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Measures for diffusion, ergodicity, & ageingRalf Metzler, U Potsdam & Wroc law U Science & Technology
— Lanzhou, 4th August 2018 —
– Typeset by FoilTEX– 1
Microscopical observations
on the particles contained in the pollen of plants; and on the generalexistence of active molecules in organic and inorganic bodies
Rocks of all ages, including thosein which organic remains havenever been found, yielded themolecules in abundance. Theirexistence was ascertained in eachof the constituent molecules ofgranite, a fragment of the Sphinxbeing one of the specimens ex-amined.
R Brown, Edinburgh new Philosophical Journal 358-371 (1828) 2
Brownian motion
P (r,∆t) =1
(4πK∆t)d/2
exp
(−
r2
4K∆t
)
Einstein-Smoluchowski relation:
K =kBT
mη=
(R/NA)T
mη∆t=30 sec
J Perrin, Comptes Rendus (Paris) 146 (1908) 967: NA = 70.5× 1022 4
Ivar Nordlund: 100+ years of SPT with time series analysis
Mercury droplet in aqueous solution
I Nordlund, Z Physik (1914): NA = 5.91× 1023 7
Eugen Kappler: ultimate diffusion measurements
Obiturary by L Reimer, Physikalische Blatter Feb 1978 pp 86 8
Eugen Kappler: ultimate diffusion measurements
E Kappler, Ann d Physik (1931): NA = 60.59× 1022 ± 1% 9
Brownian motion & Kappler’s diffusion measurements
Distribution @ fixed photographic plate
〈r2(t)〉 = 2dKt
P (r, t) = (4πKt)−d/2
exp(−r2/[4Kt])
E Kappler, Ann d Physik (1931): NA = 60.59× 1022 ± 1% 10
Single molecule imaging, state-of-the-art
10–6 10–5 10–4 10–3
Time (s)
10–6 10–5 10–4 10–3
Time (s)
GFP in water
GFP in cytoplasm
Diffraction limit
10–3
10–2
10–1
iM
SD
(µ
m2)
100
10–3
10–2
10–1
100
iM
SD
(µ
m2)
GFP dimer in water
GFP dimer in cytoplasm
Diffraction limit
100 µs
0.5 µs
C di Renzo et al, Nat Comm (2014)
Cou
rtes
yY
uva
lG
arin
i
S Shao, B Xue & Y Sun, Biophys J (2018) 11
When Brownian diffusion is not Gaussian
A B
-1 0
-1
0
1
2
log
<x
2>
/σ2
log t (s)
-60 -30 0 30 60
-3
-2
-1
0
log
Gs(x
,t)
x/σ
C
1
0.0 0.5 1.0
-1
0
1
log
Gs(r
,t)
r (μm)
C
-1 0 1 2-1
0
1
2
log
<Δ
r2>
/ξ2
log t (s)
B
1
A
ξ
2a
Colloidal beads diffusing on nanotubes Nanospheres diffusing in entangled actin
Wang et al, PNAS (2009); Nature Mat (2012) 12
Heterogeneous diffusion in population of nematodes
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
2468
101214(a)
diffusion coefficient (mm2 s–1)
probab
ility d
ensity
funct
ion
(b)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.800.10.20.30.40.50.60.70.80.91.0
cumula
tive p
robabi
lity fu
nction
0
0.1
0.2
0.3
0.4
–3 –2 –1 0radians
relativ
e freq
uency
1 2 3
Soy
bea
ncy
stn
emat
od
e&
egg
Mal
eC
.el
egan
sn
emat
od
e
S Hapca, JW Crawford & IM Young, Roy Soc Interface (2009) 13
Non-Gaussian diffusion of Dictyostelium cells
0 100 200 300 400 500 600
0
100
200
300
400
500
600
x, m
y,m
1.21 t 20 sec
3 2 1 0 1 2 3
104
103
102
101
100
x, m
Nx
1.15 t 60 sec
5 0 5
104
103
102
101
100
x, m
Nx
1.09 t 200 sec
20 10 0 10 20
104
103
102
101
100
x, m
Nx
1.06 t 600 sec
30 20 10 0 10 20 30
104
103
102
101
100
x, m
Nx
AG Cherstvy, O Nagel, C Beta & RM (2018) 14
Non-Gaussian diffusion of Dictyostelium cells
4 2 0 2 4
0.0
0.2
0.4
0.6
0.8
z x s2 3
pz
e1.93
0 0.5 1 1.5 2
100
10 1
10 2
10 3
K,m2sec
nfit 5
e2.55
0 0.5 1 1.5 2
100
10 1
10 2
10 3
K,m2sec
nfit 15
Similarity function Kβ ' exp(−c1β + c2)
AG Cherstvy, O Nagel, C Beta & RM (2018) 15
Fickian, non-Gaussian diffusion with diffusing diffusivity
B Wang, J Kuo, SC Bae & S Granick, Nat Mat (2012): 〈x2(t)〉 = 2K1t, yet P (x, t)
non-Gaussian. Superstatistical approach P (x, t) =∫∞
0G(x, t)p(D)dD
MV Chubinsky & G Slater, PRL (2014); R Jain & KL Sebastian, JPC B (2016): diffusing
diffusivity
Our minimal model for diffusing diffusivity:
x(t) =√
2D(t)ξ(t)
D(t) = y2(t)
y(t) = −τ−1y + ση(t)
0.001
0.01
0.1
1
-4 -2 0 2 4
P(x,t)
x
Sim t=0.5Sim t=0.2Sim t=0.1Eq (27) t=0.5Eq (27) t=0.2Eq (27) t=0.1
0.001
0.01
0.1
1
-20 -15 -10 -5 0 5 10 15 20
P(x,t)
x
Sim t=100Sim t=10Sim t=1IFT t=100IFT t=10IFT t=1
0
4000
8000
12000
16000
20000
0 4000 8000 12000 16000 20000
<x2 (t)>
t 2
3
4
5
6
7
8
9
0.01 0.1 1 10 100 1000 10000
kurto
sist
TheorySimulation
AV Chechkin, F Seno, RM & IM Sokolov, PRX (2017); generalised γ(D): V Sposini, AV Chechkin, G Pagnini, F Seno & RM, NJP (2018) 16
Passive motion of submicron tracers in cells is viscoelastic
! ! !" ! !
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Tracer
bead
sin
worm
likem
icellarsolu
tion−→
Lip
idgran
ules
inlivin
gyeast
cells↓
JH Jeon, . . . L Oddershede & RM, PRL (2011); JH Jeon, N Leijnse, L Oddershede & RM, NJP (2013) 17
Superdiffusion in supercrowded Acanthamoeba castellani
inhibits myosin II motors
depolymerises microtubules
inhibits actin polymerisation
y motors cause superdiffusion
Centrosome
MicrotubulusNucleus
GranuleAc n filaments
Vesicle
Diges ve
vacuole
Contrac le
vacuoleAcanthapodium
A B
JF Reverey, J-H Jeon, H Bao, M Leippe, RM & C Selhuber-Unkel, Sci Rep (2015) 18
Non-Gaussian diffusion in viscoelastic systems
So far consensus: submicron tracer motion in cytoplasm is FBM-like, i.e., Gaussian
RNA-protein particles in E.coli & S.cerevisiae perform exponential anomalous diffusion:
5 4 3 2 1 0 1 2 3 4 510
3
10 2
10 1
100
!x"/#
"
P(!
x"/#
")
Laplace
Gaussian
1.5 1.0 0.5 0.0 0.5 1.0 1.510
2
10 1
100
101
!x"
(µm)
P(!
x")
(µm
1)
"=1 s
"=10 s
"=2 min
"=17 min
a
c
E. coli
5 4 3 2 1 0 1 2 3 4 510
3
10 2
10 1
100
!x"/#
"
P(!
x"/#
")
Laplace
Gaussian
1.5 1.0 0.5 0.0 0.5 1.0 1.510
2
10 1
100
101
!x"
(µm)
P(!
x")
(µm
1)
"=0.015 s
"=0.15 s
"=1.5 s
"=15 s
b
d
S. cerevisiae
E. coli S. cerevisiae
0 1 2 3 4 5 6 7 810
3
10 2
10 1
100
D/<D>
P(D
/<D
>)
E. coli 1 15 s
E. coli 1 15 min
S. cerevisiae 0.015 0.75 s
Exponential
10-710-610-510-410-310-210-1100
-10 -5 0 5 10
P(x,t)
x
Modelling based on grey GLE: J Slezak, RM & M Magdziarz, NJP (2018)
TJ Lampo, S Stylianidou, MP Backlund, PA Wiggins & AJ Spakowitz, BPJ (2017); N&V: RM, BPJ (2017) 19
Single lipid motion in bilayer membrane MD simulations
Liquid disordered Liquid ordered Gel phase
J-H Jeon, H Martinez-Seara Monne, M Javanainen & RM, PRL (2012) 20
Sample trajectories for the lipid & cholesterol motion
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Liquid disordered Liquid ordered Gel phase
� � �Liquid ordered/gel phases: extended anomalous diffusion
J-H Jeon, H Martinez-Seara Monne, M Javanainen & RM, PRL (2012) 21
Tempered FBM & FLE motion: sub- to normal diffusion
Consider tempered fGn:
〈ξ(t)ξ(t+ τ)〉 =
C
Γ(2H − 1)τ
2H−2e−τ/τ?
C
Γ(2H − 1)τ
2H−2
(1 +
τ
τ?
)−µ
10-2 100 102 104 106
0.2
1
5
20
t
<x2HtL>�t
10-2 10-1 100 101 102
10-2
10-1
100
101
t, ns
<x2HtL>
,nm
2
D Molina-Garcia, T Sandev, G Pagnini, AV Chechkin & RM (2018) 22
Reproducible TA MSD & antipersistent correlations
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Rattling dynamics: exptl first passage PDF y FLE motion
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D Molina-Garcia, T Sandev, G Pagnini, AV Chechkin & RM (2018) 23
Protein crowded membranes reduce effective mobility
1:50
1:75
1:100
1:150
1:200
1:INF
0
2
4
·10−7
0.10.5
1.01.01.4
2.0 0.61.2
1.52.0
2.4
3.3Diffusioncoeffi
cientof
lipids(cm
2 /s)
1:50
1:75
1:100
1:150
1:200
1:INF
0
20
40
60
10−9
0.41.52.74.610.9
40.6
2.411.513.0
28.3
40.2
59.8
Diffusioncoeffi
cientof
proteins
(cm
2 /s)
Protein crowding effects anomalous lipid diffusion
1 ns 10 ns 100 ns 1 µs 10 µs 100 µs
0.6
0.8
1
Time
α
1 ns 10 ns 100 ns 1 µs 10 µs 100 µs
0.8
0.9
1
1.1
Time
α
Left: DPPC (protein-aggregating) case. Right: DLPC protein non-aggregating case.
Blu
e:D
LP
C.
Red
:D
PP
C
:::: ::::
M Javanainen, H Hammaren, L Monticelli, JH Jeon, RM & I Vattulainen, Faraday Disc (2013) 24
Crowding in membranes increases dynamic heterogeneity
0
0.5
1
α
0
0.5
1
0.8
0.9
1
100
101
102
103
104
105
lag time ∆ (ns)
10-2
100
102
104
100
101
102
103
104
105
lag time ∆ (ns)
10-2
100
102
104
100
101
102
103
104
lag time ∆ (ns)
10-2
100
102
104
δ2(∆
)
0.7
0.8
0.9
1
0.6
0.8
1
0
0.5
1
Single NaK channel DLPC (non-aggregating) DPPC (aggregating)
↓B
lue:
lipid
s.R
ed:
protein(s)
0 100 200 300 400 500
lag time ∆ (ns)
-0.2
0
0.2
0.4
Cδt(∆
)/C
δt(0
)
DPPC (crowded membrane)
FLE theoryLipids & proteins behave quite differently
Increment correlation no longer simple FBM −→
J-H Jeon, M Javanainen, H Martinez-Seara, RM & I Vattulainen, PRX (2016) 25
Crowding in membranes: non-Gaussian lipid/protein diffusion
!""# $"%&'()*
+,-$*.
10-2
100
102
104
10-1
100
101
-log(1
-Π)
10-2
10-1
100
101
102
103
10410
-1
100
101
-log(1
-Π)
10-2
10-1
100
101
102
103
104
r2 (nm
2)
10-1
100
101
-log(1
-Π)
~r2
~r1.6
~r1.6
~r1.4~r
1.6
~r1.6
~r1.6
~r1.7
Dilute membrane: P (r, t) Gauss
Crowded membrane (δ ≈ 1.3 . . . 1.7):
P (r, t) ∝ exp
(−[ r
ctα/2
]δ)10-2 10-1 100 101 102
r2 (nm2)
10-1
100
101
-log(1
-
0 1 2 3 4 5r (nm)
-6-4-2024
log P r(r,
)
10-3 10-2 10-1 100 101
r2 (nm2)10-1
100
101
-log(1
-0 0.5 1 1.5 2
r (nm)-8-6-4-2024
log P r(r,
)
10-3 10-2 10-1 100 101 102
r2 (nm2)10-1
100
101
-log(1
-Π)
0 2 4 6r (nm)
-5
0
5
log P r(r,∆
)
2=1.691=1.56
δ2=1.58 δ2=1.28
δ1=1.39
~r2
~r2
δ2=1.58
δ1=1.93~r2
δ2=1.28δ2=1.53
δ1=1.67
1=1.83
1=1.60
2=1.26 2=1.16
J-H Jeon, M Javanainen, H Martinez-Seara, RM & I Vattulainen, PRX (2016) 26
Crowding in membranes increases dynamic heterogeneity
0 5 10 15 20 25
0.1
0.2K
α(t
)
0 5 10 15 20 25time t (µs)
0
0.1
0.2
Kα(t
)
0 0.05 0.1 0.15 0.2 0.25K
α
0
5
10
15
20
25
P(K
α)
DPPC lipids (dilute)
0 20 40 60 800
0.05
0.1
Kα(t
)
0 20 40 60 80time t (µs)
0
0.05
0.1
Kα(t
)
0 0.05 0.1 0.15 0.2K
α
0
5
10
15P
(Kα)
Diffusivity(t) for two lipids
Lipid diffusivity, dilute membrane
Lipid diffusivity, crowded membrane
J-H Jeon, M Javanainen, H Seara Monne, RM & I Vattulainen, PRX (2016) 27
Confinement in argon system shows geometric origin
a b c
d e
f g h
J-H Jeon, M Javanainen, H Seara Monne, RM & I Vattulainen, PRX (2016) 28
Geometry-induced violation of Saffman-Delbruck relation
P12K1L P21QJP P32F1C P41NQG P52W5J P61LDF P73TDP
0
0.2
0.4
0.6
0.8Dilute
D/D
0
CG2DLJ
1 3 100.030.10.3
0 5 10 15 200
0.2
0.4 Crowded
r/r0
D/D
0
D∝R−1 D∝ln R−11 3 100.01
0.030.10.3
Dilute system: Saffman-Delbruck law
D(R) ' log(1/R)
Crowded membrane & 2DLJ discs:
D(R) ' 1/R
M Javanainen, H Seara Monne, RM & I Vattulainen, JPC Lett (2017) 29
CTRW-like motion of Ka channels in plasma membraneB CA B
○
ψ(τ) ' τ−1−α scale free
δ2(∆) apparently random
δ2(∆) 6= 〈r2(∆)〉 WEB
T
AV Weigel, B Simon, MM Tamkun & D Krapf, PNAS (2011) 30
Time averaged MSD & weak ergodicity breaking (WEB)
Time averaged MSD ' ∆ is pseudo-Brownian and ageing (〈x2(t)〉 ' Kαtα):⟨
δ2(∆)⟩∼
1
N
N∑i
δ2i (∆) ∼
2dKα
Γ(1 + α)
∆
T 1−α ∴ Kα ≡〈δr2〉2τα
Amplitude distribution δ2 of trajectories (ξ ≡ δ2/〈δ2〉):
φα(ξ) ∼Γ1/α(1 + α)
αξ1+1/αL
+α
(Γ1/α(1 + α)
ξ1/α
)
φ1/2(ξ) =2
πexp
(−ξ2
π
); φ1(ξ) = δ(ξ − 1)
Confinement does not effect a plateau (〈x2(t)〉 ' const(T )):⟨δ2(∆)
⟩∼(⟨
x2⟩B−〈x〉2B
) 2 sin(πα)
(1− α)απ
(∆
T
)1−α
;1
(Kαλ1)1/α� ∆� T
0 1 2 3 4 50
0.5
1
0 1 2 3 4 50
0.5
1
ξ
α = 0.5
α = 0.75
Y He, S Burov, RM & E Barkai, PRL (2008); S Burov, RM, & E Barkai, PNAS (2010) 31
Granule subdiffusion in harmonic optical tweezer potential
!" ! ! !
!
!"
#
!"
#
!"
#
!"
#
!"
!"
!"#
$%
!"
!"#
!! "#$ !
$%&'()*+ !"#$
%&
〈r2(t)〉 ' tα 〈r2(t)〉 ' const
α ≈ 0.80..0.86
J-H Jeon, V Tejedor, S Burov, E Barkai, C Selhuber-Unkel, K Berg-Sørensen, L Oddershede & RM, PRL (2011) 32
Ageing effects in single trajectory time averages
atageing period measurement t
Ageing mean squared displacement (Λ(z) = (1 + z)α − zα)⟨δ2(∆)
⟩a
=Λα(ta/T )
Γ(1 + α)
g(∆)
T 1−α < 〈x2(t)〉a '
{tα, ta � t
tα−1a t, ta � t
Probability to make at least one step during [ta, ta + T ]: population splitting
mα(T/ta) ' (T/ta)1−α
, T � ta
10−1
100
10110
−5
10−3
10−1
mα=1
∆
δ2
10−1
100
10110
−5
10−3
10−1
mα=0.063
∆
δ2
J Schulz, E Barkai & RM, PRL (2013), PRX (2014) 33
Self-similar internal protein dynamics: 13 decades of ageing
10−5
−2−12 −14
−5 −4 −3 −2 −1 0
−12 −10
−10
−8
−6
−6
−2
2
6−4
−2
0
2
0
0
PGK domains
PGK residues
K-RAS segments
ePepN, dom. 1−2
ePepN, dom. 2−3
ePepN, dom. 2−4
ePepN, dom. 4-rest
Ref. 3
Ref. 4
∝t0.9
2 4 6 8 10 12 14 16
2
4
6
8
log
10(
c/
10−
12 s
)τ
log10(t/10−12 s)
log
10[S
(f)/
(nm
2 1
0−
12 H
z−1 )
]
log10(f/10 12 Hz)
10
12 Experiments
Experiments
MD data
100 101 102 103 104 105 106
10−4100 ps10 ns500 ns
17 µs
100 ps
100 ps
10 ns
500 ns
17 µs
10 ns
500 ns
17 µs
∝ f−1
∝ f−3.6
Supplementary Equation 9
10−3
10−2a
10−2
10−1
100
1/e
C(
; t
)
1.5
(ps)
10−1 100 101 102 103 104 105 107106
0.2
0.1
b
c d
∆
2 (
; t)
δ∆
∆
∆
∆
∆
(ps)∆
d(t)
local conformational
change
1
2
3
4
5
6
7
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9
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101
102
103
104
105106
107
108
109
110
111
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120
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141142143
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© 2016 Macmillan Publishers Limited. All rights reserved
X Hu, L Hong, MD Smith, T Neusius, X Cheng & JC Smith, Nature Phys (2016); N&V RM Nature Phys (2016) 34
Power spectral density of a single Brownian trajectory
-30-20-10
0 10 20 30 40
2 4 6 8 10 12
ln(f-2)
ln S(f
)
ln f 0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8
P(A(γ S
))
A(γS)
)
)
Power spectrum /w conserved slope Amplitude distribution for various dT
heory
Exp
erimen
t
D Krapf, E Marinari, RM, G Oshanin, A Squarcini & X Xu, NJP (2018); Perspective: ND Schnellbacher & US Schwarz, NJP (2018) 35
Power spectral density of a single FBM trajectory
✶ ✶! ✶!! ✶!!! ✶!✹
✶!✺
✁✶✂!
✶✂✶
✶✂✄
✶✂☎
✶✂✆
✝❙✭❍✱✁✮
✞❂✸✴✟
✞❂✷✴✸
✞❂✠✴✷
✞❂✠✴✸
✞❂✠✴✟
✞❂✼✴✽
γ =
(〈S2
T (f)〉 − 〈ST (f)〉2)1/2
〈ST (f)〉
D Krapf, N Lukat, E Marinari, RM, G Oshanin, C Selhuber-Unkel, A Squarcini, L Stadler, M Weiss & X Xu (2018) 36
Power spectral density of a single FBM trajectory
Agaroseα = 0.8
Telomeres α = 0.5
Vacuolesα = 1.3
Amoebaα = 2
0 2 4 60.0
0.5
1.0
P(A
)
A0 2 4 6
0.0
0.4
0.8
P(A
)
A0 2 4 6
0.0
0.5
1.0
1.5
2.0
P(B
)
B
S(2
) (f =
0)
time (s)
0.1 1 10 1001E-6
1E-4
0.01
1
0.01 0.1 1 100.01
1
100
1E4
1E61E8
0.1 1 10
1E-4
0.01
1
100
0.01 0.1 1 10 100
1E-3
1
1000
1E6 agarose
telomeres vacuoles amoeba
0.01 0.1 1 10
0.01
1
PS
D (µ
m2 /H
z)
f (Hz)f (Hz)f (Hz)f (Hz)
PS
D (µ
m2 /H
z)
PS
D (µ
m2 /H
z)
PS
D (µ
m2 /H
z)1E-4
Agarose Telomeres Vacuoles
D Krapf, N Lukat, E Marinari, RM, G Oshanin, C Selhuber-Unkel, A Squarcini, L Stadler, M Weiss & X Xu (2018) 37
First-past-the-post: few-encounter limit & geometry control
r0
Θ
1
θ0
0 0.2 0.4 0.6 0.8 1ω
0.0
0.5
1.0
1.5
2.0
P(
ω)
(b)10
-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
10-4
10-2
100
102
℘(t
)
t
(a)
10-3
10-2
10-1
�
T Mattos, C Mejıa-Monasterio, RM & G Oshanin, PRE (2012); A Godec & RM, PRX (2016); Sci Rep (2016) 38
Geometry control, defocusing, & finite target reactivity
10-4 10-2 100 10210-4
10-2
100
102 (a)
10-3 10-1 101 103 10510-6
10-4
10-2
100(b)
10-7 10-5 10-3 10-1 101 103
10-4
10-2
100
102
104 (a) r/R = 0.02r/R = 0.1r/R = 0.5r/R = 1
10-7 10-5 10-3 10-1 101 103 10510-6
10-4
10-2
100
102 (b) r/R = 0.02r/R = 0.1r/R = 0.5r/R = 1
10-3 10-1 101 103reaction time (a.u.)
10-4
10-2
100
102
104
reactiv
ity (a.
u.)
10-1410-1210-1010-810-610-4
10-7 10-5 10-3 10-1 101 103 105reaction time (a.u.)
0
0.2
0.4
0.6
0.8
1
distan
ce to
targe
t (a.u.
)
10-14
10-12
10-10
10-8
10-6
10-4
10-2
D Grebenkov, RM & G Oshanin, NJP (2017); PCCP (2018); E-print (2018) 39
Maximum likelihood implementation of diffusing diffusivity
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9 10
Pro
b(M
odel|D
ata
)
∆t
BMBM+Noise
FBMFBM+Noise
DD
S Thapa, AG Cherstvy & RM (2018); model see AV Chechkin, F Seno, RM & IM Sokolov, PRX (2017) 40
Application to mucin data
0
0.2
0.4
0.6
0.8
1
5 10 15 20
Pro
b(M
odel
|Dat
a)
Trajectory Number
BMBM+Noise
FBMFBM+Noise
DD
S Thapa, AG Cherstvy & RM (2018); model see AV Chechkin, F Seno, RM & IM Sokolov, PRX (2017) 41
Stochasticity of fixational eye movementsM
SD
Discp
lacemen
tA
CF
CJJ Herrmann, RM & R Engbert, Sci Rep (2017) 42
Time averages & ageing in financial market time series
dX(t) = µX(t)dt+ σX(t)dW (t)
δ2d(∆) =
T−∆∫td
[X(t+ ∆)−X(t)]2 dt
T − td −∆
∼∆
T − tdX
20
(eσ2T − eσ
2td
)
log[⟨δ2d(∆, td)
⟩/⟨δ2(∆)
⟩]∼ td/T
AG Cherstvy, D Vinod, E Aghion, AV Chechkin & RM, NJP (2017) 43
Overview articles
: Single particle manipulation & tracking:C Nørregaard, RM, CM Ritter, K Berg-Sørensen & LB Oddershede,Chem Rev 117, 4342 (2017)
:: Anomalous diffusion models, WEB & ageing:RM, JH Jeon, AG Cherstvy & E Barkai, Phys Chem Chem Phys 16,24128 (2014)
::: Ageing renewal theory:JHP Schulz, E Barkai & RM, Phys Rev X 4, 011028 (2014)
:::: Anomalous diffusion in membranes:RM, JH Jeon & AG Cherstvy, Biochimica et Biophysica Acta - Biomem-branes 1858, 2451 (2016)
; Polymer translocation:V Palyulin, T Ala-Nissila & RM, Soft Matter 10, 9016 (2014)
44