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

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Lévy-stable distributions Fractional calculus Simple applications Nonequilibrium phase transitions Applications to Local Scale Inv Lévy flights, fractional calculus and non-equilibrium phase transitions with long-range interactions Haye Hinrichsen University of Würzburg, Germany MBSFF09 – MPIPKS Dresden February 2009 Haye Hinrichsen, University of Würzburg

Transcript of 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

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

    Outline

    1 Lévy-stable distributions

    2 Fractional calculus

    3 Simple applications

    4 Nonequilibrium phase transitions

    5 Applications to Local Scale Invariance

    Haye Hinrichsen, University of Würzburg

  • Lévy-stable distributions Fractional calculus Simple applications Nonequilibrium phase transitions 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.

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    Lévy-stable probability densities

    Textbook form:

    P(x ; α, β, c, µ) =1

    ∫ +∞−∞

    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.

    Haye Hinrichsen, University of Würzburg

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

    Symmetric Lévy-stable distributions:

    (Wikipedia)

    Haye Hinrichsen, University of Würzburg

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

    Lévy-stable distributions have algebraic tails:

    P(x) ' αcα(1 + β) sin(πα/2) Γ(α)

    π|x |1+α∼ |x |−1−α

    Haye Hinrichsen, University of Würzburg

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

    Asymmetric Lévy-stable distributions:

    Haye Hinrichsen, University of Würzburg

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

    Corresponding cumulative distribution function:

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    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)

    Haye Hinrichsen, University of Würzburg

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

    Outline

    1 Lévy-stable distributions

    2 Fractional calculus

    3 Simple applications

    4 Nonequilibrium phase transitions

    5 Applications to Local Scale Invariance

    Haye Hinrichsen, University of Würzburg

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

    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 .

    Haye Hinrichsen, University of Würzburg

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

    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

    Haye Hinrichsen, University of Würzburg

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

    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...

    Haye Hinrichsen, University of Würzburg

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

    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)

    Haye Hinrichsen, University of Würzburg

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

    Fractional derivatives

    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)

    )

    Haye Hinrichsen, University of Würzburg

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

    Fractional derivatives

    Desired properties of a fractional derivative:

    1 Linearity2 Translational invariance3 Homogeneity4 Dνx D

    µx = D

    ν+µx

    Haye Hinrichsen, University of Würzburg

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

    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)

    Haye Hinrichsen, University of Würzburg

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

    Fractional derivatives - general form

    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.

    Haye Hinrichsen, University of Würzburg

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

    Fractional derivatives - general form

    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

    Haye Hinrichsen, University of Würzburg

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

    Outline

    1 Lévy-stable distributions

    2 Fractional calculus

    3 Simple applications

    4 Nonequilibrium phase transitions

    5 Applications to Local Scale Invariance

    Haye Hinrichsen, University of Würzburg

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

    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 .

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    Lévy flights in space

    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)

    Haye Hinrichsen, University of Würzburg

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

    Lévy flights in space

    Haye Hinrichsen, University of Würzburg

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

    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

    )

    Haye Hinrichsen, University of Würzburg

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

    Application to empirical data

    Haye Hinrichsen, University of Würzburg

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

    Application to empirical data

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

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

    Application to empirical data

    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.

    Haye Hinrichsen, University of Würzburg

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

    Application to empirical data

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

    Haye Hinrichsen, University of Würzburg

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

    Application to empirical data

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

    Haye Hinrichsen, University of Würzburg

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

    Application to empirical data

    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

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    Fractional derivative in time:

    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

    Haye Hinrichsen, University of Würzburg

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

    Fractional derivative in time:

    Simplest example:∂κt ρ(t) = δ(t)

    ⇒ (iω)κρ̃(ω) = 1/√

    ⇒ ρ(t) ∝{

    0 for t ≤ 0tκ−1 for t ≥ 0

    Differential equations of this type describenon-Markovian processes.

    Haye Hinrichsen, University of Würzburg

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

    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)

    Haye Hinrichsen, University of Würzburg

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

    Outline

    1 Lévy-stable distributions

    2 Fractional calculus

    3 Simple applications

    4 Nonequilibrium phase transitions

    5 Applications to Local Scale Invariance

    Haye Hinrichsen, University of Würzburg

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

    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)

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    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.

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    C-Code:

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    This model exhibits a continuous phase transition:

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing 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) .

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    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 .

    Haye Hinrichsen, University of Würzburg

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

    Simplest absorbing phase transition

    Minimal space dimension needed to observe a continuousphase transition:

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

    Haye Hinrichsen, University of Würzburg

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

    Boundary-induced phase transition

    This process is related to a boundary-induces phasetransition

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

    Haye Hinrichsen, University of Würzburg

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

    Boundary-induced phase transition

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

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

    Boundary-induced phase transition

    Haye Hinrichsen, University of Würzburg

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

    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)

    Haye Hinrichsen, University of Würzburg

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

    Long-range DP in higher dimensions

    Fluctuation effects displace the naive short-range threshold.

    Haye Hinrichsen, University of Würzburg

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

    Outline

    1 Lévy-stable distributions

    2 Fractional calculus

    3 Simple applications

    4 Nonequilibrium phase transitions

    5 Applications to Local Scale Invariance

    Haye Hinrichsen, University of Würzburg

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

    Local Scale Invariance

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

    Haye Hinrichsen, University of Würzburg

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

    Local Scale Invariance

    Equilibrium in 2d:

    Haye Hinrichsen, University of Würzburg

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

    From Scaling to Conformal Invariance

    Example: Conformally transformed lattice:

    Conformaltransformations

    are generated byLn = zn+1∂z

    Haye Hinrichsen, University of Würzburg

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

    Conformal generator L−1 = ∂z : Translations

    L−1 = ∂z

    Haye Hinrichsen, University of Würzburg

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

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

    L0 = z∂z

    Haye Hinrichsen, University of Würzburg

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

    Conformal generator L1 = z2∂z

    L1 = z2∂z

    Haye Hinrichsen, University of Würzburg

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

    Conformal generator L2 = z3∂z

    L2 = z3∂z

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    Anisotropic Conformal Invariance ?

    Nonequilibrium in (1+1)d:

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

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

    X−1 = −∂t

    Haye Hinrichsen, University of Würzburg

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

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

    X0 = −t∂t−12r∂r

    Haye Hinrichsen, University of Würzburg

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

    LSI generator X1 = −t2∂t − tr∂r

    X1 = −t2∂t−tr∂r

    Haye Hinrichsen, University of Würzburg

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

    Local scale transformation visualized

    Original lattice:

    Haye Hinrichsen, University of Würzburg

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

    Local scale transformation visualized

    Lattice mapped by exp(X0):

    Haye Hinrichsen, University of Würzburg

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

    Local scale transformation visualized

    Lattice mapped by exp(X1):

    Haye Hinrichsen, University of Würzburg

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

    Local scale transformation visualized

    Lattice mapped by exp(X2):

    Haye Hinrichsen, University of Würzburg

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

    Local scale transformation visualized

    Main problem of LSI:

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

    Haye Hinrichsen, University of Würzburg

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

    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

    Haye Hinrichsen, University of Würzburg

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

    How shear is compensated in the Gaussian case:

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    How general is LSI ?

    Good news: LSI works with fractional derivatives!

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    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 .

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    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.

    Haye Hinrichsen, University of Würzburg

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

    Invitation

    - Take any partial differential equation.

    - Replace ∇2 by ∇̃σ and ∂t by ∂̃κt .

    - Look what happens....

    Haye Hinrichsen, University of Würzburg

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

    Thank you !

    Haye Hinrichsen, University of Würzburg

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