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The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. [ 1 ] [ 2 ] Of the four parameters defining the family, most attention has been focused on the stability parameter, α {\displaystyle \alpha } (see panel).
Diffusion process is stochastic in nature and hence is used to model many real-life stochastic systems. Brownian motion , reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diffusion processes.
When an orbital diagram is drawn for the logistic map, it is possible to see how the branch representing the stable periodic orbit splits, which represents a cascade of period-doubling bifurcations. Bifurcation diagram of the period-doubling bifurcation cascade occurring between parameters r 1 = 3 {\displaystyle r_{1}=3} and a ∞ = 3.56994 ...
In recent times, reaction–diffusion systems have attracted much interest as a prototype model for pattern formation. [20] The above-mentioned patterns (fronts, spirals, targets, hexagons, stripes and dissipative solitons) can be found in various types of reaction–diffusion systems in spite of large discrepancies e.g. in the local reaction ...
The term "Lévy flight" was coined after Paul Lévy by Benoît Mandelbrot, [3] who used this for one specific definition of the distribution of step sizes. He used the term Cauchy flight for the case where the distribution of step sizes is a Cauchy distribution , [ 4 ] and Rayleigh flight for when the distribution is a normal distribution [ 5 ...
The convection–diffusion equation can be derived in a straightforward way [4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: + =, where j is the total flux and R is a net ...
Under an idealized reaction condition for A + B → product in a diluted solution, Smoluchovski suggested that the molecular flux at the infinite time limit can be calculated from Fick's laws of diffusion yielding a fixed/stable concentration gradient from the target molecule, e.g. B is the target molecule holding fixed relatively, and A is the ...
Since diffusion goes as the Laplacian of velocity, the dissipation rate may be written in terms of the energy spectrum as: = (), with ν the kinematic viscosity of the fluid. From this equation, it may again be observed that dissipation is mainly associated with high wavenumbers (small eddies) even though kinetic energy is associated mainly ...