Outline Langevin dynamics Fluctuation Relations Non-Gaussian dynamics Experiments Summary Anomalous Transport and Fluctuation Relations: From Theory to Biology Aleksei V. Chechkin 1 , Peter Dieterich 2 , Rainer Klages 3 1 Institute for Theoretical Physics, Kharkov, Ukraine 2 Institute for Physiology, Dresden University of Technology, Germany 3 Queen Mary University of London, School of Mathematical Sciences Nonequilibrium Processes at the Nanoscale Kavli Institute for Theoretical Physics, Beijing, 5 August 2016 Anomalous Transport and Fluctuation Relations Rainer Klages 1
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Anomalous Transport and Fluctuation Relations: From Theory ...klages/talks/afr_kitpc.pdfAnomalous Transport and Fluctuation Relations: From Theory to Biology Aleksei V. Chechkin1,
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Mori, Kubo (1965/66): generalize ordinary Langevin equation to
mv = −∫ t
0dt ′ κ(t − t ′)v(t ′) + k ζ(t)
by using a time-dependent friction coefficient κ(t) ∼ t−β;applications to polymer dynamics (Panja, 2010) and biologicalcell migration (Dieterich, RK et al., 2008)
solutions of this Langevin equation:
position pdf is Gaussian (as the noise is still Gaussian)
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 :
why important? of very general validity and1 generalizes the Second Law to small systems in noneq.2 connection with fluctuation dissipation relations3 can be checked in experiments (Wang et al., 2002)
Anomalous Transport and Fluctuation Relations Rainer Klages 8
with constant field F and Gaussian white noise ζ(t)
entropy production ξt is equal to (mechanical) work Wt = Fx(t)with ρ(Wt) = F−1(x , t); remains to solve the correspondingFokker-Planck equation for initial condition x(0) = 0
the position pdf is again Gaussian, which impliesstraightforwardly:
(work) TFR holds if < x >= Fσ2x/2
hence FDR1 ⇒ TFR
see, e.g., van Zon, Cohen, PRE (2003)
Anomalous Transport and Fluctuation Relations Rainer Klages 10
Fluctuation relation for anomalous Langevin dynamics
check TFR for overdamped generalized Langevin equation
∫ t
0dt ′x(t ′)κ(t − t ′) = F + ζ(t)
both for internal and external power-law correlated Gaussiannoise κ(t) ∼ t−β
1. internal Gaussian noise:• as FDR2 implies FDR1 and ρ(Wt) ∼ (x , t) is Gaussian, itstraightforwardly follows the existence of the transientfluctuation relation
for correlated internal Gaussian noise ∃ TFR• diffusion and current may both be normal or anomalousdepending on the memory kernel
Anomalous Transport and Fluctuation Relations Rainer Klages 11
means by plotting R for different t the slope might change.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
0.4χ
-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)similar results for other glassy systems (Sellitto, PRE, 2009)
Anomalous Transport and Fluctuation Relations Rainer Klages 18