brownian motion with time- dependent diffusion coefficient · dependent diffusion and friction...
TRANSCRIPT
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Anna Bodrova
Skolkovo Institute of Science and Technology
Brownian motion with time-
dependent diffusion coefficient
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Acknowledgement
Nikolay Brilliantov
University of Leicester,
United Kingdom
University of Potsdam, Germany
Ralf Metzler Andrey Cherstvy Alexei Chechkin
Igor Sokolov
Humboldt University,
Berlin, Germany
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Normal diffusionA. Einstein (1905)
D – diffusion coefficient
2(t) 6R Dt=
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Anomalous diffusion
1 𝑅2(t) ∼ 𝑡𝛼
Superdiffusion
Subdiffusion
Normal diffusion
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Anomalous diffusion in biology
➢ Flight of albatroses.
➢ Movement of spider monkeys.
➢ Molecular motors: active motion of motor proteins
with cargo along the filaments in the cytoskeleton.
Superdiffusion
Advantages of superdiffusion in biology
➢ Superdiffusion leads to the effective search
strategy for finding randomly located objects.
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Anomalous diffusion in biology
➢ Diffusion of channel proteins in cell membranes.
➢ Diffusion of telomers (chromosomal end parts) in
human cell nuclei.
➢ Motion of messenger RNA in bacteria cells.
➢ Anomalous diffusion of large molecules due to
high density of the cell environment.
Subdiffusion
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Some sources of anomaly
Geometrical constraints
➢ Diffusion in crowded systems
➢ Diffusion on fractal structures
Diffusion in inhomogeneous environment
➢ Heterogeneous diffusion process (motion with
space-dependent diffusion coefficient)
Diffusion in non-stationary environment
➢ Scaled Brownian motion (motion with
time-dependent diffusion coefficient)
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Ergodicity breakingEnsemble averaged mean-squared displacement
(MSD) is not equal to the ensemble-averaged MSD.
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Time-averaged mean-squared
displacement (MSD)
Δ – lag time t – trajectory length
t
Single particle tracking experiments
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Time-dependent
diffusion coefficient D(t)
I. Brownian motion in a bath with
time-dependent temperature
II. Snow melt dynamics
III. Diffusion in turbulence
IV. Water diffusion in brain tissue
V. Free cooling granular gases
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Powders
Granular systems
Sand
StonesNuts
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Granular solids
Sand
Stones
For small loads granular systems behave like solids:
resist the load and preserve their form
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Granular liquids
For lager loads they
flow like liquids and
preserve their volume
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Granular gases
For larger loads they
form rapid granular flows,
where they behave like gases
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Granular gas – rarefied system of macroscopic
particles, which collide with loss of energy
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Granular gas: space
Planetary rings
Protoplanetary disc
Interstellar dust
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Free granular gas
If no external forces act on the
granular system, it evolves freely
and the particles slow down.
D
t
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Magnetic levitation
Georg Maret, University of Konstanz, Germany
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The Bremen Drop-tower146 m
9.3 s of weightlessness
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Low gravity environment
Parabolic flights(30 parabolas,
20 s of reduced gravity)
Eric Falcon, Univ Paris Diderot,
Sorbonne Paris Cité, Paris, France
Rocket experiments (12 min of microgravity)
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1 2 12
1 2 12
V V Vε
V V V
−= =
−
( ) 2
12
24
1 1 mVε=ΔE −−
Restitution coefficient ɛ
V 1 V 2
1V
2V
Energy loss:
perfectly inelastic
collision
0 1
perfectly elastic
collision
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Restitution coefficient ɛDependence of ɛ on the relative velocity V12:
( ) 1/5 2/5
12 1 12 2 121ε V CV C V= − +
1ε →
12 0V →For small collisional relative velocities
collisions become more elastic:
Coefficients C1, C2 depend on Young's modulus,
Poisson ratio, viscosity, density and sizes of
colliding particles
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Event-driven simulations
of granular gases
( )( )1 1 12
11
2v v v n n = − +
( )( )2 2 12
11
2v v v n n = + +
( )
( )12
12
v n
v n
= −
Restitution
coefficient
Instantaneous binary
collisions of particles
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22
( ) ( )3 2
mvT t dv f = v
Granular temperature
The mean kinetic energy (granular temperature) in
a granular gas decreases due to inelastic collisions
according to the Haff’s law:
( )0
2
0
( )1 /
TT t
t =
+
10-2
10-1
100
101
102
103
104
105
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
T
t
~ 1/t2
( )0
5/3
0
( )1 /
TT t
t =
+
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Diffusion coefficient of granular
particles is time-dependent:
Diffusion coefficient
- velocity correlation time
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Diffusion coefficient
of rough granular partcles
Smooth particles
Rough particles
Moment of inertia Tangential restitution
coefficient
Tangential velocity
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Diffusion coefficient of granular gases
Constant
restitution coefficient
Velocity-dependent
restitution coefficient
Granular temperature
(mean kinetic energy of granular particles)
Diffusion coefficient
N. V. Brilliantov and T. Poeschel, Phys. Rev. E 61, 1716 (2000)
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2 (t) (t)v D =
Overdamped Langevin equation
( )1 2 1 2(t ) (t ) t t = −
0
0
1(t) ~
1 /
DD
t t=
+
Diffusion coefficient
White Gaussian noise
Scaled Brownian
motion (SBM)Ultraslow SBM
0
1(t)
DD
t
−
= 0
Mean-squared displacement (MSD)
( )2
0 0(t) 2 log 1 / 0x D t = +2
0(t) 2x D t
=
0 =
(t) 2 (t) (t) (t)dv
m v Ddt
+ =
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Underdamped Langevin equation with
time-dependent temperature
Mean-squared displacement
Friction coefficient Temperature Diffusion coefficient
For large times SBM result
For small times ballistic behavior
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Time-averaged MSD
Underdamped Overdamped
For large lag times
the underdamped and overdamped limits become comparable:
For intermediate lag times
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MSD and time-averaged MSD
1/ 2 =
Underdamped SBM
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10-2
100
102
104
106
108
10-11
10-8
10-5
10-2
101
104
107
a)
()
()
~
~
(
)
Time-averaged MSD: overdamped
and underdamped limits
1/ 2 =
1/ 6 =
10-2
100
102
104
106
108
10-11
10-8
10-5
10-2
101
104
b)
()
()
~
~
~
(
)
Underdamped Overdamped
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Underdamped Langevin equation:
ultraslow limit
Mean-squared displacement
Time-averaged mean-squared displacement
Friction coefficient Temperature Diffusion coefficient
for
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Ultraslow underdamped SBM
MSD and time-averaged MSD
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Time-averaged MSD:
overdamped and underdamped limits
Underdamped
Overdamped
Ultraslow underdamped SBM
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Granular gas with constant
restitution coefficient
Mean-squared
displacement (MSD):
( )12ε v const=
𝑅2(𝑡) ∼ log 𝑡
Time-averaged MSD:
10-2
10-1
100
101
102
103
104
10-2
100
102
104 ~ log(t)
~ t2
~
a)
R2(t)
()
t
0 = 33
v = 3.7
D0 = 3.7
R2(t
),
(
)
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( ) 1/5 2/5
12 1 12 2 121ε v C v C v= − +
Mean-squared
displacement (MSD):
𝑅2(𝑡) ∼ 𝑡1/6
Time-averaged MSD:
Granular gas with velocity-dependent
restitution coefficient
Crossover from
𝛿2(Δ) ∼ Δ7/6/𝑡
𝛿2(Δ) ∼ Δ/𝑡5/6to10-1
101
103
105
107
10-2
100
102
104
~ t2
~ t1/6
~
~
b)
R2(t)
()
0 = 20
v = 3.0556
D0 = 3.0556
R2(t
),
(
)
t,
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Resettnig
If one searches for a goal and is lost, sometimes it is
useful to return to starting point and to start
the search from the beginning.
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Resetting
• Foraging animals
• Optimizing
search algorithms
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Typical trajectories for scaled
Brownian motion with resetting
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t
D
D
t
t
x
Renewal process
Non-renewal process
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Diffusion coefficient
White Gaussian noise
Scaled Brownian motion
Mean-squared displacement
Probability distribution function
Superdiffusion Subdiffusion
Ordinary Brownian motion
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Distribution of waiting times between the
resetting eventsExponential (Poissonian)
resetting
Survival probability
Power-law resetting
The probability that an event occurs at time t
constant rate
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Probability distribution function
Mean-squared displacement
The probability to find the particle at location x at time t:
No resetting Last resetting at time t’
Renewal process: diffusion coefficient also resets to initial value D0
Non- renewal process: diffusion coefficient is not affected by the
resetting events
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Mean-squared displacementExponential resetting
Renewal process:
Non-renewal process:
Steady state:
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Probability distribution functionExponential resetting: non-renewal process
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Probability distribution functionExponential resetting: renewal process
Steady state
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First passage time
Exponential resetting: renewal process
0,01 0,1 1 10 100
1
10
100
=1.2
=1
=0.8
r10
-210
-110
010
110
1
103
105
=1.5
=1
=0.5
r
x0=1 x0=5
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Mean-squared displacement
Power-law resetting
non-renewal process
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Mean-squared displacementPower-law resetting
non-renewal process
10-2
10-1
100
101
102
103
104
10-1
100
101
102
~ t-0.5
~ t-0.1
~ t0.1
= 0.5
= 1.4
= 1.5
= 1.6
= 2.5
~ t0.5
x
2
t
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Mean-squared displacementPower-law resetting: renewal process
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Mean-squared displacement
10-2
10-1
100
101
102
103
104
105
100
101
102
103
exp resetting
= 0.5
= 2.5
~ t0.5
x
2
t
Power-law resetting
renewal process
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First passage timePower-law resetting: renewal process
x0=101 2 3 4
102
103
104
=2
=1
=0.5
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Conclusions➢ The underdamped scaled Brownian motion, a novel anomalous
diffusion process governed by Langevin equation with time-
dependent diffusion and friction coefficients, describes the diffusion
in bath with time-dependent temperature and in granular gases.
➢ The overdamped limit of the Langevin equation does not capture
all properties of the system
➢ Resetting significantly affects the behavior of Brownian motion with
time-dependent diffusion coefficient
1. Bodrova A., Chechkin A. V., Cherstvy A. G., Metzler R., Quantifying non-ergodic
dynamics of force-free granular gases. PCCP 17, 21791 (2015).
2. Bodrova A., Chechkin A. V., Cherstvy A. G., Metzler R., Ultraslow scaled
Brownian motion. New J. Phys. 17, 063038 (2015)
3 . Bodrova A., Chechkin A. V., Cherstvy A. G., Safdari H., Sokolov I.M., Metzler R.,
Underdamped scaled Brownian motion: (non-)existence of the overdamped limit in
anomalous diffusion. Sci. Rep. 6, 30520 (2016)
4. Bodrova A.S., Chechkin A. V., Sokolov I.M. Non-renewal resetting of scaled
Brownian motion. Submitted to Phys. Rev. E (2018)
5. Bodrova A.S., Chechkin A. V., Sokolov I.M. Scaled Brownian motion with renewal
resetting. Submitted to Phys. Rev. E (2018)