Energy and information of chaotic dynamical systems

We postulate that a basic process in nature operates on a minimal time-scale, Δ0>0, and that on this time-scale the dynamics is governed by a deterministic, discrete-time nonlinear transformation τΔ0:X→X, which is manifested as a (chaotic) process τΔ=τΔ0N on a larger (observable) time-scale Δ=NΔ0. We assume that τΔ possesses an observable measure μΔ and define the average energy of the dynamical system (X,τΔ,μΔ) by where m denotes mass if the process represents the motion of a particle. With this definition, we prove a conservation of energy principle for each Δ. In the case where Δ can be made arbitrarily small, this definition reduces to the classical definition of average energy for a mechanical system governed by a differential equation. Let EΔ and IΔ denote the energy and information content of the dynamical system observed at the time-scale Δ. Using elementary concepts from dynamical systems, it is shown that energy and information are related by EΔ=KIΔ2, where K is a constant. This allows the estimation of the information content of matter.