Relate continuum coefficients to slope distribution in oscillator lattices

Determine whether, and in what quantitative form, the continuum coefficients ν and λ in the effective Kardar–Parisi–Zhang/Edwards–Wilkinson equation for the phase field φ(x,t) of a one-dimensional oscillator lattice (∂tφ = η* + ν∇2φ + (λ/2)(∇φ)2) can be expressed in terms of statistical moments of the distribution of nearest-neighbor phase differences (local slopes) under time-dependent noise.

Background

In the time-dependent noise setting, the authors analyze the local slope distribution and note it stabilizes early during the growth regime, paralleling observations in an etching model within the KPZ universality class where macroscopic parameters appear linked to slope statistics.

Motivated by these parallels, they explicitly raise whether the parameters of the continuum approximation for synchronizing oscillators—derived in Section 2.2—could be related to moments (e.g., variance or higher moments) of the slope distribution measured at the lattice level.