Field-induced superdiffusion in models with a glassy dynamics - - PowerPoint PPT Presentation

Field-induced superdiffusion in models with a glassy dynamics

Giacomo Gradenigo

Eric Bertin (LIPhy, Grenoble), Giulio Biroli (IPhT,Saclay)

Padova, Dipartimento di Fisica ‘’G. Galilei’’ , 09-03-2016

Einstein Relation for a Brownian particle

Colloidal particle in an equilibrium fluid Diffusion Drift

hx(t)iE = µE t hx2(t)i hx(t)iE = 2 βE

Einstein relation

µ = βD m¨ x = −γ ˙ x + p 2γT η + E hx2(t)i0 = 2 Tγ−1t = 2Dt

η = white noise E = force

‘’Out-of-equilibrium’’ Einstein relation for a Brownian particle

Colloidal particle in an equilibrium fluid Diffusion Drift

hx(t)iE = µE t m¨ x = −γ ˙ x + p 2γT η + E hx2(t)iE = 2Dt + µ2E2t2

Einstein relation recovered by subtracting the squared drift

Einstein relation ?

hx2(t)iE hx(t)i2

E

hx(t)iE = 2 βE

η = white noise E = force

Standard transport properties

Diffusion Drift

hx(t)iE = µE t hx2(t)i0 = 2 Tγ−1t = 2Dt

?

Anomalous diffusion = ? Heterogeneous medium = ?

‘’Non-anomalous diffusion is not always Gaussian’’

[G. Forte, F. Cecconi, A. Vulpiani, Eur. Phys. J. B 87 (2014)]

hx2(t)iE hx(t)i2

E ⇠ tν

ν 6= 1

SUPERDIFFUSION ‘’Einstein relation in superdiffusive system’’

• G. Gradenigo, A. Sarracino, D. Villamaina, A. Vulpiani, JSTAT (2012)

Non-anomalous diffusion is not always Gaussian: An example from Levy-Walks

p(τ) = 1 τ 1+α 2 < α < 4

g(v) ∼ e−v2/(2σ) Xt =

N

X

i=1

 viτi + E 2 τ 2

i

• Velocity: narrow symmetric

distribution Duration of flights: broadly (but not too much) distributed Linear diffusion

hX2

t i ⇠ t

Linear drift

hXtiE ⇠ t

hX2

t iE hXti2

E ⇠ tν

ν > 1

t =

N

X

i=1

τi

Finite displacement in a finite time

Field-induced superdiffusion

• D. Winter, J. Horbach, P. Virnau, K. Binder,
• Phys. Rev. Lett. 108, 028303 (2012)
• O. Bénichou et al, Phys. Rev. Lett. 111, 260601 (2013)

Yukawa fluid-binary mixture in 3D

10−5 10−4 10−3 10−2 10−1 100 10−8 10−6 10−4 10−2 100 102 104 σx(t)2/t ρ2

0t

ρ0 = 0.1 ρ0 = 0.05 ρ0 = 10−2 ρ0 = 10−3 ρ0 = 10−4

Hard-core lattice gas

Non-anomalous diffusion is not always Gaussian: Glassy and crowded environments

Linear diffusion

hX2

t i ⇠ t

Linear drift

hXtiE ⇠ t

hX2

t iE hXti2

E ⇠ t3/2

Negative differential mobility

Glassy systems (figure above): R.-L. Jack, D. Kelsey, J. P. Garrahan, D. Chandler, Phys. Rev. E 78, 011506 (2008). Crowded systems: O. Bénichou et al,

• Phys. Rev. Lett. 113, 268002 (2014).

Trap model (figure above): M. Baiesi,

• A. Stella, C. Vanderzande, Phys. Rev.

E 92, 042121 (2015).

Non-anomalous diffusion is not always Gaussian: Glassy and crowded environments

5 10 15 20 0.5 1 1.5 TLG KA

(a)

4

(b)

v/v0 f

Z = wE

E + wO E + wN + wS

pO

E = wO E /Z

E

Field-induced anomalous diffusion in glassy systems: short road map

• Kinetically constrained models (KCM): Fredrickson-Andersen (discrete

variables), Bertin-Bouchaud-Lequex models (continuous variables)

• Exchange and persistence times: breaking of the Stokes-Einstein relation

(supercooled liquids VS ordinary life)

• Population Splitting for P(x,t): even a linear MSD may be non trivial
• Field-induced superdiffusion
• Exponent 3/2 of field-induced superdiffusion from population splitting
• Population splitting in glasses: strong anomalous diffusion
• Probe in a KCM: diffusion in a random environment

The Fredrickson-Andersen model

(non-interacting spins in a magnetic field) Active site Inactive site ni = 0

ni = 1 P(0 → 1) = e−β P(1 → 0) = 1

Equilibrium dynamics Constraint No update allowed Two updates allowed

H =

N

X

i=1

ni

Energy: number of active spins

Slow dynamics at low temperatures

τeq = exp(3β)

The Fredrickson-Andersen model

(non-interacting spins in a magnetic field) Active site Inactive site ni = 0

ni = 1 H =

N

X

i=1

ni

Energy: number of active spins Constraint No update allowed Two updates allowed

‘’Ghost’’ probes on the FA model

Tracer particle no no no yes no no Motion is allowed only in presence of ‘’two’’ mobility defects No feedback: Activity field does not feel the probe Distance at t=0 from a defect HETEROGENEOUS DYNAMICS Active site: ‘’mobility defect’’

1 2 + E 1 2 − E E ∈ [−1/2, 1/2]

ψ(τ)

Exchange times distribution: time between two jumps Persistence times distribution: time before the first jump (from arbitrary intial time)

p(t) = ⇒ τeq ∼ e3β p(t) = R ∞

t

ds ψ(s) R ∞ ds s ψ(s)

Diffusion of the probe Diffusion looks non-anomalous Einstein relation

Dynamical heterogeneities: broad distribution of waiting times

ψ(t) = ⇒ D ∼ e−2β hx(t)iE ⇠ hx2(t)i0 ⇠ t hx2(t)i0 hx(t)iE = const

(Low temperature)

ψ(τ)

Exchange times distribution: time between two jumps Persistence times distribution: time before the first jump (from arbitrary intial time)

p(t) = ⇒ τeq ∼ e3β p(t) = R ∞

t

ds ψ(s) R ∞ ds s ψ(s)

Diffusion of the probe

Dynamical heterogeneities: broad distribution of waiting times

D τeq 6= const

Breakdown of Stokes-Einstein

ψ(t) = ⇒ D ∼ e−2β

(Low temperature)

Dynamical heterogeneities: broad distribution of waiting times

P(t) = Z ∞

t

ds p(s)

Persistence: prob to be at rest until time t

ψ(τ)

Exchange times distribution: time between two jumps Persistence times distribution: time before the first jump (from arbitrary intial time)

p(t) = ⇒ τeq ∼ e3β p(t) = R ∞

t

ds ψ(s) R ∞ ds s ψ(s)

Diffusion of the probe

ψ(t) = ⇒ D ∼ e−2β

(Low temperature)

Inspection paradox

τ1 τ2

ψ(τ)

Exchange times: waiting times between two buses When I arrive to the bus stop I do NOT KNOW the time elapsed since the last departure …and I measure the time before the next arrival

t p(t) ∼ Z ∞

t

ψ(s)ds

From a random arrival at the bus stop I sample the Persistence time distribution

hti > hτi

My average waiting time (arriving at station at arbitrary time) Average time between two bus passages

ψ(τ) = decay slower than exponential

Dynamical heterogeneities: broad distribution of waiting times

Tracer particle

Mobility defect:

Random Walker (diffusion coefficient D and concentration c0 depend on temperature)

Persistence: survival probability of a target in a sea of predators

[O. Bénichou et al., Phys. Rev. Lett. 111, 260601 (2013)]

P(t) = e−c0

√ t

p(t) = −dP(t) dt = c0 e−c0

√ t

t1/2 ψ(t) = −dp(t) dt = c0 e−c0

√ t

t3/2 + O(c2

0)

The Bertin-Bouchaud-Lequex model

ρi ρi−1 ρi−2 ρi+1 ρi+2

ρnew

i+1 = q (ρi+1 + ρi)

ρnew

i

= (1 − q) (ρi+1 + ρi) ψ(q) ∼ 1 [q(1 − q)]µ−1

ρi ∈ [0, 1]

Density of a coarse-grained variable Elementary step of dynamics (ρi, ρi+1) =

⇒ (ρnew

i

, ρnew

i+1)

The Bertin-Bouchaud-Lequex model

ρi ρi−1 ρi−2 ρi+1 ρi+2

Elementary step of dynamics (ρi, ρi+1) =

⇒ (ρnew

i

, ρnew

i+1)

ρnew

i+1 = q (ρi+1 + ρi)

ρnew

i

= (1 − q) (ρi+1 + ρi) ψ(q) ∼ 1 [q(1 − q)]µ−1

KINETIC CONSTRAINT

(ρi + ρi+1)/2 < ρth Active links

ρi ∈ [0, 1]

Density of a coarse-grained variable

The Bertin-Bouchaud-Lequex model

ρi ρi−1 ρi−2 ρi+1 ρi+2

ρnew

i+1 = q (ρi+1 + ρi)

ρnew

i

= (1 − q) (ρi+1 + ρi) ψ(q) ∼ 1 [q(1 − q)]µ−1

KINETIC CONSTRAINT

(ρi + ρi+1)/2 < ρth Active links

Tracer particle no yes

ρi ρi−1 ρi−2 ρi+1 ρi+2

ρnew

i+1 = q (ρi+1 + ρi)

ρnew

i

= (1 − q) (ρi+1 + ρi) ψ(q) ∼ 1 [q(1 − q)]µ−1

KINETIC CONSTRAINT

(ρi + ρi+1)/2 < ρth Active links

no yes

µ = 0.3

Mobility links have a diffusive dynamics

Tracer particle

Population splitting scenario: why a linear MSD is not trivial

P(x, t) = hδ(x [x(t) x(0)])i =

∞

X

m=0

πm(t)φ(m)(x)

φ(m)(x) = probability to be at x after m jumps

P(x, t) = π0(t)φ(0)(x) +

∞

X

n=1

πn(t)φ(n)(x) = P(t)δ(x) + Z t p(t − s)P1st(x, s)ds

πm(t) = probability of m jumps up to t

Persistence Probability distribution of those who made at least one step

Population splitting scenario: why a linear MSD is not trivial

= P(t)δ(x) + Z t p(t − s)P1st(x, s)ds

Persistence Probability distribution of those who made at least one step

0.001 0.01 0.1 1 10 10-5 10-3 10-1 10 103

c <x2(t)>1st c2 (t/τmicro) x1/2 x

hx2(t)i = Z t ds p(t s)hx2(s)i1st

P(x, t) ∼ t

Population splitting scenario: why a linear MSD is not trivial

0.001 0.01 0.1 1 10 10-5 10-3 10-1 10 103

c <x2(t)>1st c2 (t/τmicro) x1/2 x

hx2(t)i = Z t ds p(t s)hx2(s)i1st

CTRW But … Preasymptotic regime

ψ(τ) ⇠ 1 τ 3/2 = ) hx2(s)i1st ⇠ s1/2 p(t − s) ∼ c0 √t − s

hx2(t)i1st ⇠ c0 Z t ps pt sds ⇠ t c0

t ⌧ c−2 ∼ t

Population splitting scenario: why a linear MSD is not trivial

= P(t)δ(x) + Z t p(t − s)P1st(x, s)ds

Persistence Probability distribution of those who made at least one step

0.001 0.01 0.1 1 10 10-5 10-3 10-1 10 103

c <x2(t)>1st c2 (t/τmicro) x1/2 x

hx2(t)i = Z t ds p(t s)hx2(s)i1st

P(x, t)

CTRW And … Therefore … [p ⇤ hx2i1st](t) ⇠ t Asymptotic regime

ψ(τ) ⇠ e−c0

√τ

= ) hx2(s)i ⇠ s p(t − s) ∼ e−c0

√t−s

t c−2 ∼ t

Anomaly is evident in field-induced superdiffusion

1e-05 0.0001 0.001 0.01 0.1 1 10 100 1000 1e-4 1e-2 1.0 100

c2 σx

2

c2 ( t / τmicro) t3/2 t

c = 1.9 10-3 2.8 10-3 3.8 10-3 5.2 10-3 6.7 10-3 1.0 10-2 1e-2 1.0 100 1e-2 1.0 c <x>ε c2 ( t / τmicro)

t

10-6 10-2 102 1e-4 1e-2 1.0 100

c2 σx

2

c2 t t3/2

c = 2.8 10-4 2.7 10-3 1.6 10-2 5.1 10-2 1.1 10-1 2.0 10-1 10-5 10-2 10 1e-4 1e-2 1.0 100 c <x>ε c2 t

t

Fredrickson-Andersen Bertin-Bouchaud-Lequeux

Breaking of Einstein relation for ‘’out-of-equilibrium’’ fluctuations

hx(t)iE ⇠ t σ2

x(t) = hx2(t)iE hx(t)i2

E ⇠ t3/2

Field-induced superdiffusion

The longer is the elapsed time the larger the population of particles which ‘’have jumped at least once’’

hx2(s)iE,1st ⇠ s σ2

x(t) = hx2(t)iE hx(t)i2

E = c0

Z t ds hx2(s)iE,1st (t s)1/2 + O(c2

0)

σ2

x(t) ∼ c0

Z t ds s √t − s ∼ c0 t3/2 P(x, t) = P(t)δ(x) + Z t p(t − s)P1st(x, s)ds

The preasymptotic linear drift is non trivial

‘’Velocity anomaly of a driven tracer in a confined crowded environment’’,

• P. Illien, O. Benichou, G. Oshanin, R. Voituriez, PRL, 030603 (2014)

p(t − s) ∼ (t − s)−1/2 p(t − s) ∼ e−c0

√ t

Asymptotic regime Preasymptotic regime hx(t)iE = Z t ds p(t s)hx(s)i1st,E ⇠ c0 t

hx(s)i1st,E ⇠ s1/2 hx(s)i1st,E ⇠ s

0.01 0.1 1 10 100 0.01 0.1

t1/2 P(x / t1/2) x / t1/2

0.0001 0.001 0.01 0.1 10 100 1000

P(x,t) x

105 106 5 • 106 107 5 • 107

P1step(x, t)

Strong anomalous diffusion

P E

1st(x, t) =

1 t1/2 F ⇣ x t1/2 ⌘ `(t) = t1/2 hx(t)iE ⇠ t 6= `(t) P(x, t) = P(t)δ(x) + Z t p(t − s)P1st(x, s)ds

h|x(t)|ni ⇠ `n(t) h|x(t)|ni ⌧ `n(t)

Anomalous Strong anomalous (multiscaling)

• P. Castiglione, A. Mazzino, P. Muratore-Gianneschi, A. Vulpiani, Physica D 134, 75 (1999)

P(x,t) cannot be collapsed with a single scaling length. Transport in turbulent flows

FIELD ON FIELD OFF

t ⌧ c−2 t c−2

hx(t)i ⇠ t hx(t)i ⇠ t hx2(t)i ⇠ t hx2(t)i ⇠ t

Kinetically constrained models

hx2(t)iE hx(t)i2

E ⇠ t3/2

hx2(t)iE hx(t)i2

E ⇠ t

ψ(τ) ∼ e−c0

√τ

τ 3/2

Probability distribution of times between jumps

Conclusions

• Fredrick-Anderson (FA) and Bertin-Bouchaud-Lequex features field-induced

superdiffusion with exponent 3/2

• The field-induced superdiffusion is a preasymptotic regime well

characterized numerically and well understood theoretically

• Strong anomalous diffusion easy understandable in terms of population splitting
• Population splitting is also typical typical of more realistic

(3d offlattice) models with slow heterogeneous dynamics

P(x, t) = P(t)δ(x) + Z t p(t − s)P1st(x, s)ds

THANKS FOR YOUR ATTENTION