scholarly journals Anomalous Diffusion in Systems with Concentration-Dependent Diffusivity: Exact Solutions and Particle Simulations

2020 ◽  
Vol 8 ◽  
Author(s):  
Alex Hansen ◽  
Eirik G. Flekkøy ◽  
Beatrice Baldelli
Keyword(s):  
Diffusion Equation ◽  
Anomalous Diffusion ◽  
Kolmogorov Equation ◽  
Particle Model ◽  
Initial Condition ◽  
Linear Diffusion ◽  
Exact Analytical Solutions ◽  
Normal Diffusion ◽  
Particle Simulations ◽  
Concentration Dependent Diffusivity

We explore the anomalous diffusion that may arise as a result of a concentration dependent diffusivity. The diffusivity is taken to be a power law in the concentration, and from exact analytical solutions we show that the diffusion may be anomalous, or not, depending on the nature of the initial condition. The diffusion exponent has the value of normal diffusion when the initial condition is a step profile, but takes on anomalous values when the initial condition is a spike. Depending on the sign of the exponent in the diffusivity the diffusive behavior will then be either sub-diffusive or super-diffusive. We introduce a particle model that behaves according to the non-linear diffusion equation in the macroscopic limit. This correspondence is demonstrated via kinetic theory, i.e. by means of Chapman-Kolmogorov equation, as well as by direct simulations.

scholarly journals Hyperballistic Superdiffusion and Explosive Solutions to the Non-Linear Diffusion Equation

2021 ◽  
Vol 9 ◽  
Author(s):  
Eirik G. Flekkøy ◽  
Alex Hansen ◽  
Beatrice Baldelli
Keyword(s):  
Exact Solution ◽  
Diffusion Equation ◽  
Particle Concentration ◽  
Planck Equation ◽  
Particle Dynamics ◽  
Particle Model ◽  
Linear Diffusion ◽  
Fokker Planck Equation ◽  
Non Linear ◽  
Particle Simulations

By means of a particle model that includes interactions only via the local particle concentration, we show that hyperballistic diffusion may result. This is done by findng the exact solution of the corresponding non-linear diffusion equation, as well as by particle simulations. The connection between these levels of description is provided by the Fokker-Planck equation describing the particle dynamics. PACS numbers:

scholarly journals Caputo Fractional Derivative and Quantum-Like Coherence

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 211
Author(s):  
Garland Culbreth ◽  
Mauro Bologna ◽  
Bruce J. West ◽  
Paolo Grigolini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Caputo Fractional Derivative ◽  
Quantum Coherence ◽  
Time Derivative ◽  
The Other ◽  
Diffusion Equations ◽  
Self Organization ◽  
Ordinary Time

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.

Anomalous diffusion equation using a new general fractional derivative within the Miller–Ross kernel

2020 ◽  
Vol 34 (27) ◽  
pp. 2050289
Author(s):  
Yiying Feng ◽  
Jiangen Liu
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Fractional Integral ◽  
Liouville Type

In view of the generalization of Miller–Ross kernel in the sense of Riemann–Liouville type, we propose the new definitions of the general fractional integral (GFI) and general fractional derivative (GFD) to discuss the anomalous diffusion equation, which is distinct from those classic calculus operators. The obtained analytical solution of the application described in the graph is effective and accurate making the use of Laplace transform.

scholarly journals Unique continuation principle for the one-dimensional time-fractional diffusion equation

2019 ◽  
Vol 22 (3) ◽  
pp. 644-657 ◽  
Author(s):  
Zhiyuan Li ◽  
Masahiro Yamamoto
Keyword(s):  
Diffusion Equation ◽  
Anomalous Diffusion ◽  
Function Method ◽  
Unique Continuation ◽  
Fractional Diffusion ◽  
Uniqueness Of Solutions ◽  
One Dimensional ◽  
Continuation Principle ◽  
Time Fractional Diffusion Equation ◽  
The One

Abstract This paper deals with the unique continuation of solutions for a one-dimensional anomalous diffusion equation with Caputo derivative of order α ∈ (0, 1). Firstly, the uniqueness of solutions to a lateral Cauchy problem for the anomalous diffusion equation is given via the Theta function method, from which we further verify the unique continuation principle.

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