A Lévy flight is a
random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
in which the step-lengths have a
Lévy distribution
In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is kn ...
, a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
that is
heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
random directions. Later researchers have extended the use of the term "Lévy flight" to also include cases where the random walk takes place on a discrete grid rather than on a continuous space.
[
The term "Lévy flight" was coined by ]Benoît Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
, 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
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) func ...
, and Rayleigh flight for when the distribution is a normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
(which is not an example of a heavy-tailed probability distribution).
The particular case for which Mandelbrot used the term "Lévy flight"[ is defined by the survivor function of the distribution of step-sizes, ''U'', being
:
Here ''D'' is a parameter related to the ]fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is mea ...
and the distribution is a particular case of the Pareto distribution
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial ...
.
Properties
Lévy flights are, by construction, Markov processes Markov ( Bulgarian, russian: Марков), Markova, and Markoff are common surnames used in Russia and Bulgaria. Notable people with the name include:
Academics
* Ivana Markova (born 1938), Czechoslovak-British emeritus professor of psychology a ...
. For general distributions of the step-size, satisfying the power-like condition, the distance from the origin of the random walk tends, after a large number of steps, to a stable distribution
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stab ...
due to the generalized central limit theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...
, enabling many processes to be modeled using Lévy flights.
The probability densities for particles undergoing a Levy flight can be modeled using a generalized version of the Fokker–Planck equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as ...
, which is usually used to model Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position ins ...
. The equation requires the use of fractional derivatives. For jump lengths which have a symmetric probability distribution, the equation takes a simple form in terms of the Riesz fractional derivative. In one dimension, the equation reads as
:
where ''γ'' is a constant akin to the diffusion constant, ''α'' is the stability parameter and ''f''(''x'',''t'') is the potential. The Riesz derivative can be understood in terms of its Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
.
:
This can be easily extended to multiple dimensions.
Another important property of the Lévy flight is that of diverging variances in all cases except that of ''α'' = 2, i.e. Brownian motion. In general, the θ fractional moment of the distribution diverges if ''α'' ≤ ''θ''. Also,
:
The exponential scaling of the step lengths gives Lévy flights a scale invariant
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.
The technical ter ...
property, and they are used to model data that exhibits clustering.
Applications
The definition of a Lévy flight stems from the mathematics related to chaos theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have ...
and is useful in stochastic measurement and simulations for random or pseudo-random natural phenomena. Examples include earthquake
An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, from ...
data analysis, financial mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
, cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adve ...
, signals analysis as well as many applications in astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galax ...
, biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
, and physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...
.
Another application is the Lévy flight foraging hypothesis The Lévy flight foraging hypothesis is a hypothesis in the field of biology that may be stated as follows:
''Since Lévy flights and walks can optimize search efficiencies, therefore natural selection should have led to adaptations for Lévy fligh ...
. When sharks and other ocean predators cannot find food, they abandon Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position ins ...
, the random motion seen in swirling gas molecules, for Lévy flight — a mix of long trajectories and short, random movements found in turbulent fluids. Researchers analyzed over 12 million movements recorded over 5,700 days in 55 data-logger-tagged animals from 14 ocean predator species in the Atlantic and Pacific Oceans, including silky shark
The silky shark (''Carcharhinus falciformis''), also known by numerous names such as blackspot shark, gray whaler shark, olive shark, ridgeback shark, sickle shark, sickle-shaped shark and sickle silk shark, is a species of requiem shark, in the ...
s, yellowfin tuna
The yellowfin tuna (''Thunnus albacares'') is a species of tuna found in pelagic waters of tropical and subtropical oceans worldwide.
Yellowfin is often marketed as ahi, from the Hawaiian , a name also used there for the closely related bigeye ...
, blue marlin and swordfish. The data showed that Lévy flights interspersed with Brownian motion can describe the animals' hunting patterns. Birds and other animals (including humans) follow paths that have been modeled using Lévy flight (e.g. when searching for food).[ Biological flight data can also apparently be mimicked by other models such as composite correlated random walks, which grow across scales to converge on optimal Lévy walks.] Composite Brownian walks can be finely tuned to theoretically optimal Lévy walks but they are not as efficient as Lévy search across most landscapes types, suggesting selection pressure for Lévy walk characteristics is more likely than multi-scaled normal diffusive patterns.
Efficient routing in a network can be performed by links having a Levy flight length distribution with specific values of alpha.[
]
See also
*Anomalous diffusion
Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement (MSD), \langle r^(\tau )\rangle , and time. This behavior is in stark contrast to Brownian motion, the typical diffusion process descri ...
*Fat-tailed distribution
A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and Heavy-tailed distributi ...
*Heavy-tailed distribution
In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distrib ...
*Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which disp ...
* Lévy alpha-stable distribution
*Lévy flight foraging hypothesis The Lévy flight foraging hypothesis is a hypothesis in the field of biology that may be stated as follows:
''Since Lévy flights and walks can optimize search efficiencies, therefore natural selection should have led to adaptations for Lévy fligh ...
Notes
References
*
Further reading
*
*
External links
A comparison of the paintings of Jackson Pollock to a Lévy flight model
{{DEFAULTSORT:Levy Flight
Fractals
Markov processes
Paul Lévy (mathematician)