Influence of an environment changing in time on crucial events: From geophysics to biology

Abstract

This paper is devoted to the study of the interaction between two distinct forms of non-stationary processes, which we will refer to as non-stationarity of the first and second kind. The non-stationarity of the first kind is caused by criticality-generated events that we call crucial events. Crucial events signal ergodicity breaking emerging from the interaction between the units of the complex system under study, indicating that the non stationarity of first kind has an internal origin. The non-stationarity of second kind is due to the influence on the system of interest of an environment changing in time, thereby implying an external origin. In this paper we show that the non-stationarity of first kind, measured by an inverse power law index $\mu$ is characterized by singularities at $\mu =2$ and $\mu =3$. We realize the interaction between the non-stationarity of first kind and the non-stationarity of second kind with a model frequently adopted to study earthquakes, namely, a system of main-shocks assumed to be crucial events, generating a cascade of after-shocks simulating the changing in time environment. We prove that the after-shocks significantly affect the detection of anomalous scaling, but in the case $\mu =2.5$, which is sufficiently far from $\mu =2$ and $\mu =3$, both sources of singularities being strongly affected by the non-stationarity of second kind. To explain why it is possible to detect for earthquakes the value $\mu =2.06$, proposed in earlier work and very close to the singularity $\mu =2$, we advocate a new theoretical perspective, involving a deviation from the traditional concept of criticality in physics and borrowing suggestions from biology. This new approach is based on the truncation of inverse power laws that has the effect of shifting the intermediate asymptotics to short-time region, without weakening their role for information transmission. This leads us to the conclusion that the whole planet is a living system.