Post on 18-Jul-2020
Anna Bodrova
Skolkovo Institute of Science and Technology
Brownian motion with time-
dependent diffusion coefficient
Acknowledgement
Nikolay Brilliantov
University of Leicester,
United Kingdom
University of Potsdam, Germany
Ralf Metzler Andrey Cherstvy Alexei Chechkin
Igor Sokolov
Humboldt University,
Berlin, Germany
Normal diffusionA. Einstein (1905)
D – diffusion coefficient
2(t) 6R Dt=
Anomalous diffusion
1 𝑅2(t) ∼ 𝑡𝛼
Superdiffusion
Subdiffusion
Normal diffusion
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.
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
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)
Ergodicity breakingEnsemble averaged mean-squared displacement
(MSD) is not equal to the ensemble-averaged MSD.
Time-averaged mean-squared
displacement (MSD)
Δ – lag time t – trajectory length
t
Single particle tracking experiments
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
Powders
Granular systems
Sand
StonesNuts
Granular solids
Sand
Stones
For small loads granular systems behave like solids:
resist the load and preserve their form
Granular liquids
For lager loads they
flow like liquids and
preserve their volume
Granular gases
For larger loads they
form rapid granular flows,
where they behave like gases
Granular gas – rarefied system of macroscopic
particles, which collide with loss of energy
Granular gas: space
Planetary rings
Protoplanetary disc
Interstellar dust
Free granular gas
If no external forces act on the
granular system, it evolves freely
and the particles slow down.
D
t
Magnetic levitation
Georg Maret, University of Konstanz, Germany
The Bremen Drop-tower146 m
9.3 s of weightlessness
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)
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
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
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
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 =
+
Diffusion coefficient of granular
particles is time-dependent:
Diffusion coefficient
- velocity correlation time
Diffusion coefficient
of rough granular partcles
Smooth particles
Rough particles
Moment of inertia Tangential restitution
coefficient
Tangential velocity
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)
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
+ =
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
Time-averaged MSD
Underdamped Overdamped
For large lag times
the underdamped and overdamped limits become comparable:
For intermediate lag times
MSD and time-averaged MSD
1/ 2 =
Underdamped SBM
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
Underdamped Langevin equation:
ultraslow limit
Mean-squared displacement
Time-averaged mean-squared displacement
Friction coefficient Temperature Diffusion coefficient
for
Ultraslow underdamped SBM
MSD and time-averaged MSD
Time-averaged MSD:
overdamped and underdamped limits
Underdamped
Overdamped
Ultraslow underdamped SBM
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
),
(
)
( ) 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,
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.
Resetting
• Foraging animals
• Optimizing
search algorithms
Typical trajectories for scaled
Brownian motion with resetting
t
D
D
t
t
x
Renewal process
Non-renewal process
Diffusion coefficient
White Gaussian noise
Scaled Brownian motion
Mean-squared displacement
Probability distribution function
Superdiffusion Subdiffusion
Ordinary Brownian motion
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
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
Mean-squared displacementExponential resetting
Renewal process:
Non-renewal process:
Steady state:
Probability distribution functionExponential resetting: non-renewal process
Probability distribution functionExponential resetting: renewal process
Steady state
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
Mean-squared displacement
Power-law resetting
non-renewal process
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
Mean-squared displacementPower-law resetting: renewal process
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
First passage timePower-law resetting: renewal process
x0=101 2 3 4
102
103
104
=2
=1
=0.5
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)