### Abstract

A linear Boltzmann equation with non-autonomous collision operator is rigorously derived in the Boltzmann-Grad limit for the deterministic dynamics of a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with heterogeneously distributed background particles, which do not interact among each other. The validity of the linear Boltzmann equation holds for arbitrary long times under moderate assumptions on spatial continuity and higher moments of the initial distributions of the tagged particle and the heterogeneous, non-equilibrium distribution of the background. The empiric particle dynamics are compared to the Boltzmann dynamics using evolution systems for Kolmogorov equations of associated probability measures on collision histories.

Original language | English |
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Pages (from-to) | 3299-3355 |

Number of pages | 57 |

Journal | Discrete and Continuous Dynamical Systems - Series A |

Volume | 38 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Jul 2018 |

### Fingerprint

### Keywords

- Boltzmann equation
- Derivation
- Evolution system
- Rayleigh gas.
- Semigroup

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

**Derivation of a Nonautonomous Linear Boltzmann Equation from a Heterogeneous Rayleigh Gas.** / Matthies, Karsten; Stone, George.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems - Series A*, vol. 38, no. 7, pp. 3299-3355. https://doi.org/10.3934/dcds.2018143, https://doi.org/10.3934/dcds.2018143

}

TY - JOUR

T1 - Derivation of a Nonautonomous Linear Boltzmann Equation from a Heterogeneous Rayleigh Gas

AU - Matthies, Karsten

AU - Stone, George

PY - 2018/7/1

Y1 - 2018/7/1

N2 - A linear Boltzmann equation with non-autonomous collision operator is rigorously derived in the Boltzmann-Grad limit for the deterministic dynamics of a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with heterogeneously distributed background particles, which do not interact among each other. The validity of the linear Boltzmann equation holds for arbitrary long times under moderate assumptions on spatial continuity and higher moments of the initial distributions of the tagged particle and the heterogeneous, non-equilibrium distribution of the background. The empiric particle dynamics are compared to the Boltzmann dynamics using evolution systems for Kolmogorov equations of associated probability measures on collision histories.

AB - A linear Boltzmann equation with non-autonomous collision operator is rigorously derived in the Boltzmann-Grad limit for the deterministic dynamics of a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with heterogeneously distributed background particles, which do not interact among each other. The validity of the linear Boltzmann equation holds for arbitrary long times under moderate assumptions on spatial continuity and higher moments of the initial distributions of the tagged particle and the heterogeneous, non-equilibrium distribution of the background. The empiric particle dynamics are compared to the Boltzmann dynamics using evolution systems for Kolmogorov equations of associated probability measures on collision histories.

KW - Boltzmann equation

KW - Derivation

KW - Evolution system

KW - Rayleigh gas.

KW - Semigroup

UR - http://www.scopus.com/inward/record.url?scp=85046376300&partnerID=8YFLogxK

U2 - 10.3934/dcds.2018143

DO - 10.3934/dcds.2018143

M3 - Article

VL - 38

SP - 3299

EP - 3355

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 7

ER -