A Short Introduction to the Theory of Lévy Flights

A Short Introduction to the Theory of Lévy Flights

A. Chechkin

School of Chemistry, Tel Aviv University, ISRAELand

Akhiezer Institute for Theoretical Physics, National Science Center «Kharkov Institute of Physics and Technology», Kharkov, UKRAINE

QMUL, 10 Jul 2009QMUL, 10 Jul 2009

LÉVY motion(LÉVY motion(--s) s) →→ “LÉVY flight(“LÉVY flight(--s)” s)” P. P. LévyLévy →→ B. Mandelbrot:B. Mandelbrot:

Paradigm of nonParadigm of non--Brownian random motionBrownian random motion

2. Properties of α-stable Lévy probability laws

2. Properties of Gaussian probability laws

1. Generalized Central

Limit Theorem1. Central Limit Theorem

Lévy MotionBrownian Motion

Mathematical Foundations:

Paul Pierre Lévy (1886-1971)

Theory of Lévy stable probability distributions in mathematical monographs:

Lévy (1925, 1937)Khintchine (1938)Gnedenko & Kolmogorov (1949-53)Feller (1966-71)Lukacs (1960-70)Zolotarev (1983-86, 1997)Janiki & Weron (1994)Samorodnitski & Taqqu (1994)

B.V. B.V. GnedenkoGnedenko, A.N. , A.N. KolmogorovKolmogorov, “Limit , “Limit DistributionsDistributions forfor SumsSums of Independent of Independent RandomRandom

Variables“ (1949)Variables“ (1949)

The prophecy: “All these distribution laws, called stable, deserve the most serious attention. It is probable that thescope of applied problems in which they play an essential role will become in due course rather wide.”(but did not mention any !)

The guidance for those who write books on statisticalphysics: “… “normal” convergence to abnormal (that is, different from Gaussian –A. Ch.) stable laws … , undoubtedly, has to be considered in every large textbook, which intends to equip well enough the scientist in thefield of statistical physics.”(fragmentary exposition in Balescu, Ebeling&Sokolov,...)

First remarkable propertyof stable distributions

1 21

. . .n d

n i ni

X X X X c X=

+ + + = =∑

( )2 2 2 2

1

n

ii

X X X n X t=

∆ = = =∑

1 1

1

, 1/ 1/ 2n

ii

X X n tα α α=

∆ ∝ ∝ >∑∼

1. Stability: distribution of sum of independent identically distributed stable random

variables = distribution of each variable (up to scaling factor)

ccnn = = nn1/a1/a , 0 < , 0 < αααααααα ≤≤≤≤≤≤≤≤ 2, 2, αααααααα is is LèvyLèvy indexindex

Gauss: Gauss: αααααααα = 2, = 2, ccnn = = nn1/21/2

CorollaryCorollary 2.2. Normal diffusion lawNormal diffusion law, , Brownian motionBrownian motion1 2 1 2

1

n

ii

X X n t=

∆ ∝ ∝∑∼

or the rule of summingor the rule of summingvariancesvariances

CorollaryCorollary 3.3. SuperdiffusionSuperdiffusion, , LL éévyvy

CorollaryCorollary 1.1. Statistical selfStatistical self--similaritysimilarity , , random fractal:random fractal:

When does the whole look like its parts When does the whole look like its parts ⇒⇒⇒⇒⇒⇒⇒⇒ description of random fractal description of random fractal ptocessesptocesses

2 , 0 2x α=∞ < <

( )2 2( ,1, )

Dp x D

D xπ=

+

21( ,2, ) exp

44

xp x D

DDπ

= −

( ) 10 2

1( ; , ) exp( ) exp | | , 0 2, 0 ( , , )ˆ

| |X X

xp k D ikX D k D p x D

x

ααα

α α α +→∞< <≡ = − < ≤ > ⇒ ∝

SECOND remarkable propertySECOND remarkable propertyof stable distributionsof stable distributions

2. Power law asymptotics ∼∼∼∼ x (-1-αααα) (“heavy tails”)

⇒⇒⇒⇒⇒⇒⇒⇒

Particular casesParticular cases

1. 1. Gauss, Gauss, αααααααα = 2= 2

All moments are finiteAll moments are finite

2. Cauchy, 2. Cauchy, αααααααα = 1= 1

, 0q

x q α< ∞ < <

Moments of order q < 1 are finiteMoments of order q < 1 are finite

CorollaryCorollary :: descriptiondescription of highly nonof highly non--equilibrium processesequilibrium processespossessing large bursts and/or outlierspossessing large bursts and/or outliers

THIRDTHIRD remarkableremarkable propertypropertyof stableof stable distributionsdistributions

2 2 ( )x x f x dx∞

−∞

= < ∞∫

3. Generalized Central Limit Theorem : stable distributions are the limit ones for distributions of sums of random variables (have domain of attraction) ⇒⇒⇒⇒

appear, when evolution of the system or the result of experiment is determined by the sum of a large number of random q uantities

Gauss: attracts Gauss: attracts f(xf(x)) withwith finitefinitevariancevariance

LL éévyvy: attract : attract f(xf(x)) withwith infiniteinfinite variancevarianceand the and the samesameasymptotic behaviorasymptotic behavior

2 2 1( ) , ( ) , 0 2x x f x dx f x x α α∞

− −

−∞= =∞ < <∫ ∼

ΣΣ 11

Due to the three remarkable properties of stable distributions it is now believed that the Léévy statistics provides a framework for the description of many natural phenomena in physical, chemical, biological, economical, … systems from a general common point of view

Electric Field Distribution in an Ionized Gas Electric Field Distribution in an Ionized Gas ((HoltzmarkHoltzmark 1919)1919)

Application:Application: spectral lines broadening in plasmasspectral lines broadening in plasmas

3i

ii

erE

r=�

� Nn const

V= =

1 2 ... nE E E E= + + +� � � �

3 dimensional

symmetric

stable

distribution

32α

−

=

( ) ( ) ( )32

ikEdkW E W k e

π

∞−

−∞

= ∫��

�

�� ⌢

( )3

2a kW k e

−=�

�⌢

( ) 52

1~W E

E

�⌢

�

RelatedRelated examplesexamples::Systems Systems withwith LongLong--Range Range interactionsinteractions

• gravity field of stars (=3/2)

• electric fields of dipoles (= 1)

• velocity field of point vortices in fully developed turbulence (=3/2) ……

(to name only a few)

• also: asymmetric Levy stable distributions in the first passage theory, single molecule line shape cumulants in glasses …

( )2 10 dim

(1900), (1905)

Normal diffusion R R c Dt t

Bachelier Einstein

− = ∝� �

( )20 , 1

Anomalous diffusion

R R tµ µ− ∝ ≠� �

1

Subdiffusion

µ <

Normal Normal vsvs AnomalousAnomalous DiffusionDiffusion

transport on fractals

contamimants in

underground water

amorphous solids

convective patterns

polymeric systems

deterministic maps

••

••••

1000=nGauss

3000=nCauchy

( , , )

turbulent media

gases fluids plasmas

Hamiltonian chaotic

systems

deterministic maps

foraging movement

financial markets

•

•

•••

1

Superdiffusion

µ >

SelfSelf--Similar Structure of Brownian and Similar Structure of Brownian and LévyLévy MotionsMotions

2( )

Normal diffusion

X t t∝2( ) ,

1.66 1

Levy Superdiffusion

X t tµ

µ

∝

= >

xx100100

xx100100 xx100100

xx100100

““ If, veritably, one took the position from If, veritably, one took the position from second to second, each of these rectilinear second to second, each of these rectilinear segments would be replaced by a polygonal segments would be replaced by a polygonal contour of 30 edges, contour of 30 edges, each being as complicatedeach being as complicatedas the reproduced designas the reproduced design, and so forth.” , and so forth.”

J.B. J.B. PerrenPerren

““ The stochastic process which we call linear The stochastic process which we call linear Brownian motion is a schematic Brownian motion is a schematic representarepresenta--tiontion which describes well the properties of real which describes well the properties of real Brownian motion, observable on a small Brownian motion, observable on a small enough, but not infinitely small scale, and enough, but not infinitely small scale, and which assumes that the which assumes that the same properties existsame properties existon whatever scaleon whatever scale.” .” P.P. P.P. LL éévyvy

NORMAL NORMAL vsvs ANOMALOUS DIFFUSIONANOMALOUS DIFFUSION

Brownian motion of a small Brownian motion of a small grain of puttygrain of putty

ForagingForaging movementmovementbyby LLéévyvy flights: flights: optimal strategy to search for food optimal strategy to search for food resources distributed at random ?resources distributed at random ?

(Jean Baptiste Perrin: 1909) Gabriel Ramos-Fernandes et al. (2004)

2( )R t t∝ 2( ) , 1.7R t t µ µ∝ ≈ Spider monkeysSpider monkeys

Diffusion of tracers in fluid flows.Diffusion of tracers in fluid flows.

Large scale structures (eddies, jets or convection rolls) dominaLarge scale structures (eddies, jets or convection rolls) dominate the transport.te the transport.

Example. Experiments Example. Experiments in a rapidly rotating in a rapidly rotating annulus (annulus (SwinneySwinneyet al.).et al.).

Ordered flow:Ordered flow:Levy diffusionLevy diffusion

(flights and traps)(flights and traps)µµµµµµµµ ≈≈≈≈≈≈≈≈ 1.5 1.5 –– 1.81.8

Weakly turbulent Weakly turbulent flow:flow:

Gaussian diffusionGaussian diffusionµµµµµµµµ ≈≈≈≈≈≈≈≈ 11

Anomalous Diffusion inAnomalous Diffusion in ChannelingChanneling

Fig.1. NORMAL DIFFUSION: Fig.1. NORMAL DIFFUSION: randomlyrandomlydistributeddistributedatomatomstringsstrings

Fig.2. SUPERDIFFUSION: Fig.2. SUPERDIFFUSION: positivelypositivelyand and negativelynegativelychargedchargedparticlesparticlesin a in a periodicperiodicfieldfield of of atomatomstringsstringsalongalongthetheaxisaxis<100> in a <100> in a siliconsilicon crystalcrystal..

“Paradoxical” Diffusion in Chemical Space (Sokolov et al.1997)

Optimal Target Search on a FastOptimal Target Search on a Fast--Folding Polymer Chain (Folding Polymer Chain (LomholtLomholt et al.2005)et al.2005)

•• IntersegmentalIntersegmental jumps permitted jumps permitted at chain contact points due to polymer loopingat chain contact points due to polymer looping

2

2( , ) ( , ) ( ) ( )

| |B L off on bulkn x t D D k n x t k n j t x

t x x

α

α δ ∂ ∂ ∂= + − + − ∂ ∂ ∂

•• Contoured length |x| stored Contoured length |x| stored in a loop between contact pointsin a loop between contact points

Protein diffusion to next Protein diffusion to next neighborneighbor sitessitesonon foldingfolding DNA DNA ((ddeoxyribonucleiceoxyribonucleic acidacid))

1( ) | |

1/ 2

x x αλα

− −

=∼

•• Motion of binding proteins or enzymes Motion of binding proteins or enzymes along DNA: detach to a volume along DNA: detach to a volume →→→→→→→→ reattach reattach before reaching the targetbefore reaching the target(Berg(Berg –– von von HippelHippel model)model)

SuperdiffusionSuperdiffusion of of bankbank notesnotes in in thethe US, US, µµ ≈≈ 22(Brockmann, Hufnagel, Geisel, Nature 2006)

www.wheresgeorge.comwww.wheresgeorge.com

FigureFigure: : trajectoriestrajectoriesof of banknotesbanknotesoriginatedoriginatedfromfrom 4 4 placesplaces::reminiscentreminiscentof Lof Léévyvy flightsflights

Geographical displacementr between two reportlocation of a bank note

( , ) ( , ) , 0.6, 2 /| |

W t r D W t rt r

β α

β α α β µ β α∂ ∂= ≈ ≈ =∂ ∂

2 1r x x= −� �

A Lévy Flight for Light(P. Barthelemy et al, Nature 2008)

•••••••• New optical material in New optical material in which light performs a which light performs a LL éévyvy flightflight

•••••••• Ideal experimental Ideal experimental system to study system to study LL éévyvyflights in a controlled wayflights in a controlled way

•• Precisely chosen Precisely chosen distribution of glass distribution of glass microspheresmicrospheresof of different diameters ddifferent diameters d

P(dP(d) ) ~ d~ d--((2 + 2 + αααααααα))

Figure: Figure: LL éévyvy walker trajectory in a scale walker trajectory in a scale –– invarianinvarian Levy glassLevy glass

Lévy flights of photons in hot atomic vapours(Mercadier et al. Nature Physics 2009)

( )20

1( )

ln /P x

x x l≈

•••• Measuring the step size distribution of photons

Figure: Double cell configuration to measure the s tep size PDF

•••• Under Gaussian assumptions of emission and absorption spectra (Doppler broadening)

Figure: power law exponent αααα of the step size distribution vs number n of scattering events (multiply scattering)

Impulsive Impulsive NoisesNoises ModeledModeled withwith thethe Lévy Lévy StableStableDistributionsDistributions

Naturally serve for the description of the processesNaturally serve for the description of the processeswithwith large large outliersoutliers , far , far fromfrom equilibriumequilibrium

Examples include:

•••• economic stock prices and current exchange rates (1963)•••• radio frequency electromagnetic noise•••• underwater acoustic noise•••• noise in telephone networks•••• biomedical signals•••• stochastic climate dynamics•••• turbulence in the edge plasmas of thermonuclear devices (Uragan2M, ADITYA, 2003; Heliotron J, 2005 ...)

Examples include:

•••• economic stock prices and current exchange rates (1963)•••• radio frequency electromagnetic noise•••• underwater acoustic noise•••• noise in telephone networks•••• biomedical signals•••• stochastic climate dynamics•••• turbulence in the edge plasmas of thermonuclear devices (Uragan2M, ADITYA, 2003; Heliotron J, 2005 ...)

LLéévy vy noisesnoises and Land Léévy vy motionmotion

LÉVY NOISES

Lévy index ↓↓↓↓ ⇒⇒⇒⇒ outliers ↑↑↑↑

LÉVY MOTION:successive additions of the noise values

Lévy index ↓↓↓↓ ⇒⇒⇒⇒ “flights” becomelonger

EXPERIMENTAL OBSERVATION OF LEXPERIMENTAL OBSERVATION OF LÉÉVY FLIGHTS IN WIND VY FLIGHTS IN WIND VELOCITYVELOCITY ((LammefjordLammefjord, , denmarkdenmark, , 20062006; ; GoncharGonchar et al.et al.))

Measuring masts Velocity increments Distribution of increments

250

min

250

min

25 m

in25

min

2.5

min

2.5

min

Important for reliable prediction of fatique loads

««LévyLévy TurbulenceTurbulence» » in boundaryin boundary plasmaplasma of of

stellaratorstellarator ««UraganUragan 3М»3М» ((GoncharGonchar et al., 2003)et al., 2003)

Helical magnetic coilsHelical magnetic coils((from abovefrom above))

Poincare section of magnetic linePoincare section of magnetic line, ЛЗ , ЛЗ –– Langmuir proveLangmuir prove

eV100eV,500)0(,m10

ms,60kW,200

0.3,π2/)(T,7.0

m1.0,m1,9,3

ie3-18

e

RF

φ

0

<≈≈

=≈

≈=≈===

TTn

P

aB

aRml

τι

Ion saturation currentIon saturation currentat different probe positionsat different probe positions::

аа) ) RR = = 111 с111 сmm ; ; бб) ) RR = 112 с= 112 сmm ;;

вв) ) RR = 112.5 с= 112.5 сmm ; ; гг) ) RR = 113 с= 113 сmm..

Boundary region is important for confinementBoundary region is important for confinement

««LévyLévy TurbulenceTurbulence» » in boundaryin boundary plasmaplasma of of

stellaratorstellarator ««UraganUragan 3М» 3М»

1. 1. Kurtosis and “chiKurtosis and “chi--square” square” criteriumcriterium : : evidence of nonevidence of non--GaussianityGaussianity..

2. 2. Modified method of percentilesModified method of percentiles: : LL éévyvy index of turbulent fluctuationsindex of turbulent fluctuations

Ion saturation currentIon saturation current ∼∼∼∼∼∼∼∼ δδδδδδδδnn Floating potentialFloating potential ∼∼∼∼∼∼∼∼ δϕδϕδϕδϕδϕδϕδϕδϕ

•••••••• Fluctuations of density and potential measured in boundary plasmFluctuations of density and potential measured in boundary plasma of a of stellaratorstellarator““ UraganUragan 3M” obey 3M” obey LL éévyvy statisticsstatistics

•••••••• similar conclusions about the similar conclusions about the LL éévyvy statistics of plasma fluctuations have been statistics of plasma fluctuations have been drawn fordrawn for

•••••••••••••••• ADITYAADITYA•••••••••••••••• LL --22MM•••••••••••••••• LHDLHD , , TJTJ--IIII•••••••••••••••• HeliotronHeliotron J J

““ burstybursty” character of fluctuations in other devices ” character of fluctuations in other devices ((DIIIDIII --DD, , TCABRTCABR , …), …)

““ Levy turbulence”Levy turbulence” is a widely spread phenomenonis a widely spread phenomenon⇒⇒⇒⇒⇒⇒⇒⇒ Models are strongly needed !Models are strongly needed !

Two advanced conceptsTwo advanced concepts

Truncated Levy Flights: Truncated Levy Flights: PDFsPDFsresembles resembles LévyLévy stable stable distribution in the central part, however at greater scales thedistribution in the central part, however at greater scales theasymptoticsasymptoticsdecay faster, than the decay faster, than the LévyLévy stable ones, stable ones, ⇒⇒⇒⇒⇒⇒⇒⇒the Central Limit Theorem is applied the Central Limit Theorem is applied ⇒⇒⇒⇒⇒⇒⇒⇒at large times the PDF tends to Gaussian, however, at large times the PDF tends to Gaussian, however, sometimes sometimes very slowlyvery slowly

Levy Walks: finite velocity when the motion of a massive Levy Walks: finite velocity when the motion of a massive object is consideredobject is considered

Might be important for the bumblebees in a finite box ???Might be important for the bumblebees in a finite box ???

2x < ∞

A. Chechkin, V. Gonchar, J. Klafter and R. Metzler, Fundamentals of Lévy Flight Processes. Adv. Chem. Phys. 133B, 439 (2006).

A. Chechkin, R. Metzler, J. Klafter, V. Gonchar, Introduction to the Theory of Lévy Flights. In R. Klages, G. Radons, I.M. Sokolov (Eds), Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim (2008).

R. Metzler, A. Chechkin, J. Klafter, Lévy statistics and anomalousstatistics and anomaloustransport:transport: Lévy flights and flights and subdiffusionsubdiffusion.. Encyclopaedia of Complexity and System Science. Springer-Verlag, 2009(ArXiv:0706.3553v1).

Recent reviewsRecent reviews

Thank you for attention !Thank you for attention !