Information flow and causality as rigorous notions ab initio

TL;DR

This paper rigorously derives the concept of information flow from first principles, establishing its connection to causality and providing explicit formulas for various dynamical systems, thereby clarifying the causation-correlation relationship.

Contribution

It introduces a rigorous, first-principles derivation of information flow and causality, with explicit formulas applicable to deterministic and stochastic systems of arbitrary dimension.

Findings

Information flow is explicitly derived and shown to obey causality.

In linear systems, causation implies correlation, but not vice versa.

Applications to benchmark systems demonstrate the tractability of the information flow structure.

Abstract

Information flow (or information transfer as may be called) the widely applicable general physics notion can be rigorously derived from first principles, rather than axiomatically proposed as an ansatz. Its logical association with causality and, particularly, the most stringent one-way causality, if existing, is firmly substantiated and stated as a fact in proved theorems. Established in this study are the information flows among the components of time-discrete mappings and time-continuous dynamical systems of arbitrary dimensionality, both deterministic and stochastic. They have been obtained explicitly in closed form, and all possess the property of causality, which reads: if a component, say $x_i$, has an evolutionary law independent of $x_j$, then the information flow from $x_j$ to $x_i$ vanishes. These results have been put to applications with benchmark systems, such as the Kaplan-Yorke map, the R\"ossler system, the baker transformation, the H\'enon map, and a stochastic potential flow. Besides recovering the properties as expected from the respective systems, some of the applications show that the information flow structure underlying a complex trajectory pattern could be tractable. For linear systems, the resulting remarkably concise formula asserts analytically that causation implies correlation, while correlation does not imply causation, resolving unambiguously the long-standing debate over causation versus correlation.