Caputo Fractional Derivative and Quantum-Like Coherence

Abstract

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

Biophotons and Emergence of Quantum Coherence—A Diffusion Entropy Analysis

“…Our analysis clearly shows the presence of crucial events, at least in the early stages of the germination process, and the presence of a transition to a stage where the scaling remains significantly anomalous, but the main origin of this anomaly is due to FBM which, according to Ref. [ 48 ], may be a sign of quantum coherence. The advocates of quantum mechanics to explain the origin of cognition may use these results to support their view.…”

“…The authors of Ref. [ 48 ] with the help of Ref. [ 43 ] demonstrated that the dynamical origin of FBM is a generalized Langevin equation that is derived using the formalism of quantum mechanics [ 49 ], thereby suggesting that FBM may be a manifestation of quantum coherence.…”

“…We believe that the results of this paper will lead to future research work where the role of quantum coherence can be established with more precision, by evaluating in detail the spectrum and deepening the quantum coherence properties discussed in Ref. [ 48 ].…”

“…The transition from regions with crucial events to regions where FBM dominates suggests the influence of quantum coherence [ 48 ]. Therefore, we are forced to adopt a physical interpretation of the transport of information that cannot rest only on the role of crucial events.…”

Cauchy Problem for a Stochastic Fractional Differential Equation with Caputo-Itô Derivative

“…Mathematical models based on ordinal or partial differential equations have been successfully used to describe the behavior of systems through the space and time. On the other hand, although the origin is very old, it was not until recent years that fractional order differential equations have gained more attention in different areas of science; for example, they are very useful to describe complex systems with memory effects (see [1][2][3][4][5] and the references therein).…”