Operator-Based Forward Sensitivity of Transient Interfacial Fluxes Under Localized Relaxation Contrasts

Abstract

We investigate the transient interfacial-flux response of a spatially localized relaxation contrast embedded in a diffusive dissipative medium. A discrete operator formulation is combined with a forward-sensitivity expansion in the relaxation contrast, allowing the macroscopic response to be computed deterministically through first-, second-, and third-order Taylor sensitivity coefficients rather than through phenomenological fitting. For a reference configuration with $N=800$, $R=10$, $\kappa=0.1$, and $\lambda_{\mathrm{out}}=0.5$ evaluated at the reference observation time $T=0.6$, the resulting sensitivity coefficients are $j_1 = 0.315299$, $j_2 = -0.094110$, and $j_3 = 0.018616$. Direct comparison with full nonlinear simulations of the same discrete time-stepping scheme shows that the third-order perturbative prediction reproduces the measured response with a residual scaling consistent with $\vert{}\Delta\lambda\vert{}^4$ over the investigated small-contrast regime. The normalized higher-order coefficients exhibit an emergent temporal structure with $Q_2$ and $Q_3$ remaining close to the exponential Taylor values $1/2$ and $1/6$, while exhibiting systematic temporal drift. We further identify an optimal sensitivity window, $T_\ast \approx 0.469$, at which the leading-order interfacial response is maximal. This optimum and the corresponding leading sensitivity are approximately invariant under moderate radius variations at fixed interface width, while their coupled dependence on diffusion and background relaxation demonstrates that no single universal diffusion-time scaling captures the full dynamics. These results establish an operator-based framework for quantitatively connecting localized microscopic relaxation anomalies to transient macroscopic interface measurements, while explicitly distinguishing exact operator-level relations from emergent effective scaling laws.