Analytic derivation of the gradient-dependent deviation time

Establish analytically the relation between the deviation time of finite-gradient moment-ratio attractors and the transverse wave number, including the range of validity and subleading corrections to the observed scaling \(\exp(\tau_D/\tau_R)\propto (k\tau_R)^{-(m-1)}\), equivalently \(\tau_D/\tau_R\simeq (m-1)\ln[1/(k\tau_R)]\), for the spherical-harmonic sectors with index \(m\).

Background

For finite transverse wave number $k$, gradient-induced couplings mix neighboring azimuthal spherical-harmonic sectors. A moment ratio in a non-hydrodynamic sector therefore first approaches the constant attractor of the corresponding zero-gradient sector and later crosses over to the global hydrodynamic attractor. The crossover or deviation time $\tau_D$ decreases as the gradient strength increases and grows for sectors with larger $m$, because coupling them to the hydrodynamic sectors requires more successive nearest-neighbor transitions.

Numerical results for the ratios $L_{m+4,m}/L_{m,m}$ with $m=2,3,4$ give exponents close to $m-1$ in the small-gradient scaling $\exp(\tau_D/\tau_R)\propto(k\tau_R)^{-a_m}$. The paper attributes this pattern qualitatively to the leading mixing amplitude being proportional to $k^{m-1}$, but leaves an analytic derivation, the precise domain of applicability, and subleading corrections unresolved.