutions for every absolutely continuous initial datum of finite energy and, for bounded initial densities supported on velocities that are large compared with the field, uniqueness of the weak solution; under the finite horizon assumption, the upper bound on the velocities can be replaced by an exponential tail of the initial density. In these regimes, we prove convergence of the empirical measures of the particle system to the solution of the kinetic equation. In general, convergence holds along subsequences.
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BibTeX
@misc{castorrini2026meanfield,
title = {{Mean-field limit and Vlasov equation for a Gaussian-thermostatted Lorentz gas}},
author = {Roberto Castorrini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27992},
eprint = {2610.27992},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27992},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mean-field limit and Vlasov equation for a Gaussian-thermostatted Lorentz gas
AU - Roberto Castorrini
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27992
UR - https://arxiv.org/abs/2610.27992
AB - We study N particles moving in a dispersing billiard under an electric field and coupled through a Gaussian thermostat. The corresponding mean-field equation is a Vlasov-type equation with specular reflection. We reformulate the equation as a fixed point problem for the time-integrated thermostat coefficient. This yields existence of weak solutions for every absolutely continuous initial datum of finite energy and, for bounded initial densities supported on velocities that are large compared with the field, uniqueness of the weak solution; under the finite horizon assumption, the upper bound on the velocities can be replaced by an exponential tail of the initial density. In these regimes, we prove convergence of the empirical measures of the particle system to the solution of the kinetic equation. In general, convergence holds along subsequences.
ER -