A spectral model of power-law decay in natural and engineered systems

We present a first-principles spectral mechanism for the emergence of nonextensive $q$-exponential dilution and power-law relaxation in non-ideal transport systems. By modeling an incompletely mixed reactor as a layered diffusion matrix with an absorbing boundary, we demonstrate that macroscopic power-law tails depend on the geometric interaction between the initial tracer placement and the domain's boundary configuration. For a one-dimensional system, an asymmetric, volumetrically distributed initial concentration profile projects onto the low-wavenumber eigenmodes, generating an emergent Gamma distribution of relaxation rates; at an infinitesimal boundary layer thickness ($\Delta z \to 0$), this profile yields the nonextensive $q$-exponential decay function exactly across the entire temporal domain with $q = 5/3$. Extended to $d$ dimensions under a highly localized, boundary-adjacent singular initial condition, the resulting scaling exponents and corresponding $q$ values depend explicitly on the spatial configuration of the absorbing boundaries. However, in the one-dimensional limit ($d=1$), these distinct initial states and boundary formulations intersect, rendering the $q=5/3$ exponent geometrically invariant. Our approach establishes a clear connection between linear diffusion transport and nonextensive statistical mechanics, showing how heavy-tailed transport can be derived from boundary geometry and spectral dimensionality.