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 η = white noise p m ¨ x = − γ ˙ x + 2 γ T η + E E = force Colloidal particle in an equilibrium fluid Diffusion h x 2 ( t ) i 0 = 2 T γ − 1 t = 2 Dt h x ( t ) i E = µ E t Drift h x 2 ( t ) i = 2 h x ( t ) i E β E Einstein relation µ = β D

‘’Out-of-equilibrium’’ Einstein relation for a Brownian particle η = white noise p m ¨ x = − γ ˙ x + 2 γ T η + E E = force Colloidal particle in an equilibrium fluid h x 2 ( t ) i E = 2 Dt + µ 2 E 2 t 2 Diffusion h x ( t ) i E = µ E t Drift Einstein relation ? Einstein relation h x 2 ( t ) i E � h x ( t ) i 2 = 2 recovered by E subtracting the h x ( t ) i E β E squared drift

Standard transport properties Diffusion h x 2 ( t ) i 0 = 2 T γ − 1 t = 2 Dt h x ( t ) i E = µ E t Drift ? Heterogeneous medium = ? Anomalous diffusion = ? ‘’Non-anomalous diffusion is not always Gaussian’’ [G. Forte, F. Cecconi, A. Vulpiani, Eur. Phys. J. B 87 (2014)] h x 2 ( t ) i E � h x ( t ) i 2 E ⇠ t ν ν 6 = 1

Non-anomalous diffusion is not always Gaussian: An example from Levy-Walks N N  � v i τ i + E Finite displacement X 2 τ 2 X X t = t = τ i in a finite time i i =1 i =1 1 g ( v ) ∼ e − v 2 / (2 σ ) p ( τ ) = 2 < α < 4 τ 1+ α Velocity: narrow symmetric Duration of flights: broadly (but distribution not too much) distributed h X 2 Linear drift t i ⇠ t h X t i E ⇠ t Linear diffusion h X 2 t i E � h X t i 2 E ⇠ t ν ν > 1 SUPERDIFFUSION ‘’ Einstein relation in superdiffusive system ’’ G. Gradenigo, A. Sarracino, D. Villamaina, A. Vulpiani, JSTAT (2012)

Non-anomalous diffusion is not always Gaussian: Glassy and crowded environments h X 2 E ⇠ t 3 / 2 t i ⇠ t h X 2 t i E � h X t i 2 Linear diffusion Linear drift h X t i E ⇠ t Field-induced superdiffusion - D. Winter, J. Horbach, P. Virnau, K. Binder, Phys. Rev. Lett. 108, 028303 (2012) Yukawa fluid-binary mixture in 3D 10 0 10 − 1 -O. Bénichou et al , Phys. Rev. Lett. 111, 260601 (2013) 10 − 2 σ x ( t ) 2 /t Hard-core lattice gas ρ 0 = 0 . 1 ρ 0 = 0 . 05 10 − 3 ρ 0 = 10 − 2 10 − 4 ρ 0 = 10 − 3 ρ 0 = 10 − 4 10 − 5 10 − 8 10 − 6 10 − 4 10 − 2 10 0 10 2 10 4 ρ 2 0 t

Non-anomalous diffusion is not always Gaussian: Glassy and crowded environments Negative differential mobility 1.5 (a) TLG KA 1 v/v 0 E 0.5 0 0 5 10 15 20 f 4 (b) Trap model (figure above) : M. Baiesi, Glassy systems (figure above): A. Stella, C. Vanderzande, Phys. Rev. R.-L. Jack, D. Kelsey, J. P. Garrahan, D. E 92, 042121 (2015). Chandler, Phys. Rev. E 78, 011506 (2008). p O E = w O E / Z Crowded systems: O. Bénichou et al , Phys. Rev. Lett. 113, 268002 (2014). E + w N + w S Z = w E E + w O

Field-induced anomalous diffusion in glassy systems: short road map - Kinetically constrained models (KCM): Fredrickson-Andersen (discrete variables), Bertin-Bouchaud-Lequex models (continuous variables) - Probe in a KCM: diffusion in a random environment - 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

The Fredrickson-Andersen model (non-interacting spins in a magnetic field) Inactive site n i = 0 N Active site X H = n i Energy: number of active spins n i = 1 i =1 P (0 → 1) = e − β Equilibrium dynamics P (1 → 0) = 1 No update allowed Constraint Two updates allowed

The Fredrickson-Andersen model (non-interacting spins in a magnetic field) Inactive site n i = 0 N Active site X H = n i Energy: number of active spins n i = 1 i =1 τ eq = exp(3 β ) Slow dynamics at low temperatures No update allowed Constraint Two updates allowed

‘’Ghost’’ probes on the FA model 1 1 E ∈ [ − 1 / 2 , 1 / 2] 2 − E 2 + E Active site: ‘’mobility defect’’ Tracer particle no no HETEROGENEOUS DYNAMICS Distance at t=0 from a defect no yes no no Motion is allowed only in presence of ‘’two’’ mobility defects No feedback: Activity field does not feel the probe

Dynamical heterogeneities: broad distribution of waiting times ψ ( τ ) Exchange times distribution: time between two jumps ⇒ D ∼ e − 2 β ψ ( t ) = Diffusion of the probe R ∞ ds ψ ( s ) Persistence times distribution: time t p ( t ) = before the first jump R ∞ ds s ψ ( s ) (from arbitrary intial time) 0 ⇒ τ eq ∼ e 3 β p ( t ) = (Low temperature) h x 2 ( t ) i 0 h x ( t ) i E ⇠ h x 2 ( t ) i 0 ⇠ t = const Einstein relation h x ( t ) i E Diffusion looks non-anomalous

Dynamical heterogeneities: broad distribution of waiting times ψ ( τ ) Exchange times distribution: time between two jumps ⇒ D ∼ e − 2 β ψ ( t ) = Diffusion of the probe R ∞ ds ψ ( s ) Persistence times distribution: time t p ( t ) = before the first jump R ∞ ds s ψ ( s ) (from arbitrary intial time) 0 ⇒ τ eq ∼ e 3 β p ( t ) = (Low temperature) D τ eq 6 = const Breakdown of Stokes-Einstein

Dynamical heterogeneities: broad distribution of waiting times ψ ( τ ) Exchange times distribution: time between two jumps ⇒ D ∼ e − 2 β ψ ( t ) = Diffusion of the probe R ∞ ds ψ ( s ) Persistence times distribution: time t p ( t ) = before the first jump R ∞ ds s ψ ( s ) (from arbitrary intial time) 0 ⇒ τ eq ∼ e 3 β p ( t ) = (Low temperature) Z ∞ P ( t ) = ds p ( s ) Persistence: prob to be at rest until time t t

Inspection paradox ψ ( τ ) = decay slower than exponential Average time between two My average waiting time h t i > h τ i bus passages (arriving at station at arbitrary time) τ 1 τ 2 t Exchange times: waiting ψ ( τ ) times between two buses Z ∞ When I arrive to the bus p ( t ) ∼ ψ ( s ) ds stop I do NOT KNOW the time t elapsed since the last From a random arrival at the bus stop I sample departure …and I measure the Persistence time distribution the time before the next arrival

Dynamical heterogeneities: broad distribution of waiting times Tracer particle Mobility defect : Random Walker (diffusion coefficient D and concentration c 0 depend on temperature) Persistence: survival probability of a target in a sea of predators √ P ( t ) = e − c 0 t [O. Bénichou et al. , Phys. Rev. Lett. 111, 260601 (2013)] √ e − c 0 t p ( t ) = − d P ( t ) = c 0 t 1 / 2 dt √ e − c 0 t ψ ( t ) = − dp ( t ) + O ( c 2 = c 0 0 ) t 3 / 2 dt

The Bertin-Bouchaud-Lequex model ρ i ∈ [0 , 1] Density of a coarse-grained variable ρ i ρ i +1 ρ i − 1 ρ i +2 ρ i − 2 ⇒ ( ρ new , ρ new Elementary step of dynamics ( ρ i , ρ i +1 ) = i +1 ) i 1 ρ new i +1 = q ( ρ i +1 + ρ i ) ψ ( q ) ∼ [ q (1 − q )] µ − 1 ρ new = (1 − q ) ( ρ i +1 + ρ i ) i

The Bertin-Bouchaud-Lequex model Active links ρ i ∈ [0 , 1] Density of a coarse-grained variable ρ i ρ i +1 ρ i − 1 ρ i +2 ρ i − 2 ⇒ ( ρ new , ρ new Elementary step of dynamics ( ρ i , ρ i +1 ) = i +1 ) i 1 ρ new i +1 = q ( ρ i +1 + ρ i ) ψ ( q ) ∼ [ q (1 − q )] µ − 1 ρ new = (1 − q ) ( ρ i +1 + ρ i ) i ( ρ i + ρ i +1 ) / 2 < ρ th KINETIC CONSTRAINT

The Bertin-Bouchaud-Lequex model Tracer particle Active links no yes ρ i ρ i − 1 ρ i +1 ρ i +2 ρ i − 2 1 ρ new i +1 = q ( ρ i +1 + ρ i ) ψ ( q ) ∼ [ q (1 − q )] µ − 1 ρ new = (1 − q ) ( ρ i +1 + ρ i ) i ( ρ i + ρ i +1 ) / 2 < ρ th KINETIC CONSTRAINT

Mobility links have a µ = 0 . 3 diffusive dynamics Tracer particle Active links no yes ρ i ρ i − 1 ρ i +1 ρ i +2 ρ i − 2 1 ρ new i +1 = q ( ρ i +1 + ρ i ) ψ ( q ) ∼ [ q (1 − q )] µ − 1 ρ new = (1 − q ) ( ρ i +1 + ρ i ) i ( ρ i + ρ i +1 ) / 2 < ρ th KINETIC CONSTRAINT

Population splitting scenario: why a linear MSD is not trivial ∞ X π m ( t ) φ ( m ) ( x ) P ( x, t ) = h δ ( x � [ x ( t ) � x (0)]) i = m =0 π m ( t ) = probability of m jumps up to t φ ( m ) ( x ) = probability to be at x after m jumps ∞ P ( x, t ) = π 0 ( t ) φ (0) ( x ) + X π n ( t ) φ ( n ) ( x ) n =1 Z t = P ( t ) δ ( x ) + p ( t − s ) P 1st ( x, s ) ds 0 Persistence Probability distribution of those who made at least one step

Population splitting scenario: why a linear MSD is not trivial Z t h x 2 ( t ) i = ds p ( t � s ) h x 2 ( s ) i 1st ∼ t 0 x 10 c <x 2 (t)> 1st 1 0.1 x 1/2 0.01 0.001 10 -5 10 -3 10 -1 10 3 � 10 c 2 (t/ τ micro ) Z t P ( x, t ) = P ( t ) δ ( x ) + p ( t − s ) P 1st ( x, s ) ds 0 Persistence Probability distribution of those who made at least one step

Population splitting scenario: why a linear MSD is not trivial Z t h x 2 ( t ) i = ds p ( t � s ) h x 2 ( s ) i 1st ∼ t t ⌧ c − 2 Preasymptotic regime 0 0 CTRW x 10 c <x 2 (t)> 1st 1 h x 2 ( s ) i 1st ⇠ s 1 / 2 ψ ( τ ) ⇠ = ) 1 τ 3 / 2 0.1 x 1/2 c 0 But … p ( t − s ) ∼ √ t − s 0.01 0.001 10 -5 10 -3 10 -1 10 3 � 10 c 2 (t/ τ micro ) p s Z t h x 2 ( t ) i 1st ⇠ c 0 p t � sds ⇠ t c 0 0