Lévy flights, fractional calculus and non-equilibrium phase ...mbsffe09/talks/Hinrichsen.pdfKenneth...

Lévy-stable distributions Fractional calculus Simple applications Nonequilibrium phase transitions Applications to Local Scale Invariance

Lévy flights, fractional calculus andnon-equilibrium phase transitions with

long-range interactions

Haye Hinrichsen

University of Würzburg, Germany

MBSFF09 – MPIPKS DresdenFebruary 2009

Haye Hinrichsen, University of Würzburg


Outline

1 Lévy-stable distributions

2 Fractional calculus

3 Simple applications

4 Nonequilibrium phase transitions

5 Applications to Local Scale Invariance



Stable probability densities

A probability density P(x) is called stable if the sum ofuncorrelated random variables distributed according to P is(up to rescaling) again distributed according to P.

In other words: Noise that becomes more intense but notqualitatively different when added from independent sources.



Stable probability densities

More precisely:Let X1 and X2 be uncorrelated random variables distributedaccording to PX . This distribution is said to be stable if theprobability density PY of any linear combinationY = a1X1 + a2X2, given by the weighted convolution product

PY (y) =1

a1a2

∫ +∞−∞

PX(x − y

a1

)PX( y

a2

)coincides with the original distribution PX up to rescaling, i.e.

PY (y) =1R

PX( y

R

)where R(a1,a2) is the corresponding scale factor.



Lévy-stable probability densities

Using the convolution theorem the stability condition in Fourierspace reads

P̃(a1k)P̃(a2k) =1√2π

P̃(

R(a1,a2)k).

An elementary calculation shows that stable distributions are ofthe form

P̃(k) =

B+ exp(−(+k)α) for k > 01 for k = 0B− exp(−(−k)α) for k < 0

with a parameter 0 < α ≤ 2, called Lévy-stable distributions.



Lévy-stable probability densities

Textbook form:

P(x ; α, β, c, µ) =1

2π

∫ +∞−∞

dk e−ik(x−µ)−|ck |α(1−iβsgn(k) tan(πα/2))

α = exponent (controlling power-law tails)β = skewness (asymmetry parameter)c = widthµ = mean

...includes the special case of a Gaussian distribution for α = 2.



Symmetric Lévy-stable distributions:

(Wikipedia)



Lévy-stable distributions have algebraic tails:

P(x) ' αcα(1 + β) sin(πα/2) Γ(α)

π|x |1+α∼ |x |−1−α



Asymmetric Lévy-stable distributions:



Corresponding cumulative distribution function:



Generalized central limit theorem

Lévy-stable distributions are attractive fixed points, i.e.,probability density functions with the asymptotic behaviorP(x) ∼ |x |−1−α converge under repeated convolution to aLévy-stable probability density.

Proof: Analyze the maximum at k = 0 in Fourier space in thelimit of a very large power.



Remark on Tsallis statistics:

The q-Gaussian probability density

Pq(x) = N−1(1− (1− q)βx2)1/(1−q)

is unstable and will be attractedby the usual Gaussian fixed point.

P̃q(k) =2

11−q +

32 (q − 1)

q−34(q−1) k

1q−1−

12 K 1

2 +1

1−q

„k√q−1

«Γ

“1

q−1 −12

” ∼ 1− k2

C. Tsallis, J. Stat. Phys. 52, 1988H. J. Hilhorst and G. Schehr, J. Stat. Mech. P06003 (2007)Thierry Dauxois, J. Stat. Mech. N08001 (2007)



Outline








Fractional derivatives - Heuristics

A fractional derivative is a non-integral derivative

∂nx =∂n

∂xn→ Dνx =

∂ν

∂xνn ∈ N, ν ∈ R

which behaves algebraically like an ordinary differentialoperator

Example:∂

1/2x x ∝

√x .



Fractional derivatives - Heuristics

A fractional derivative is linear and should act on functions inthe usual way:

• on power series:

∂nx xm =

m!(m − n)!

xm−n → Dνx xµ =Γ(µ+ 1)

Γ(µ− ν + 1)xµ−ν

• on waves:

∂nx eikx = (ik)neikx → Dνx eikx = (ik)νeikx



Fractional derivatives - Confusing history

Leibniz (1695): What is ∂1/2x ?Lacroix (1819): Derivative of arbitrary orderLaplace (1820): Fractional integration D−nuxFourier (1822): Fractional operators like (ik)µ

Liouville (1832): Fractional derivative of power seriesLagrange (1849): ∂µx ∂νx = ∂

µ+νx

Center (1850): Dνx 1 is different for different definitionsGrunwald (1867): Fractional derivative as a discrete sumRiemann (1892): Dνx = ∂nx D

ν−1x

Weyl (1917): Forward-fractional integrationMarchaud (1926?): Definition related to Levy flights...



Fractional derivatives

Present consensus:

There are several equally legitimate definitions of fractionalderivatives.These definitions are valid on different function spaces.Whenever these function spaces overlap they leadconsistent results.

Keith B. Oldham and Jerome Spanier, Dover (2006)Kenneth S. Miller and Bertram Ross, Wiley (1993)




Sometimes fractional derivatives are difficult to handle:

Ordinary Leibniz product rule:

∂nx

(f (x)g(x)

)=

n∑k=0

n!k !(n − k)!

(∂kx f (x)

)(∂n−kx g(x)

)

The fractional counterpart does not close:

Dνx(

f (x)g(x))

=∞∑

k=0

Γ(ν + 1)Γ(k + 1)Γ(ν − k + 1)

(Dkx f (x)

)(Dν−kx g(x)

)




Desired properties of a fractional derivative:

1 Linearity2 Translational invariance3 Homogeneity4 Dνx D

µx = D

ν+µx



Fractional derivatives - general form

1. Linearity:

Dνx(αf (x) + βg(x)

)= αDνx f (x) + βD

νx g(x)

⇒ Dνx can be written as an integral

Dνx f (x) =∫ +∞−∞

dy Kν(x , y) f (y)

where Kν is a Kernel function (distribution).

Examples:

Identity 1 K (x , y) = δ(x − y)First derivative ∂x K (x , y) = −δ′(x − y)Second derivative ∂2x K (x , y) = δ′′(x − y)




2. Translational invariance:

Taf (x) = f (x − a)

[Ta,Dνx ] = 0

implies that the kernel Kν depends only on the difference:

Dνx f (x) =∫ +∞−∞

dy Kν(x − y)f (y)

This leads to a convolution product.




3. Homogeneity (scale invariance):

x̃ = ax

Dνx f (ax) = aνDx̃ f (x̃)

⇒ The kernel Kν is a generalized homogeneous function:

Kν(ax) = a−1−νKν(x) for ∀a > 0, x 6= 0

⇒ For x 6= 0 the kernel is a power law:

Kν(x) =

K+ (+x)−1−ν for x > 0undefined for x = 0K− (−x)−1−ν for x < 0



Outline








Fractional derivative in physical space

Fractional derivatives in 2D/3D ?

∇νx = (Dνx ,Dνy ,Dνz )

not really clear:

∇νxei~k ·~x = (i~k)νei

~k ·~x

Generalized Laplacian:

∇̃2ν = ∇νx · ∇νx = D2νx + D2νy + D2νz

Works if and only if the components are symmetric: K+ = K−:

∇̃2νei~k ·~x = |~k |2νei~k ·~x .



Lévy flights in space

The kernel representation of ∇̃σ reads

∇̃σ ρ(x, t) := 1N⊥

∫ddx ′ |x′|−d−σ[ρ(x + x′, t)− ρ(x, t)]

with the prefactor N⊥(σ) = −πd/2Γ(−σ2 )2σΓ( d+σ2 )

.

It generates isotropic Lévy flights over a distance r which isdistributed as

P(r) ∼ r−d−σ.




Random walks by Lévy flights are described by theequation for anomalous diffusion

∂tρ(x, t) = ∇̃σρ(x, t)

The solution for an initially localized particle is a Lévy-stabledistribution:

ρ(x, t) =1

(2π)d

∫ddk exp

(ik · x− |k|σt

). (0 < σ < 2)



Ordinary versus Anomalous Diffusion

Ordinary diffusion Lévy flights

∂tρ(x, t) = D∇2ρ(x, t) ∂tρ(x, t) = D̃∇̃σρ(x, t)

∇2eik·x = −k2 eik·x ∇̃σ eik·x = −|k|σ eik·x

Gaussian distribution: Lévy-stable distributions:ρ(x, t) = 1

(4πDt)d/2 exp(− x24Dt

)ρ = 1

(2π)d∫

ddk exp(ik · x− D|k |σt

)



Application to empirical data




This is not a random walk.Haye Hinrichsen, University of Würzburg



Mainstream economics:

log (asset price) = random walk

This assumption provides the basis for the Black-Scholesmodel for option pricing (1975, nobel prize 1997)

Most people continue to use such “Gaussian tools”for risk management.




Example:Online data of an option traded by a major bank in Dresden:




For the S&P 500 stock indexone finds α ≈ 1.4.




Although people know thatGaussian statistics does notapply, they continue to use it.

...nicely described by former trader Nicholas Taleb in 2006.Haye Hinrichsen, University of Würzburg


Fractional derivative in time:

Fractional derivatives ∂κt used as time evolution generatorshave to be fully asymmetric in order to respect causality

Interpretation: Algebraically distributed waiting times ∆t withprobability distribution P(∆t) ∼ t−1−κ.




Integral kernel representation (Marchaud):

∂κt ρ(t) =1N‖(κ)

∫ ∞0

dt ′ t ′−1−κ[ρ(t)− ρ(t − t ′)] ,

with N‖(κ) = −Γ(−κ).

Action in momentum space:

∂κt eiωt = (iω)κeiωt




Simplest example:∂κt ρ(t) = δ(t)

⇒ (iω)κρ̃(ω) = 1/√

2π

⇒ ρ(t) ∝{

0 for t ≤ 0tκ−1 for t ≥ 0

Differential equations of this type describenon-Markovian processes.



Fractional derivative in time and space:

Diffusion with waiting times:

∂κt ρ(x, t) = ∇2xρ(x, t)

Anomalous diffusion with waiting times:

∂κt ρ(x, t) = ∇̃σρ(x, t)

H. C. Fogedby, Phys. Rev. E (1994)



Outline








Simplest absorbing phase transition

Stochastic fractional differential equation for asingle degree of freedom ρ(t):

∂κt ρ(t) = aρ(t)− ρ2(t) +√ρ(t)ξ(t)

where ξ(t) is a Gaussian noise,

Interpretation:Spreading process on a timeline.Directed percolation in zero space dimensions.

O. Deloubrière and F. van Wijland, Phys. Rev. E (2002)A. C. Barato and H. H., J. Stat. Mech (2009)




Update rule:

1 Select the lowest t for which s(t) = 1.2 With probability µ generate waiting time ∆t and set

s(t + ∆t) := 1.3 Otherwise set s(t) := 0.




C-Code:




This model exhibits a continuous phase transition:




Nevertheless the process is described by a nontrivial fieldtheory.

Partition sum:Z ∝

∫DφDφ̄e−S[φ,φ̄]

Field-theoretic action S = S0 + Sint with

S0[φ, φ̄] =∫ +∞−∞

dt φ̄(t)[τ ∂̃κt − a

]φ(t)

and

Sint[φ, φ̄] =g2

∫ +∞−∞

dt φ̄(t)[φ(t)− φ̄(t)

]φ(t) .




Field-theoretic renormalization group:

Main results:Lower critical threshold: κc = 13Exact scaling relation: βν‖ =

1−κ2

Critical exponents:

ν‖ = 3−94ε+O(ε2)

β = 1− 94ε+O(ε2)

where ε = κ− κc .




Minimal space dimension needed to observe a continuousphase transition:

equilibrium short-range 2equilibrium long-range 1nonequilibrium short-range 1nonequilibrium long-range 0



Boundary-induced phase transition

This process is related to a boundary-induces phasetransition

A.C. Barato and H.H., Phys. Rev. Lett. (2008)




Here we have κ = 1/2.Haye Hinrichsen, University of Würzburg


Long-range DP in higher dimensions

DP Langevin equation with Lévy flights and waiting times:

∂κt ρ(x, t) = aρ(x, t)− ρ(x, t)2 + D∇σρ(x, t) +√ρ(x, t)ξ(t) .

H. K. Janssen et al, Eur. Phys. B, (1999 and 2008) H. H. and M. Howard,Eur. Phys. B, (1999) J. Adamek and H. H., JSTAT (2005)A. Jimenez-Dalmaroni, Phys. Rev. E (2006)



Long-range DP in higher dimensions

Fluctuation effects displace the naive short-range threshold.



Outline








Local Scale Invariance

Local Scale Invariance (LSI) is a concept that aims atgeneralizing conformal invariance to anisotropicdynamical systems.



Local Scale Invariance

Equilibrium in 2d:



From Scaling to Conformal Invariance

Example: Conformally transformed lattice:

Conformaltransformations

are generated byLn = zn+1∂z



Conformal generator L−1 = ∂z : Translations

L−1 = ∂z



Conformal generator L0 = z∂z : Dilatations (scaling)

L0 = z∂z



Conformal generator L1 = z2∂z

L1 = z2∂z



Conformal generator L2 = z3∂z

L2 = z3∂z



Dynamical Systems

Time-dependent systems:

... involve time.

Time is directed while space is not.Generally there is no symmetry between space and time.Space and time scale differently.One has to respect causality.

⇒ Anisotropic scaling.



Anisotropic Conformal Invariance ?

Nonequilibrium in (1+1)d:



The basic idea of LSI

Perform ’conformal’ transformation in 1d on the time axis.Perform comoving time-dependent dilatation of space.Carry out time dependent rescaling of the order parameter.



LSI generators and commutation relations:

General structure: The system under consideration is assumedto be invariant under the action of the generators Xn,Ym whichobey the commutation relations

[Xn,Xm] = (n −m)Xn+m[Xn,Ym] =

(nz−m

)Yn+m ,

where z is the dynamical exponent.



LSI generator X−1 = −∂t : Time translations

X−1 = −∂t



LSI generator X0 = −t∂t − 12r∂r : Anisotropic Scaling

X0 = −t∂t−12r∂r



LSI generator X1 = −t2∂t − tr∂r

X1 = −t2∂t−tr∂r



Local scale transformation visualized

Original lattice:




Lattice mapped by exp(X0):




Main problem of LSI:

Process under consideration has to be in some senseinvariant under shear.



For a random walk (z = 2) it works:

General structure:

Xn = −tn+1∂t −n + 1

2tnr∂r − fX (n, r , t)

Yn = −tm+1/2∂r − fY (n, r , t)Mn = −fM(n, r , t)

[Xn,Xm] = (n −m)Xn+m[Xn,Ym] = (n/2−m)Yn+m[Xn,Mm] = −mMn+m[Yn,Ym] = (n −m)Mn+m[Yn,Mm] = [Mn,Mm] = 0



How shear is compensated in the Gaussian case:



How general is LSI ?

On the predictive power of Local Scale Invariance, JSTAT(2008).

Knowing one generator (X2), all other generators can beconstructed recursively.All generators have a kernel representation.Compensation of shear requires to choose particularkernel functions.No predictive power regarding two-point functions.Probably LSI makes nontrivial predictions for three-pointfunctions.



How general is LSI ?

Good news: LSI works with fractional derivatives!



Spatially nonlocal representation of LSI generators

Fractional representation for arbitrary z in 1+1 dimensions-

Type I: Non-local in space:

X1 = −t2∂t −2z

tr∂r −2xz

t (1)

−(β + γ)r2∂2−zr − 2γ(2− z)r∂1−zr−γ(2− z)(1− z)∂−zr

Y1−1/z = −t∂r − (β + γ)zr∂2−zr − γz(2− z)∂1−zr . (2)

M. Henkel and F. Baumann, 2007.



Temporally nonlocal representation of LSI generators

Type II: Non-local in time:

X1 = −t2∂t −2z

tr∂r −2xz

t − αr2∂2/z−1t

Y1−1/z = −t∂r − αzr∂2/z−1t .



Conclusion

1 Lévy-stable distributions and fractional calculus are closelyrelated.

2 Fractional derivatives are non-local linear operators withpower-law characteristics. Various definitions exist.

3 Spatial and temporal derivatives in physics.

4 Simple non-equilibrium phase transition on a timeline

5 LSI is valid for systems based on linear fractionaldifferential equations.



Invitation

- Take any partial differential equation.

- Replace ∇2 by ∇̃σ and ∂t by ∂̃κt .

- Look what happens....



Thank you !


Lévy-stable distributionsFractional calculusSimple applicationsNonequilibrium phase transitionsApplications to Local Scale Invariance

Lévy flights, fractional calculus and non-equilibrium phase ...mbsffe09/talks/Hinrichsen.pdfKenneth...

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