Two-level system dynamical coarse-graining of annual rings

The tree annual ring is defined and geometrically described by two new physical quantities designated as golden ring cube (GRC) and golden ring volume (GRV). The quantities were used to derive a two-level system dynamical coarse-graining framework for annual rings by applying non-equilibrium equality of free energy differences of dynamical repetitive transitions from earlywood to latewood and vice versa. The scaling properties of growth rings interaction parameter β, GRC and GRV were then numerically quantified and tested for Douglas-fir and white spruce. Although parameter β was different for Douglas-fir and white spruce, its invariant property was tested and found to be the same for both species. The parameter β was thereafter used to derive a novel heterogeneity test criterion that was described and tested. β is an important parameter that provides integrated information for the dependence of the GRV and GRC on species, growth factors, and environmental interaction signals.