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)