Advanced Course on Stochastic Processes and Applications

Aleksei Chechkin

J.B. Perrin, 1909

Brockmann, Hufnagel, Geisel, Nature2006

Brownian motion, or Brownian Random Walk, or Normal Diffusion

“Anomalous”Random Walk, or

Anomalous diffusion

Stochastic Processes for Physicists

…

The presentation will be given at a “physical” level of accuracy, with the use of exercises and particular examples.

Why “advanced ” course ?Nowadays the area of applications of stochastic processes is expanding drastically.

Novel classes of stochastic phenomena have recently been observed in a wide variety of complex systems far from equilibrium

The coherent description of such phenomena poses a fundamental challenge to a modern statistical physics of non-equilibrium state and requires the use of several concepts that go beyond the classical university courses on the theory of probability, stochastic processes and kinetic theory.

Advanced course : Introductory overview of some modern concepts in the theory of stochastic processes and random walk phenomena, which are not included in the famous treatises

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“Важно не то, что строго, а то что верно” (It is not important whether this statement is strictly proved or not, important that it is correct).

Andrey Kolmogorov

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Objects to study

Stochastic processes / random processes evolving with time

More precisely: stochastic process is a collection of random variables indexed with time

Time: discrete and continuous

More specifically: going beyond the Brownian motion

Two paradigmatic examples of stochastic processes:

Stationary process:

white Gaussian noise

Non-stationary Gaussian process:

Brownian motion / random walk,

or Wiener process (note terminology)

Non-Gaussian and/or non-Markovian Noises and Random Walks

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Markovian

processes

Random walkis succession of random steps

Generalized Central Limit Theorem and alpha-stable Lévy probability lawsAlpha-stable Lévy processes, Lévy motion (Lévy flights); (ordinary) Brownian motion as a limit case of the alpha-stable Lévy family

Non-Markovian processes. Fractional Brownian , fractional Langevinand fractional Lévy motions

Renewal theory, continuous time random walk model, generalized master equation, fractional Fokker-Planck equations.

Subordinated stochastic processes with random time, subordinatedstochastic Langevin equationsLévy walks: Lévy flights with finite velocity

Random walks on fractal structures

Ergodicity breaking and aging

First passage processes

Random walks in non-stationary and inhomogeneous media: scaled Brownian motion and heterogeneous diffusion processes

Contents

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Wonderful World of Random Walks:

From Lucretius ’s “De rerum natura ”

to Optimal Search Strategies

Lecture # 1

An image, a type goes on before our eyesPresent each moment; for behold wheneverThe sun's light and the rays, let in, pour downAcross dark halls of houses: thou wilt seeThe many mites in many a manner mixedAmid a void in the very light of the rays,And battling on, as in eternal strife,And in battalions contending without halt,In meetings, partings, harried up and down.

De Rerum Natura, Book II: “The Dance of Atoms”.Titus Lucretius Carus

1st cent. BC

First (documented) observation of random walk

The random motion of particles is called Brownian Motion

R. Brown (1827)

Phil. Mag.

(1828)

What Robert Brown saw in his microscope ?

• very rapid

• highly irregular

• zigzag motion

•••• living pollen

•••• dead pollen

•••• fine inorganic particles

Display identical motion having a physical origin (but which ???)

This motion is not caused by currents (flows) in the liquid, nor by convection, nor by evaporation of the solvent

“VitalForce”(?)

Pulverized fragments of Egyptian sphinx !

A.Fick: The Diffusion Equation (1855)Experiment: diffusions of salts in water between two reservoirs through tubes

“It was quite natural to suppose, that this law for the diffusion of a salt in its solvent must be identical with that, according to which the diffusion of heat in a conducting body takes place; upon this law Fourier founded his celebrated theory of heat, and it is the same which Ohm applied with such extraordinary success, to the diffusion of electricity in a conductor. According to this law, the transfer of salt occurring in a unit of time, between two elements of space filled with differently concentrated solutions of the same salt, must be directly proportional to the difference of concentration, and inversely proportional to the distance of elements from one another.” Adolf Fick, On Liquid Diffusion (German original), Poggendorff’s Annalen der Physik and Chemie(1855)

( , ) ( , )q r t T r tκ= − ∇� � �

Fourier’s law of heat transfer (1822)

( , )q r t� �

Heat flux density

•

First Fick’s law of mass transfer (1855)

( , ) ( , )J r t D n r t= − ∇�

� �

•

Diffusion flux

( , )J r t�

�

Amount of substance that flow through unit area during unit time

A.Fick: The Diffusion Equation (1855)

Continuity equation:

( , ) ( , ) 0n r t divJ r tt

∂ + =∂

�

� �

+( , ) ( , )J r t D n r t= − ∇�

� �

( , ) ( , )n r t D n r tt

∂ = ∆∂� �

2 2 2

2 2 2divgrad

x y z

∂ ∂ ∂∆ ≡ ≡ + +∂ ∂ ∂

Second Fick’s law, or

Diffusion equation

1st Fick’s law :

A. Einstein (1879-1955)

Solution to a nearly 80 years old puzzle:

Explanation of origin and properties of Brownian mo tion

M. Smoluchowsky (1872-1917)

Annalen der Physik:

Cornerstone of Probability Calculus and theory of S tochastic Processes

t

( )t�

r

t=0

t

( )t�

r

2( )t Dt≈�

r

♦Linear in time t

♦♦♦♦ D: depends on→ properties of the solvent and → properties of a Brownian particle

Explanation of Brownian motion Einstein and Smoluchowsky

♦ Movements of Brownian particles are caused by collisions withmolecules of the solvent

♦ These molecules move erratically in display of their thermalenergy, of which the temperature is a certain measure

♦ Molecules of the solvent are too little to be observed directly

♦ But particles of the suspension, even though they are tiny from a humanpoint of view, are true giants in comparison with the molecules of thesolvent, and can be observed directly

position

PDF is Gaussian( , )p t�

r

∆�ir

∆�i+1r

1( )

N

it

== ∆∑

� �

ir r

( )R t�

( )R t�

”Can any of you readers refer me to a work wherein I should find a solution of the following problem, or failing the knowledge of any existing solution provide me with an original one? I should be extremely grateful for the aid in the matter. A drunkard starts from the point O and walks l yards in a straight line; he then turns through any angle whatever and walks another lyards in a second straight line. He repeats this process n times. Inquire the probability that after n stretches he is at a distance between r and r + δ rfrom his starting point O”. K. Pearson, The Problem of the Random Walk. Nature, July27,1905

”Can any of you readers refer me to a work wherein I should find a solution of the following problem, or failing the knowledge of any existing solution provide me with an original one? I should be extremely grateful for the aid in the matter. A drunkard starts from the point O and walks l yards in a straight line; he then turns through any angle whatever and walks another lyards in a second straight line. He repeats this process n times. Inquire the probability that after n stretches he is at a distance between r and r + δ rfrom his starting point O”. K. Pearson, The Problem of the Random Walk. Nature, July27,1905

Appearance of the “Random Walk” term (Pearson, 1905 ):

Studying spatial/temporal evolution of mosquito pop ulations invading cleared jungle regions

The problem is too complex to model deterministically, a random walk model should be conceptualized

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Lord Rayleigh’s response (August 3, 1905, one week after):

“This problem, proposed by Prof. Karl Pearson in the current number of Nature, is the same as that of the composition of n iso-periodic vibrations of unit amplitude and of phases distributed at random, considered in Phil.Mag., X, p.73, 1880.”

“This problem, proposed by Prof. Karl Pearson in the current number of Nature, is the same as that of the composition of n iso-periodic vibrations of unit amplitude and of phases distributed at random, considered in Phil.Mag., X, p.73, 1880.”

Lord Rayleigh (1880): statistics of multimode rando m oscillations

1( ) cos( ) cos

N

nn

t a tξ ω ϕ ρ=

= + = Φ∑

1( ) ,

2n nw ϕ π ϕ ππ

= − ≤ ≤

Phases homogeneously distributed

and statistically independent

cos , sinx yρ ρ= Φ = ΦIn Cartesian coordinate system (x,y)

2 2

2 21

( , ) exp( )

Nx y

p x yNa Naπ

+ = −

Probability density for the resulting N-mode oscillation, N>>1

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Gaussian probability density function :

“The lesson of Lord Rayleigh's solution is that in open country themost probable place of finding a drunken man who is at all capableof keeping on his feet is somewhere near his starting point !”

“The lesson of Lord Rayleigh's solution is that in open country themost probable place of finding a drunken man who is at all capableof keeping on his feet is somewhere near his starting point !”

“I ought to have known it, but my reading of late years has drifted into other channels, and one does not expect to find the first stage of a biometric problem provided in a memoir on sound…”

“I ought to have known it, but my reading of late years has drifted into other channels, and one does not expect to find the first stage of a biometric problem provided in a memoir on sound…”

K. Pearson’s Comment on Lord Rayleigh Response (Nature, August 10, one week after)

John William Strutt, 3rd Baron Rayleigh

(1842 – 1919)

Karl Pearson (1857-1936)

Gaussia

Stock price for GE since January 1, 2000

Dow Jones Industrial Average

Brownian random walks on financial marketsLouis Bachelier , Théorie de la Spéculation, 1900

Prices perform Brownian motion in one dimension ! “Price” = “Position”

( )x t( )x t

( )x t

Ver

tical

coo

rdin

ate

of th

e pl

anar

B

row

nian

wal

k of

a p

artic

le

2( )x t Dt≈

The same linear in time law

BUT: coefficients D’s are different

PDF is Gaussian

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Back to Einstein and Smoluchowsky:

Emergence of Normal Diffusion

i) ∃ time interval τ, so that the Brownian particle’s motion during the two consequent intervals is independent

Basic assumptions:

ii) The displacement of Brownian particle during τ is ∆x, which obeys symmetrical probability distribution function φ(∆x) = φ(-∆x), not necessarily Gaussian, but essentially

( ) ( ) ( )2 2x x x d xφ∞

−∞∆ ≡ ∆ ∆ ∆ < ∞∫

iii) Independent particles, n(x,t) → p(x,t) : phenomenological laws of particle diffusion can be applied to diffusion of probability

2

2( , ) ( , )p x t D p x t

t x

∂ ∂=∂ ∂

1-dimensional, generalization to 3D is trivial

Result ( )2

2

xD

τ

∆=

Essentially, a Random Walk Model (1880, 1900, 1905×2)

21( , ) exp

44

xp x t

DtDtπ

= −

2 2 ( , ) 2x x f x t dx Dt∞

−∞= =∫

♦

( ,0) ( )p x xδ=

- ∞ < x < + ∞

Initial condition

Mean squared displacement

Solution of the Diffusion Equation (1D)

Boundary conditions ( )( , ) , 0p t p t∞ = −∞ =

2

2p p

Dt x

∂ ∂=∂ ∂

Solution: ( , ) 1p x t dx∞

−∞=∫

Normalization:

( , ) 0x x f x t dx∞

−∞= =∫Mean displacement

♦Normal diffusion law

Langevin approach (1908): “More direct than Einstein ’s and simpler than Smulochowski’s”

♦

( ) ( ) ( )t t t tξ ξ δ′ ′= −

+ 2 ( ) ,d dx

m m D tdt dt

γ γξ= − =vv v

2v

2 2B

m k T=

Bk TD

mγ=

2( )tv

ξ(t): GaussianStokes’friction coefficient

06 r

m

πγ η=

Langevin SDE

Solve and find v(t) ⇒

Equipartitiontheorem

BA

Rk

N=

( )2 2( ) 2 1 tD

x t Dt e γγ

−= − −

FDTNA ≈ 6*1023

mol-1

2 21/ ( )t x t D tγ γ<< ⇒ ≈

21/ ( ) 2t x t Dtγ>> ⇒ ≈

ballistic

normal

J.B. Perrin, 1909

Two essential features of random walk phenomena :

♦♦♦♦ Probability distribution is Gaussian

♦♦♦♦ Mean squared displacement is linear in time

Deep mathematical foundation: Consequence of the Central Limit Theorem if

♦♦♦♦ Random increments / jumps are “small”

♦♦♦♦ Random jumps are independent

ΣΣΣΣ

2( )r t t∝�

1( )

N

it

== ∆∑

� �

ir r

∆�ir

( ) ( ) ( )2 2 dλ∆ = ∆ ∆ ∆ ∞∫� � � �

r r r r <

Anomalous diffusion law: the hallmark of anomalous transport

2( )r t t∝�

2( ) , 1r t tµ µ∝ ≠�

1µ >

1µ <

ANOMALOUS TRANSPORT / ANOMALOUS DYNAMICS

Normal diffusion

In particular, (ordinary) Brownian motion,

or Wiener process:

superdiffusion(fast)

subdiffusion(slow)

Anomalous diffusion

•••• Anomalous is normal

• Happy families are all alike; every unhappy family i s unhappyin its own way L. Tolstoi , Anna Karenina (the very first sentence)

Different sources of anomaly

Long jumps Lévy flights / Levy walks in space

fractional motionsLong temporal correlations

Long waiting times as arising from random potential models

Lévy flights in time

Diffusion on fractal structuresGeometrical constraints

Scaled Brownian motion, heterogeneous diffusion process

Inhomogeneous and/or non-stationary environment

…

First observation of anomalous diffusion law: Richardson (relative) diffusion in turbulence

2 3( )r t t∝�

How to explain the Richardson diffusion ? 2 3( )r t t∝�

Constructed by analogy with diffusion equation, can not be derived or strictly justified

D(r,t) can not be fixed on dimensional grounds alone

Space-time dependent diffusion coefficient

( , ) , 2 3 4a bD r t t r a b∝ + =

Remembering Fick…

Continuity equation (for probability density !)

( , ) ( , ) 0f r t divJ r tt

∂ + =∂

�

� �

Generalized 1 st Fick’s law :

( )( , ) , ( , )J r t D r t grad f r t= −�

� � �

+ ( )( , ) ( , ) ( , )f r t div D r t grad f r tt

∂ =∂� � �

Scaling analysis:take

2 3( )r t t∝�

Different suggestions to describe Richardson’s diffusion

Richardson (1926) :4/3( , ) ( )D r t D r r≡ ∝

Batchelor (1952) :

Hentschel and Proccacia (1984) : 2/3( , )D r t t r∝

( , ) ( , ) ( , ) ,j j

f r t D r t f r t r rt r r

∂ ∂ ∂= =∂ ∂ ∂

� � �

BUT: The distribution functions are different, and experiment might be used to distinguish between the choices

2 3( )r t t∝�

2( , ) ( )D r t D t t≡ ∝

Space-time dependent diffusion coefficient

space dependent diffusion coefficient

time dependent diffusion coefficient

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Scaled Brownian motion

Heterogeneous diffusion processes

Granular Gas of Viscoelastic Particles and Scaled Brown ian Motion

Granular particles collide inelastically and lose a fraction of their kinetic energy during collisions which transforms into heat stored in internal degrees of freedom

♦

The first stage of its force-free evolution, the granular gas is in the homogeneous cooling state characterized by uniform density and absence of macroscopic fluxes

♦

Restitution coefficient♦

♦

Self-diffusion coefficient♦

MSD: from ballistic motion to subdiffusion

2t∝2�r 1/ 6t∝2�r

( )5 / 30

5 / 30

( )1 /

TT t t

t τ−= ∝

+

Similar to scaled Brownian motion

( ) 5 / 6 5 / 60 0( ) 3 ( ) ( ) / 2 / 1 /cD t T t t m D t tτ τ − −= = + ∝

12 12v / vε ′=0 1ε< <

Granular particles collide inelastically and lose a fraction of their kinetic energy during collisions which transforms into heat stored in internal degrees of freedom

♦

The first stage of its force-free evolution, the granular gas is in the homogeneous cooling state characterized by uniform density and absence of macroscopic fluxes

♦

Constant restitution coefficient

♦

Haff’s law♦

Self-diffusion coefficient♦

MSD: from ballistic motion to ultraslow diffusion

2t∝2�r log t∝2�r

Granular Gases and Ultraslow Scaled Brownian Motion

( )20

20

( )1 /

TT t t

t τ−= ∝

+

( ) 10 0( ) 3 ( ) ( ) / 2 / 1 /cD t T t t m D t tτ τ −= = + ∝

12 12v / vε ′=0 1ε< <

Intraday fluctuations in the Euro-Dollar exchange rate: 1-min-interval tick data 1999-2004

2 1( , ) ( ) ,HHx

D x t t u uτ

−≈ =D

H ≈ 0.35

Variable diffusion processes exhibit “clustering of volatility”

Back to the famous law of turbulent Back to the famous law of turbulent

diffusiondiffusion

Monin (1955, 1956) , (1955, 1956) , Tchen (1959) :(1959) :

Space – fractional diffusion equation

1/ 3( , ) ( ) ( , )p l t D p l tt

∂ = − −∆∂

� �

Klafter, Blumen, Shlesinger (1987):

Gave inspiration to the two important models

2 3( )l t t∝�

Lévy flightsFractional Laplacian

Lévy walks: Lévy flights with finite velocity

““ LLÉÉVY flightsVY flights ”” (B. Mandelbrot): (B. Mandelbrot): paradigm of nonparadigm of non --Brownian random walkBrownian random walk

Brownian Motion Lévy Flights

1. Central Limit Theorem

1. Generalized CentralLimit Theorem

2. Properties of Gaussian probability laws

2. Properties of αααα-stable Lévy probability laws

Mathematical Foundations: Paul Pierre Lévy(1886-1971)

“Lévy Flights”: Random walks with long jumps

Brownian motion

Lévyflights

On a plane

11

( , 0 2| |x

xx αλ α+∆ →∞

∆ ∝ < <∆

)

1( )

N

it

== ∆∑

� �

ir r

( ) ( ) ( )2 2 dλ∆ = ∆ ∆ ∆ ∞∫� � � �

r r r r <

“long” in which sense ? In 1D

( ) ( ) ( )2 2 !x x x d xλ∆ = ∆ ∆ ∆ ∞∫ =

( ) ( ) ( )2 2 !dλ∆ = ∆ ∆ ∆ ∞∫� � � �

r r r r =In 2D and 3D :

Lévyindex

SelfSelfSelfSelf----Similar Structure of Brownian motion and Similar Structure of Brownian motion and Similar Structure of Brownian motion and Similar Structure of Brownian motion and LLLLéééévyvyvyvy flightsflightsflightsflights

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

Lévy noise and Lévy motion naturally serve for the description of the processes with large outliers, fa r from equilibrium

Examples of Lévy noise 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 thermonucleardevices (Uragan 2M,ADITYA,2003; Heliotron J,2005...)

WHITE LÉVY NOISE:Increments of the oLm

Lévy index ↓↓↓↓ ⇒⇒⇒⇒ outliers ↑↑↑↑

ORDINARY LÉVY MOTION or FREE LÉVY FLIGHTS :

Markovian process with stationary independent increments

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

•••••••• New optical material in New optical material in which light performs a which light performs a LL éévyvy flight flight –– LL éévyvy glassglass

•••••••• 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 microspheres of microspheres of different diameters ddifferent diameters d

P(dP(d) ) ~ d~ d--((1 + 1 + ββββββββ))Spatial distribution of the transmitted intensity

Lévy Flight for LightBarthelemy, Bertolotti, Wiersma, Nature 2008

Diffusion of tracers in fluid flows:Diffusion of tracers in fluid flows:

successfully explained with the successfully explained with the LLéévyvy walk modelwalk model

Large scale structures (eddies, jets or convection rolls) dominaLarge scale structures (eddies, jets or convection rolls) domina te the te the transporttransport

Experiments in a rapidly rotating annulus (Swinney et al., 1993)

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 spreads its wings:

Physics of ForagingLévy flights search patterns of wandering albatroses

(Viswanathan et al, Nature, 1996) : Levy index αααα ≈≈≈≈ 1

What is the best statistical strategy to adapt in order to search efficiently for randomly located objects ?

LFs also obtained for marine predators, bumblebees, deers, spider monkeys, mussels ….

Fish in Lévy – flight foraging :

55 individual fish, 100 000 000 measurements

♦♦♦♦ Prey distribution also display Lévy-like patterns …

♦♦♦♦ Strong evidence of Lévy flights, but these flights are not universal

Brownian motion versus foraging movement

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

ForagingForaging movementmovement: optimal strategy to : optimal strategy to search for food resources distributed at search for food resources distributed at random ?random ?

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

2( )t t∝rSpider monkeysSpider monkeys2 1.7( )t t∝r

Why would animals adopt a Lévy-flightforaging strategy?

Lévy – flight foraging hypothesis:

♦♦♦♦ Predators should adopt search strategy known as Levy flights where prey is sparse and distributed unpredictably

♦♦♦♦ BUT: Brownian movement is sufficiently efficient for locating abundant prey

Lévy Flights as Robotics Search Patterns

♦♦♦♦ Branch of technology that deals with the design, co nstruction, operation, development and applications of robots

♦♦♦♦ Bio-inspired robotics: making robots that are inspi red by the biological systems

♦♦♦♦ “foraging” = locating objects in an environment

The task is to incorporate Lévy search strategy into miniaturized robots

robobeeDelFly

Hummingbird robot

!

In the lecture on the first passage processes:

Exact results for the generic model of “blind” salta toryforager • Jumps from location to location without visiting poi nts in between

• Speed is not constant

• Have no knowledge of his local environment

• No memory

• Streams are allowed

Relaxing basic assumptions is possible …

♦ In case of unfavourable conditions Lévy flight searchstrategy is more efficient than Brownian search:•••• the target is far•••• the stream is oncoming

Conclusions:

What about humans ?

Money as a Proxy for Human Travel: www.wheresgeorge.com/

Trajectories of banknotes ⇒⇒⇒⇒ reminiscent of Lévy flights

ScalingScaling lawslaws of human of human traveltravel(Brockmann, Hufnagel, Geisel, Nature 2006)

www.wheresgeorge.comwww.wheresgeorge.com

Money as a Money as a proxyproxy forfor human human traveltravel: : trajectoriestrajectoriesof of banknotesbanknotesoriginatedoriginatedfromfrom 4 4 placesplaces⇒⇒⇒⇒⇒⇒⇒⇒ reminiscentreminiscentof Lof Léévyvy flightsflights in in thethe US US currencycurrency trackingtracking projectproject

Geographical displacementr between two reportlocation of a bank note

( , ) ( , ) , 0.6| |

W t r D W t rt r

β α

β α α β∂ ∂= ≈ ≈∂ ∂

2 1r x x= −� �

Understanding individual human mobility patterns:

Lévy or not Lévy ?

♦ trajectories of 100 000 anonymized mobile phone users w hoseposition is tracked for a 6 months period

♦ In contrast with the random trajectories predicted by the p revailingLévy flight, human trajectories show a high degree of tempo ral andspatial regularity: they return to a few high freque ntly locations such as home or work

♦ This regularity does not apply to banknotes: a bill always follows the trajectory of its current owner

Dollar bills perform Lévy flights, but humans ?