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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