Bounded cumulative observables from local linear relaxation

TL;DR

This paper demonstrates that bounded cumulative responses in systems with local relaxation are fundamentally due to exponential relaxation, leading to universal saturation scales regardless of system specifics.

Contribution

It establishes that local linear relaxation inherently causes saturation of cumulative observables, independent of geometry or transport mechanisms.

Findings

Cumulative observables are limited by a scale set by relaxation time.

Saturation is independent of system geometry, dimensionality, or microscopic transport.

Temporal bounds translate into spatial saturation scales via transport laws.

Abstract

Cumulative observables often exhibit saturation in systems involving propagation or spreading with local dissipation. This work shows that bounded cumulative response follows directly from local linear relaxation. Linear cumulative observables accumulated over the lifetime of a relaxing signal are limited by a scale set by the relaxation time, independent of geometry, dimensionality, or microscopic transport dynamics. When relaxation is mapped to space through transport or spreading, this temporal bound yields a corresponding spatial saturation scale determined by the transport law. The result shows that cumulative saturation follows directly from exponential local relaxation and does not depend on the specific transport mechanism.