A saturation bound for cumulative responses under local linear relaxation

TL;DR

The paper demonstrates that saturation of linear cumulative observables in systems with propagating or spreading signals is a direct consequence of local linear relaxation, independent of geometry or microscopic dynamics. Assuming exponential decay $\psi(t)=\psi_0 e^{-\nu t}$ and a linear cumulative readout $\mathcal{A}(T)=\int_0^T \psi(t)\,dt$, it derives the exact bound $\mathcal{A}(T)\le \psi_0/\nu$ and the closed-form $\mathcal{A}(T)=\frac{\psi_0}{\nu}(1-e^{-\nu T})$, revealing a two-regime behavior: linear growth at short times and saturation beyond $\tau=1/\nu$. The temporal bound maps to spatial saturation via a time–distance mapping $t_{\mathrm{eff}}(x)$ set by the transport mechanism, yielding transport-specific saturation lengths such as $L=v/\nu$ for constant-speed transport and $L=\sqrt{D/\nu}$ for diffusion, with corresponding saturated readouts. This provides a minimal, unified explanation for cumulative saturation across transport, diffusion, and stochastic systems and offers a diagnostic: persistent linear growth indicates additional dynamical ingredients beyond local exponential relaxation.

Abstract

Saturation of cumulative observables is widely observed in systems with propagating or spreading signals and is commonly modeled using system-specific mechanisms such as scattering statistics, coherence functions, or phenomenological decay laws. This work shows that such saturation follows directly from linear local relaxation alone. Any linear observable accumulated over the lifetime of a relaxing signal is bounded by a scale set by the relaxation time, independent of geometry, dimensionality, or microscopic dynamics. When relaxation is mapped to space through transport or spreading, this temporal bound yields a corresponding spatial saturation scale. A closed-form expression reveals a two-regime behavior: linear growth at short times followed by saturation beyond the relaxation time. The result provides a minimal and unified explanation for cumulative saturation across transport, diffusive, and stochastic systems.