Hayashi–Navier–Stokes Formulation: A Unified Description of Higher Angular Modes, Unresolved Degrees of Freedom, and Memory in Kinetic Hydrodynamics
Abstract
This preprint reformulates the Navier–Stokes viscosity as the effective response that remains after coarse-graining the higher angular modes and unresolved degrees of freedom of the molecular velocity distribution. Using a spherical-harmonic expansion in velocity space and a P/Q projection (P = Pl≤4, Q = I − P), eliminating the Q sector gives the Schur complement MHNS = MP − C MQ−1 D. The correction appears in the time domain as a memory kernel (Mori–Zwanzig form): unresolved degrees of freedom return to the resolved side as a non-local response. In a single-speed relaxation-time angular model (not the full Maxwellian BGK model), the first intrinsic correction from l = 4 is (η4 − η3)/η0 = (8/147) q4 + O(q6), and the first correction from the unresolved sector l ≥ 5 is (ηfull − η4)/η0 = −(64/4851) q6 + O(q8), where q = kλ is a dimensionless wavenumber. The lowest order at which an angular mode l = L contributes to the l = 2 viscous response is q2(L−2) (the round-trip order rule), proved here for the tridiagonal model. Eliminating vorticity-diffusion modes produces a √s-type non-analytic response consistent with the known hydrodynamic memory (Stokes drag, Basset history force and added mass in the unsteady Stokes problem). HNS does not replace the Navier–Stokes equations. It recovers the Navier–Stokes constitutive law in the long-wavelength, low-frequency limit and makes the higher angular degrees of freedom, the unresolved sector and the memory response explicit. The claimed novelty is the unified organization, not the discovery of higher-moment kinetic theory, projection-operator methods or hydrodynamic memory themselves; the priority relative to the existing moment, spectral-closure and non-local hydrodynamics literature remains to be checked. 日本語要旨: Navier–Stokes 粘性を、速度分布の高次角モードと未解像自由度を粗視化した後に残る有効応答として再構成する。球面調和展開と P/Q 射影により Q sector を消去すると Schur 補完が得られ、その補正項は時間領域でメモリー核として現れる。固定速度の角度模型では l=4 の固有補正 (8/147) q4 と、l≥5 の未解像 sector の最初の補正 −(64/4851) q6 を示し、高次角モード l=L の寄与が現れる最低次数 q2(L−2)(往復次数則)を三重対角模型で証明する。HNS は Navier–Stokes 方程式を置換するものではなく、長波長・低周波の極限でその構成則を回収し、高次角自由度・未解像 sector・メモリー応答を明示する拡張的な closure の定式化である。