Covariant Diffusion Equations under Velocity Suppression and Poisson Microstructure

Abstract

We investigate the combined effects of the statistical organization of particle velocities and local occupancy statistics on macroscopic transport, together with a Schr\"odinger-type representation of the resulting transport system. Starting from a model of equal-mass point particles in unbounded three-dimensional space with independent Gaussian velocity statistics, we prove that, in conditional comparisons at fixed particle number and realized total kinetic energy, increasing the group velocity contracts the residual-speed distribution by a determined scale factor while preserving the conditional isotropy of residual motion. Under independent reference velocity processes and common temporal-correlation conditions, averaging the velocities of a finite number of particles yields a $k^{-1/2}$ speed scale and a $1/k$ diffusion response. Combining these results with an instantaneous Poisson occupancy closure and weighting by elementary-particle load gives the exact occupancy-suppression function $B(\lambda)=(1-e^{-\lambda})/\lambda$, where $\lambda$ is the mean occupancy of a small lattice cell. Under common channel means, a local reciprocal-exchange constitutive law, and an effective momentum closure, we incorporate group-velocity suppression, $\chi=1-|\boldsymbol U|^2/C^2$, and Poisson occupancy suppression into channel exchange to derive coupled density--reduced-momentum transport equations that retain directed momentum advection. Here $\boldsymbol U$ is the group velocity and $C$ is the fixed root-mean-square speed budget. Retaining all spatial derivatives of the suppression coefficients, as well as changes in volume and basis vectors, we establish the spatially covariant conservation form of the system with a single absolute time. On the same suppression background, Nelson-type forward--backward statistics, supplemented by an explicit orientation-dynamics condition, yield a variable-coefficient modified Schr\"odinger equation with diffusion coefficient $d_N=\hbar\chi B(\lambda)/(2m)$. We also obtain the corresponding inverse statistical representation and conditions for orientation reconstruction. Finally, we construct a complete three-dimensional comoving geometry from local planes spanned by the momentum-density vectors at neighboring points and introduce a determined full-field amplitude--orientation transformation in a fixed finite-cell realization. Once the source dynamics, quantum target, and initial calibration are fixed, we prove that the compatibility set is preserved under evolution. This establishes a one-to-one correspondence, on a common regular time interval, between the original transport system and its complete comoving representation containing the modified Schr\"odinger readout. The correspondence permits determined changes in readout amplitude while ensuring unique reconstruction of the original state through the retained density, three-dimensional geometry, and calibration data. Under the stated statistical, constitutive, and orientation-dynamics conditions, these results connect doubly suppressed transport, spatial covariance, and reversible quantum representation within a single mathematical framework.