A geometric derivation of the linear Boltzmann equation for a particle interacting with a Gaussian random field

Sébastien Breteaux (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

The linear Boltzmann equation is derived from a microscopic quantum model of a particle interacting, in the weak coupling limit, with a translation invariant and centered Gaussian random field. The result holds in dimension $d\geq 3$ for initial states given as a pure families of states, in the sense of semiclassical measures and an assumption about renewing the stochastic potential is needed in order to force the asymptotic Markovian property, unlike in the works of Erdös and Yau. This derivation uses an isomorphism between the Hilbert space of square integrable function w.r.t. the random parameter and the bosonic Fock space over $L^{2}(\mathbb{R}^{d})$ . The solutions for the microscopic model are approximately coherent states with a parameter in the phase space, providing a geometric viewpoint.

Mots clés: Processes in random environments; Quantum field theory; Coherent states; Kinetic theory of gases

Domaines: Physique mathématique [math-ph]

Sébastien Breteaux. A geometric derivation of the linear Boltzmann equation for a particle interacting with a Gaussian random field. 2011. ⟨hal-00605919v1⟩