Fock-Space Representation of the Lebowitz--Frisch--Helfand Kinetic Model

Abstract

We develop an exact Fock-space representation of the Lebowitz--Frisch--Helfand kinetic equation. The familiar square-root Maxwellian transformation is used only as a convenient starting point: it maps the local Ornstein--Uhlenbeck relaxation sector to the bosonic number operator and thereby exposes an elementary grading. The main issue is the representation of the complete kinetic dynamics when the velocity realization depends on local macroscopic parameters. We formulate the pull-back through an arbitrary admissible realization map, show that external space--time derivatives acquire a differential connection, and prove an intertwining theorem for first-order propagation operators. The physical velocity moments are represented by dual Fock-space functionals; a transport--moment intertwining theorem then evaluates the complete propagation contribution without requiring the explicit differential connection. Compatibility between realization parameters and kinetic moments may be imposed algebraically or propagated dynamically by exact balance laws. In the hydrodynamic limit, the number grading selects the second and third Fock levels responsible for viscous stress and heat flux, leading directly to the Navier--Stokes--Fourier constitutive terms. Finally, we prove covariance under local changes of coordinate realization and illustrate it explicitly for Hermite-polynomial and Hermite-function coordinates. Thus the Hermite realization is computationally privileged, while the intertwined Fock dynamics, physical moments, and compatibility structure are representation-independent within the admissible similarity class.