A Deng–Hani–Ma-Guided Kinetic Program for Finite-Core Particles: Microscopic Dynamics, Retarded Disturbance Fields, and Fluid Limits

Abstract

The mathematical main line of this manuscript is the passage from enlarged finite-\(N\) dynamics to exact Liouville/BBGKY representations, collision-history cumulants, a Boltzmann-type mesoscopic equation, and finally model-specific fluid limits. The rigorous template is the hard-sphere theory of Yu Deng, Zaher Hani, and Xiao Ma: long-time cumulant propagation is organized by collision-history molecules, cutting algorithms, local analytic gains, and the removal of proof cutoffs. We reconstruct that template before examining which steps survive for finite-range cores, internal variables, retarded disturbance fields, baths, radiation, reactions, and boundaries. The enlarged dynamics differs from the hard-sphere model in two essential ways. Collisions may have finite duration, and the physical state may contain retained disturbance or history variables. These additions do not invalidate the Deng–Hani–Ma architecture, but they alter its elementary molecule integrals, ownership rules, bad sets, and limiting collision operator. A passive or decohering disturbance field can provide genuine propagator decay and suppress certain historical reconnections; it is an additional analytic mechanism, not a replacement for cumulant estimates or cutting. The hard-sphere branch is recovered only through a controlled reduction. The interaction duration and history-relaxation time must vanish relative to the collision time; collective history occupancy and coherent cross-packet terms must disappear; the centered collision correction must have vanishing predictable quadratic variation; and the remaining deterministic impulse must either vanish or be incorporated once into a limiting elastic spherical scattering map. Decoherence therefore selects effectively classical collision histories, while the zero-duration and scattering limits identify the particular hard-sphere dynamics. The exact finite-\(N\) flow, Liouville and BBGKY identities, conservative energy ledger, and several fixed-depth or short-time transfer modules are established for explicitly declared subclasses. The Euclidean and periodic hard-sphere results of Deng, Hani, and Ma supply the completed long-time reference template. For smooth finite-duration cores, we establish restart-compatible cutting identities and several local elementary estimates. Passive and propagating history models provide nontrivial convolution, externalization, martingale, and analytic-excess estimates. Finite mixtures, internal channels, reactions, radiation, baths, and fixed boundaries provide further transfer results for explicitly declared subclasses. Composite caustics, full molecule summation, time-slab propagation, and simultaneous removal of proof cutoffs remain open. These problems are mathematically difficult but conceptually localized: they concern composite Jacobian sublevels, decorated-molecule summation, propagation of a signed-cumulant radius, cutoff removal, and common microscopic–hydrodynamic diagonals. They do not require a new kinetic architecture, but rather completion of the Deng–Hani–Ma template for the enlarged state and verification that disturbance-field decay supplies sufficient analytic excess to dominate the additional reconstruction counts. The resulting framework provides a status-qualified microscopic foundation for downstream work on transport equations, fluid mechanics, thermodynamics, and continuum mechanics, without claiming a general global theorem for the full enlarged model class. Keywords **Boltzmann equation; Boltzmann–Grad limit; Deng–Hani–Ma theory; hard-sphere dynamics; finite-\(N\) dynamics; Liouville equation; BBGKY hierarchy; signed cumulants; collision-history molecules; cutting algorithms; finite-core particles; finite-duration collisions; retarded disturbance fields; memory effects; decoherence; analytic excess; hydrodynamic limits; transport equations; fluid mechanics; continuum mechanics.**