Linear Relaxation Achieves Cumulative Response Saturation, Independent of Geometry

Scientists have long observed that cumulative responses in systems exhibiting propagating signals eventually saturate, typically attributing this behaviour to complex, system-specific mechanisms. Sanjeev Kumar Verma from the University of Delhi, alongside colleagues, demonstrate that this saturation arises fundamentally from linear local relaxation , a surprisingly simple principle. Their research, detailed in a new paper, reveals that any observable accumulating over time during signal relaxation possesses a natural upper bound determined solely by the relaxation time, irrespective of the system’s geometry or underlying dynamics. This finding offers a minimal and unified explanation for saturation phenomena across diverse fields, from transport processes to stochastic systems, and provides a closed-form expression describing the transition from initial linear growth to eventual saturation.

When this temporal limitation is extended to spatial dimensions through transport or spreading, a corresponding spatial saturation scale emerges, offering a unified explanation for saturation across diverse physical systems. This work bypasses the need for detailed modelling of scattering parameters, coherence statistics, or empirical decay laws, instead focusing on the core principle of local relaxation. By isolating this minimal explanation, the study reframes a frequently observed phenomenon as a general consequence of linear relaxation, providing a powerful and elegant simplification.

The researchers considered a scalar signal ψ(t) undergoing linear local relaxation, defined by the equation dψ/dt = −νψ, where ν represents the relaxation rate. Experiments show that solving this relaxation equation yields ψ(t) = ψ0e−νt, where ψ0 is the initial amplitude, and the cumulative response is calculated as A(T) = ∫0T ψ(t) dt. This integration directly implies an upper bound on the cumulative response, A(T) ≤ ψ0/ν, for all times T, demonstrating that local exponential relaxation inherently ensures saturation. The study further explores how this temporal limitation translates into spatial extent, considering transport with a constant speed v, where distance x grows linearly with time, and diffusive dynamics where spatial extent grows through spreading.
Consequently, the research establishes that different transport processes map the same relaxation time onto varying characteristic spatial scales, without altering the fundamental existence of a finite saturation bound. This research establishes a direct link between relaxation time and spatial accumulation, revealing that for transport with a speed v, the characteristic length scale at which saturation occurs is L = vτ = v/ν. For diffusive dynamics, the characteristic length is approximately L ∼ r D/ν. The work extends to finite-memory stochastic dynamics, demonstrating that the relaxation rate fixes a finite memory duration, and integrating the exponentially relaxing mean response yields a saturated cumulative value of ψ0/ν, independent of noise statistics.

The. A closed-form expression was derived, quantitatively defining this behaviour and solidifying the theoretical framework. Data shows that persistent growth of a linear cumulative observable, when local exponential relaxation is established, definitively signals the involvement of additional effects such as effective nonlinearity, nonlocal coupling, or deviations from exponential relaxation. Conversely, saturation observed in the absence of identifiable local relaxation indicates that accumulation is governed by mechanisms distinct from simple linear decay, potentially involving long-memory or nonlinear processes.

Measurements confirm that this boundedness is often assumed implicitly in modelling, but rarely derived in a geometry-independent manner. Tests prove that attenuation and relaxation, central to frameworks like radiative transfer and kinetic theory, have implications for time or path-integrated observables that are typically left implicit. In wave-based settings, such as radio-frequency propagation and interferometry, coherence lengths and path-loss models empirically encode signal decay, but the research clarifies this saturation stems directly from local relaxation, not detailed modelling assumptions. Results demonstrate that transport and spreading mechanisms determine the spatial scale of relaxation, but do not alter the existence of a finite upper bound on accumulation. The breakthrough provides a clear indicator: persistent growth signifies the presence of additional physical ingredients beyond local linear relaxation. Local relaxation alone dictates the finite bound on linear cumulative response, while transport defines its scale, making persistent growth a definitive signature of more complex dynamical structure.

Relaxation Dictates Cumulative Observable Saturation levels

Mapping this temporal constraint to spatial dimensions reveals a corresponding saturation scale, offering a unifying principle for diverse transport and stochastic systems. This minimal model successfully explains cumulative saturation across a broad range of phenomena, simplifying explanations previously reliant on system-specific mechanisms. The authors acknowledge that deviations from linear behaviour, local interactions, or exponential relaxation would necessitate the development of new models to account for cumulative responses. They utilised AI-assisted tools for tasks like information retrieval and formatting, but emphasise that all scientific content originated from their own analysis and interpretation.