scholarly journals From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion

2005 ◽  
Vol 15 (2) ◽  
pp. 026103 ◽  
Author(s):  
I. M. Sokolov ◽  
J. Klafter
Keyword(s):  
Brownian Motion ◽  
Anomalous Diffusion

scholarly journals Entropic Approach to the Detection of Crucial Events

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 178 ◽  
Author(s):  
Garland Culbreth ◽  
Bruce West ◽  
Paolo Grigolini
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Anomalous Diffusion ◽  
Correlation Functions ◽  
Joint Action ◽  
Clear Distinction ◽  
Aging Effects ◽  
Recent Advances ◽  
The Brain

In this paper, we establish a clear distinction between two processes yielding anomalous diffusion and 1 / f noise. The first process is called Stationary Fractional Brownian Motion (SFBM) and is characterized by the use of stationary correlation functions. The second process rests on the action of crucial events generating ergodicity breakdown and aging effects. We refer to the latter as Aging Fractional Brownian Motion (AFBM). To settle the confusion between these different forms of Fractional Brownian Motion (FBM) we use an entropic approach properly updated to incorporate the recent advances of biology and psychology sciences on cognition. We show that although the joint action of crucial and non-crucial events may have the effect of making the crucial events virtually invisible, the entropic approach allows us to detect their action. The results of this paper lead us to the conclusion that the communication between the heart and the brain is accomplished by AFBM processes.

Correlation effects, generalized Brownian motion and anomalous diffusion

1994 ◽  
Vol 203 (1) ◽  
pp. 53-60 ◽  
Author(s):  
K.G. Wang ◽  
L.K. Dong ◽  
X.F. Wu ◽  
F.W. Zhu ◽  
T. Ko
Keyword(s):  
Brownian Motion ◽  
Anomalous Diffusion ◽  
Correlation Effects

FRACTIONAL DIFFUSION EQUATION, QUANTUM SUBDYNAMICS AND EINSTEIN'S THEORY OF BROWNIAN MOTION

2007 ◽  
Vol 21 (23n24) ◽  
pp. 3993-3999
Author(s):  
SUMIYOSHI ABE
Keyword(s):  
Brownian Motion ◽  
Quantum Theory ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Master Equation ◽  
Anomalous Diffusion ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Quantum Master Equation ◽  
Objective System

The fractional diffusion equation for describing the anomalous diffusion phenomenon is derived in the spirit of Einstein's 1905 theory of Brownian motion. It is shown how naturally fractional calculus appears in the theory. Then, Einstein's theory is examined in view of quantum theory. An isolated quantum system composed of the objective system and the environment is considered, and then subdynamics of the objective system is formulated. The resulting quantum master equation is found to be of the Lindblad type.

THE COMPLEXITY OF BROWNIAN PROCESSES RUN WITH NONLINEAR CLOCKS

2011 ◽  
Vol 25 (01) ◽  
pp. 1-10 ◽  
Author(s):  
MOONGYU PARK ◽  
JOHN H. CUSHMAN
Keyword(s):  
Brownian Motion ◽  
Fractal Dimension ◽  
Anomalous Diffusion ◽  
Mean Square Displacement ◽  
Scaling Exponent ◽  
Near Surface ◽  
Surface Ocean ◽  
Power Law Exponent ◽  
Brownian Process ◽  
Turbulent Fluids

Anomalous diffusion occurs in many branches of physics. Examples include diffusion in confined nanofilms, Richardson turbulence in the atmosphere, near-surface ocean currents, fracture flow in porous formations and vortex arrays in rotating flows. Classically, anomalous diffusion is characterized by a power law exponent related to the mean-square displacement of a particle or squared separation of pairs of particles: 〈|X(t)|2〉 ~tγ. The exponent γ is often thought to relate to the fractal dimension of the underlying process. If γ > 1 the flow is super-diffusive, if it equals 1 it is classical, otherwise it is sub-diffusive. In this work we illustrate how time-changed Brownian position processes can be employed to model sub-, super-, and classical diffusion, while time-changed Brownian velocity processes can be used to model super-diffusion alone. Specific examples presented include transport in turbulent fluids and renormalized transport in porous media. Intuitively, a time-changed Brownian process is a classical Brownian motion running with a nonlinear clock (Bm-nlc). The major difference between classical and Bm-nlc is that the time-changed case has nonstationary increments. An important novelty of this approach is that, unlike fractional Brownian motion, the fractal dimension of the process (space filling character) driving anomalous diffusion as modeled by Bm-nlc positions or velocities does not change with the scaling exponent, γ.

scholarly journals Single particle diffusion characterization by deep learning

2019 ◽  
Author(s):  
Naor Granik ◽  
Elias Nehme ◽  
Lucien E. Weiss ◽  
Maayan Levin ◽  
Michael Chein ◽  
...  
Keyword(s):  
Experimental Data ◽  
Brownian Motion ◽  
Deep Learning ◽  
Anomalous Diffusion ◽  
Single Particle ◽  
Single Particle Tracking ◽  
Cellular Systems ◽  
Accurate Analysis ◽  
Diffusion Type ◽  
Diffusion Parameters

AbstractDiffusion plays a crucial role in many biological processes including signaling, cellular organization, transport mechanisms, and more. Direct observation of molecular movement by single-particle tracking experiments has contributed to a growing body of evidence that many cellular systems do not exhibit classical Brownian motion, but rather anomalous diffusion. Despite this evidence, characterization of the physical process underlying anomalous diffusion remains a challenging problem for several reasons. First, different physical processes can exist simultaneously in a system. Second, commonly used tools to distinguish between these processes are based on asymptotic behavior, which is inaccessible experimentally in most cases. Finally, an accurate analysis of the diffusion model requires the calculation of many observables, since different transport modes can result in the same diffusion power-law α, that is obtained from the commonly used mean-squared-displacement (MSD) in its various forms. The outstanding challenge in the field is to develop a method to extract an accurate assessment of the diffusion process using many short trajectories with a simple scheme that is applicable at the non-expert level.Here, we use deep learning to infer the underlying process resulting in anomalous diffusion. We implement a neural network to classify single particle trajectories according to diffusion type – Brownian motion, fractional Brownian motion (FBM) and Continuous Time Random Walk (CTRW). We further use the net to estimate the Hurst exponent for FBM, and the diffusion coefficient for Brownian motion, demonstrating its applicability on simulated and experimental data. The networks outperform time averaged MSD analysis on simulated trajectories while requiring as few as 25 time-steps. Furthermore, when tested on experimental data, both network and ensemble MSD analysis converge to similar values, with the network requiring half the trajectories required for ensemble MSD. Finally, we use the nets to extract diffusion parameters from multiple extremely short trajectories (10 steps).

scholarly journals Urea-mediated anomalous diffusion in supported lipid bilayers

2018 ◽  
Vol 8 (5) ◽  
pp. 20180028 ◽  
Author(s):  
E. E. Weatherill ◽  
H. L. E. Coker ◽  
M. R. Cheetham ◽  
M. I. Wallace
Keyword(s):  
Brownian Motion ◽  
Polyethylene Glycol ◽  
Lipid Bilayer ◽  
Lipid Bilayers ◽  
Anomalous Diffusion ◽  
Incubation Time ◽  
Time Scale ◽  
Biological Membranes ◽  
Model Systems ◽  
Supported Lipid Bilayers

Diffusion in biological membranes is seldom simply Brownian motion; instead, the rate of diffusion is dependent on the time scale of observation and so is often described as anomalous. In order to help better understand this phenomenon, model systems are needed where the anomalous diffusion of the lipid bilayer can be tuned and quantified. We recently demonstrated one such model by controlling the excluded area fraction in supported lipid bilayers (SLBs) through the incorporation of lipids derivatized with polyethylene glycol. Here, we extend this work, using urea to induce anomalous diffusion in SLBs. By tuning incubation time and urea concentration, we produce bilayers that exhibit anomalous behaviour on the same scale as that observed in biological membranes.

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