Fock-Space Formulation of the Boltzmann Collision Operator for Maxwell Molecules

Abstract

We develop a representation-independent symmetric-Fock formulation of the nonlinear Boltzmann collision operator for Maxwell molecules. Starting from the standard velocity-space operator, we give a short derivation of Bobylev's Fourier identity and isolate its characteristic structure: two linearly related Fourier arguments followed by multiplication. We then prove, independently of collision geometry and of any coordinate realization, a canonical lift--fusion intertwining theorem: one-particle linear maps lift to the two incoming Fock legs and symmetric-algebra multiplication fuses them into a single outgoing state. Bargmann coordinates make this intrinsic construction especially transparent, but do not define it; Bobylev's collision maps subsequently select the Maxwell vertex from the already graded lift--fusion class. The resulting bilinear map obeys the exact grading \[ \widehat{\mathcal Q}(\mathscr H_p,\mathscr H_q)\subseteq\mathscr H_{p+q}, \] which gives an intrinsic origin for the triangular structures of Maxwell kinetics. Its one-vacuum restriction yields the complete Wang Chang--Uhlenbeck spectrum, while moment adaptation makes the nonequilibrium hierarchy strictly triangular and every finite compatible Fock jet an exact autonomous factor with global exponential relaxation. For the inhomogeneous problem, we extend realization covariance to the nonlinear two-leg vertex and evaluate the moving-frame connection independently through Hermite-function and Fourier realizations. The two routes give the same abstract Fock propagation and the same first Chapman--Enskog source; its traceless level-$2$ and vector level-$3$ sectors are relaxed by the exact Maxwell rates, giving $\Pr=2/3$. This provides an explicit computational demonstration that coordinate realizations may be chosen for economy without entering the structural Fock formulation.