Information Gain in Event Space Reflects Chance and Necessity Components of an Event

Abstract

Information flow for occurrences in phase space can be assessed through the application of the Lyapunov characteristic exponent (multiplicative ergodic theorem), which is positive for non-linear systems that act as information sources and is negative for events that constitute information sinks. Attempts to unify the reversible descriptions of dynamics with the irreversible descriptions of thermodynamics have replaced phase space models with event space models. The introduction of operators for time and entropy in lieu of traditional trajectories has consequently limited—to eigenvectors and eigenvalues—the extent of knowable details about systems governed by such depictions. In this setting, a modified Lyapunov characteristic exponent for vector spaces can be used as a descriptor for the evolution of information, which is reflective of the associated extent of undetermined features. This novel application of the multiplicative ergodic theorem leads directly to the formulation of a dimension that is a measure for the information gain attributable to the occurrence. Thus, it provides a readout for the magnitudes of chance and necessity that contribute to an event. Related algorithms express a unification of information content, degree of randomness, and complexity (fractal dimension) in event space.

Information Dynamics in Complex Systems Negates a Dichotomy between Chance and Necessity

“…By contrast, we are enabled to measure new information produced by an event (through the Lyapunov characteristic exponent) and the degree of complexity inherent in the event (through the information dimension) in order to gain relevant, quantitative readouts (Section 3). Rather than a dichotomy between chance and necessity, quantifiable information evolution and information dimension, as well as entropy, are suitable scientific measures for the complexity of an occurrence, which can be interpreted as being reflective of the "ordered" and "chance" components that contribute to this occurrence [5].…”

“…However, due to the reliance on the eigenvectors and eigenvalues of the operators for time and thermodynamic entropy, an assessment of the information evolution associated with an event now requires algorithms that are independent of trajectories in phase space. To achieve this, a reformulation of the Lyapunov characteristic exponent has been applied to the vector-based event space [5]. (8).…”

“…Information (the removal of uncertainty, Appendix A(2)) and complexity (non-linearity, not amenable to closed-form solutions) are intricately linked to our everyday understanding of chance, but unlike "chance" or "randomness" these physical entities can be subjected to rigorous definition and analysis. Whereas no information is yielded by equilibrium (stasis), information flow (more accurately information evolution [5]) is tied to an occurrence (Section 1). The complexity of that occurrence underlies the amount of new information generated (Sections 2 and 3), and it produces emergent phenomena (Section 4).…”