Hierarchical Log-Gaussian Relaxation on a Fixed D3Q125 Velocity Set

We develop a hierarchical order-resolved relaxation model for a fixed D3Q125 discrete-velocity kinetic formulation. Conventional adaptive collision models often use one scalar rarefaction or nonequilibrium indicator for all retained moment orders, thereby coupling distinct kinetic sectors. Here, a shared macroscopic-gradient background is combined separately with second-, third-, and fourth-order thermodynamic nonequilibrium indicators to define effective measures K2, K3, and K4, each driving its own log-Gaussian relaxation spectrum.
Pure-order perturbation tests verify selective activation, with nonmatching sectors remaining at roundoff level. Homogeneous mixed-order, amplitude, and composition tests show lower residual nonequilibrium than a common-sensor model while preserving positive populations. In a smooth periodic compression wave at the stated reference discretization and in the TNE-only sensor limit, the peak total nonequilibrium intensity is reduced by 6.565%, with reductions throughout the domain and in all retained moment sectors.
Additional timestep, transport-discretization, relaxation-spectrum, uniform-boost, long-time, and shear-wave studies show that the sign of the hierarchical correction is robust over the tested configurations, while its magnitude depends on timestep, transport scheme, sensor frame, and relaxation spectrum. The periodic benchmarks preserve the principal global invariants to floating-point accuracy and remain positive. These results establish the mechanism and numerical behavior of order-resolved activation on a fixed velocity set; independent kinetic-reference validation is still required before claiming universal accuracy improvement.