Motivation Fractional Fokker-Planck equations Experiments Summary

Fluctuation Relations for Anomalous DynamicsGenerated by Time Fractional

Fokker-Planck Equations

Peter Dieterich1, Rainer Klages2,3, Aleksei V. Chechkin2,4

1 Institute for Physiology, Technical University of Dresden, Germany

2 Max Planck Institute for the Physics of Complex Systems, Dresden, Germany

3 Queen Mary University of London, School of Mathematical Sciences

4 Institute for Theoretical Physics, Kharkov, Ukraine

DPG Spring Meeting, Regensburg, 10 March 2016

Fluctuation Relations for Anomalous Stochastic Dynamics Rainer Klages 1

Motivation Fractional Fokker-Planck equations Experiments Summary

Motivation: Fluctuation relations

Consider a (classical) particle system evolving from some initialstate into a nonequilibrium steady state.Measure the probability distribution ρ(ξt) of entropy productionξt during time t :

lnρ(ξt)

ρ(−ξt)= ξt

Transient Fluctuation Relation (TFR)

Evans, Cohen, Morriss (1993); Gallavotti, Cohen (1995)

why important? of very general validity and1 generalizes the Second Law to small systems in nonequ.2 connection with fluctuation dissipation relations (FDRs)3 can be checked in experiments (Wang et al., 2002)

Fluctuation Relations for Anomalous Stochastic Dynamics Rainer Klages 2

Motivation Fractional Fokker-Planck equations Experiments Summary

Anomalous TFR for Gaussian stochastic processes

known result:consider overdamped generalized Langevin equation

x = F + ζ(t)with force F and Gaussian power-law correlated noise

< ζ(t)ζ(t ′) >τ=t−t ′∼ (∆/τ)β for τ > ∆ , β > 0that is external (i.e., no FDR):

dynamics can generate anomalous diffusion ,σ2

x ∼ t2−β with 2 − β 6= 1 (t → ∞)

yields an anomalous work fluctuation relation ,

lnρ(Wt)

ρ(−Wt)= fβ(t)Wt

A.V.Chechkin et al., J.Stat.Mech. L11001 (2012) and L03002 (2009)

Question: what’s about non-Gaussian processes?

Fluctuation Relations for Anomalous Stochastic Dynamics Rainer Klages 3

Motivation Fractional Fokker-Planck equations Experiments Summary

Modeling non-Gaussian dynamics

• start again from overdamped Langevin equation x = F + ζ(t),but here with non-Gaussian power law correlated noise

< ζ(t)ζ(t ′) >τ=t−t ′∼ (Kα/τ)2−α , 1 < α < 2

• ‘motivates’ the non-Markovian Fokker-Planck equation

type A: ∂A(x ,t)∂t = − ∂

∂x

[

F − KαD1−αt

∂∂x

]

A(x , t)

with Riemann-Liouville fractional derivative D1−αt (Balescu, 1997)

• two formally similar types derived from CTRW theory, for0 < α < 1:

type B: ∂B(x ,t)∂t = − ∂

∂x

[

F − KαD1−αt

∂∂x

]

B(x , t)

type C: ∂C(x ,t)∂t = − ∂

∂x

[

FD1−αt − KαD1−α

t∂∂x

]

C(x , t)

They model a different class of stochastic process!

Fluctuation Relations for Anomalous Stochastic Dynamics Rainer Klages 4

Motivation Fractional Fokker-Planck equations Experiments Summary

Properties of non-Gaussian dynamics

Riemann-Liouville fractional derivative defined by

∂γ

∂tγ:=

{

∂m∂tm , γ = m∂m

∂tm

[

1Γ(m−γ)

∫ t0 dt ′ (t ′)

(t−t ′)γ+1−m

]

, m − 1 < γ < m

with m ∈ N; power law inherited from correlation decay.

two important properties:

• FDR: exists for type C but not for A and B

• mean square displacement:

- type A: superdiffusive, σ2x ∼ tα , 1 < α < 2

- type B: subdiffusive, σ2x ∼ tα , 0 < α < 1

- type C: sub- or superdiffusive, σ2x ∼ t2α , 0 < α < 1

• position pdfs: can be calculated approximately analyticallyfor A, B, only numerically for C

Fluctuation Relations for Anomalous Stochastic Dynamics Rainer Klages 5

Motivation Fractional Fokker-Planck equations Experiments Summary

Probability distributions and fluctuation relations

type A type B type C• PDFs:

• TFRs:

R(Wt) = log ρ(Wt )ρ(−Wt )

∼

{

cαWt , Wt → 0

t(2α−2)/(α−2)W α/(2−α)t , Wt → ∞

Fluctuation Relations for Anomalous Stochastic Dynamics Rainer Klages 6

Motivation Fractional Fokker-Planck equations Experiments Summary

Relations to experiments: glassy dynamics

example 1: computer simulations for a binary Lennard-Jonesmixture below the glass transition

0 3 6 9 12 15∆S

100

101

102

103

104

P t w(∆

S)

/ Pt w

(-∆S

)

tw

= 102

tw

=103

tw

=104

0 0.5 11-C0

0.2

0.4

χ

-15 0 15 30 45∆S

10-5

10-4

10-3

10-2

10-1

P t w(∆

S)

(a)

(b)

Crisanti, Ritort, PRL (2013)

• again: R(Wt) = lnρ(Wt)

ρ(−Wt)= fβ(t)Wt ; cp. with TFR type B

• similar results for other glassy systems (Sellitto, PRE, 2009)

Fluctuation Relations for Anomalous Stochastic Dynamics Rainer Klages 7

Motivation Fractional Fokker-Planck equations Experiments Summary

Cell migration without and with chemotaxis

example 2: single MDCKF cellcrawling on a substrate;trajectory recorded with a videocamera

Dieterich et al., PNAS, 2008

new experiments on murineneutrophils under chemotaxis:

Dieterich et al. (2013)

Fluctuation Relations for Anomalous Stochastic Dynamics Rainer Klages 8

Motivation Fractional Fokker-Planck equations Experiments Summary

Anomalous fluctuation relation for cell migration

preliminary experimental results:

< x(t) >∼ t and σ2x ∼ t2−β with 0 < β < 1: 6 ∃ FDR

fluctuation ratio R(Wt) is time dependent :

R(Wt) = lnρ(Wt)

ρ(−Wt)=

Wt

t1−β

Fluctuation Relations for Anomalous Stochastic Dynamics Rainer Klages 9

Motivation Fractional Fokker-Planck equations Experiments Summary

Summary

TFR tested for non-Gaussian dynamics modeled by threecases of time fractional Fokker Planck equations :

breaking FDR implies (again) anomalous TFRs

for non-Gaussian dynamics the TFR displays a nonlineardependence on the (work) variable, in contrast to Gaussianstochastic processes

anomalous TFRs appear to be important for glassy ageingdynamics

Fluctuation Relations for Anomalous Stochastic Dynamics Rainer Klages 10

Motivation Fractional Fokker-Planck equations Experiments Summary

References

P.Dieterich, RK, A.V. Chechkin, NJP 17, 075004 (2015)

Fluctuation Relations for Anomalous Stochastic Dynamics Rainer Klages 11