A field equation for induction-transduction of activation-deactivation probability on measurable space

TL;DR

This paper introduces a new field equation modeling activation-deactivation processes on measurable spaces, capturing complex behaviors like chaos, wave phenomena, and phase transitions in diverse systems.

Contribution

It develops a novel mathematical framework based on dynamics of marked random measures and derives a field equation for Bernoulli activation-deactivation processes.

Findings

Derivation of a general field equation for activation-deactivation laws.

Application to mechanisms on the unit interval demonstrating complex behaviors.

Framework applicable to various natural and artificial systems.

Abstract

Induction-transduction of activating-deactivating points are fundamental mechanisms of action that underlie innumerable systems and phenomena, mathematical, natural, and anthropogenic, and can exhibit complex behaviors such as self-excitation, phase transitions, hysteresis, polarization, periodicity, chaos, wave behavior, geometry, and energy transfer. We describe a class of primitives for induction-transduction based on dynamics on images of marked random counting measures under graphical random transformations. We derive a field equation for the law of the activation-deactivation (Bernoulli) process on an arbitrary measurable space and describe some mechanisms of action on the unit interval.