Geometry-Induced Superdiffusion in Driven Crowded Systems Carlos - PowerPoint PPT Presentation

Geometry-Induced Superdiffusion in Driven Crowded Systems Carlos Mejía-Monasterio Technical University of Madrid Galileo Galilei Institute, Arcetri Florence May 2014

Active micro-rheology Active manipulation of small probe particles by external forces, using magnetic fields, electric fields, or micro-mechanical forces. Optical tweezers Magnetic manipulation Atomic force microscopy J Pesic, et al, PRE (2012) 86 , 031403

Nonequilibrium inhomogeneity As the force increases ... a traffic jammed region in front of the intruder a wake region behind the it C M-M, G Oshanin Soft Matter (2011), 7 993

Nonequilibrium inhomogeneity As the force increases ... a traffic jammed region in front of the intruder a wake region behind the it C M-M, G Oshanin Soft Matter (2011), 7 993

Nonequilibrium inhomogeneity In the wake In front -1 10 -1 10 ∼ x − 3 / 2 -2 10 -2 10 -3 10 -3 10 -4 10 -4 10 0 1 2 10 10 10 2 0 5 10 15 10 x − x The medium remembers the passage of the intruder on large temporal and spatial scales observed in colloidal suspensions, monolayers of vibrated grains and in glass systems. C M-M, G Oshanin Soft Matter (2011), 7 993

The model Simple Exclusion Process

The model Simple Exclusion Process ◮ We consider a square lattice of L x × L y sites, of unit spacing, with P.B.C and populated with hard-core particles. ◮ Each site can be either empty or occupied by at most one particle. ◮ The system evolves in discrete time n and particles move randomly. ◮ One particle, the intruder , is subject to a Simple Exclusion Process constant force F • Bath particles move in either direction with equal jump probability 1 / 4. • The intruder moves in direction e ν with probability β 2 F · e ν , p ν = Z − 1 e where Z = 2(1 + cosh ( βσ F / 2)) and β is the inverse temperature.

Force-velocity relation β F = 10 0.3 0.2 |V| β F = 0 . 5 0.1 0 0 2 4 6 8 �� F

Force-velocity relation Stokesian regime 0.3 V = F ξ , 0.2 with friction coe ffi cient |V| 0.1 ξ = ξ mf + ξ coop where 0 0 2 4 6 8 �� F 4 τ 4 τ ( π − 2) ρ ξ mf = ξ coop = βσ 2 (1 − ρ ) and βσ 2 (1 − ρ ) 1 + (1 − ρ ) O B´ enichou, et al, PRL (2000) 84 , 511; PRB (2001) 63 , 235413 C M-M, G Oshanin, Soft Matter (2011) 7 , 993

The problem What is the probability distribution function of the intruder’s displacement at time n ? P ( R n ) We are interested in the limit of very dense lattices or very strong pulling forces.

The limit of high density 2D • Limit of small vacancy density ρ 0 = M / ( L x × L y ) � 1 • Idea: trapping of the intruder by di ff usive vacancies. O B´ enichou, G Oshanin, PRE (2001) 64 . 020103 MJAM Brummelhuis, HJ Hilhorst, Physica A (1989) 156 , 575

The limit of high density 2D many vacancies problem as many single vacancy problems. propagator of the intruder in the presence of a single vacancy is given in terms of First-Passage Time distributions of the vacancy to the site ocupied by the intruder. results in the long time limit

Intruder’s displacement � Let Z j n denote the position of the j -th vacancy at time n , j = 1 , 2 , . . . , M . � We want to compute the probability of finding the intruder at position r n at time n conditioned to { Z j n } � � P ( r n |{ Z j n + · · · + r M n , . . . , r M n |{ Z j δ ( r n , r 1 n ) P ( r 1 n } ) = · · · n } ) r 1 r M n n n |{ Z j n , . . . , r M � P ( r 1 n } ) is the conditional probability that within the time interval n the intruder moved to r 1 n due to its interaction with vacancy 1, to r 2 n due to its interaction with vacancy 2, etc. � In the lowest order in ρ 0 the vacancies contributions are independent and M � n , . . . , r M n |{ Z j P ( r n | Z j P ( r 1 n } ) ≃ n ) j =1 The problem reduces to M single vacancies, correct to O ( ρ 0 ).

Intruder’s displacement � Averaging P ( r n | Z j n ) over the initial distribution of vacancies M � � � δ ( r n , r 1 n + · · · + r M � P ( r n | Z j P ( r n ) ≃ · · · n ) n ) � r 1 r M j =1 n n � Defining the Fourier transformed distribution � exp ( − i k · r n ) � P ( r n |{ Z j P n ( k ) = n } ) � r n and summing over r n one obtains that it factorizes into � M �� exp ( − i k · r n ) � P ( r n | Z j P n ( k ) = n ) � r n

Intruder’s displacement Taking the thermodynamic limit L x , L y → ∞ with ρ 0 fixed we obtain for the characteristic function P n ( k ) ≃ exp( − ρ 0 Ω n ( k )) Ω n ( k ) is implicitly defined by n � � � F ∗ Ω n ( k ) = ∆ n − l ( k | e ν ) l ( 0 | e ν | Z ) , l =0 Z � =0 ν F ∗ l ( 0 | e ν | Z ) is the FPT conditional probability for a RW starting at Z to be at 0 at time l , given that it is at site 0 + e ν at time l − 1 and ∆ l ( k | e ν ) = 1 − p l ( k ) exp ( i ( k · e ν )) Z π 1 P ( R n ) ' d k exp ( � i ( k · R n ) � ρ 0 Ω n ( k )) 4 π 2 � π

Intruder’s displacement Ω n ( k ) can be solved explicitly in terms of its generating function ∞ � Ω n ( k ) z n Ω z ( k ) = n =0 In the large n (and ρ 0 � 1) limit z → 1 − 1 Φ ( k ) Ω z ( k ) ∼ (1 − z ) 1 − z + Φ ( k ) / χ z with π χ z ∼ − (1 − z ) ln(1 − z ) the leading asymptotic term of the generating function of the mean number of “new” (virgin) sites visited on the n -th step BD Hughes, (2005) Random walks in random environments

Intruder’s displacement Then � − 1 Φ ( k ) � 1 − ln(1 − z ) Ω z ( k ) ∼ Φ ( k ) , (1 − z ) 2 π with Φ ( k ) = − ia 0 k x + a 1 k 2 x / 2 + a 2 k 2 y / 2 sinh( β F / 2) a 0 = (2 π − 3) cosh( β F / 2) + 1 , cosh( β F / 2) a 1 = (2 π − 3) cosh( β F / 2) + 1 , 1 a 2 = cosh( β F / 2) + 2 π − 3 .

Intruder’s displacement In the large n (and ρ 0 � 1) Z π 1 P ( R n ) ' d k exp ( � i ( k · R n ) � ρ 0 Ω n ( k )) 4 π 2 � π Φ ( k ) = − ia 0 k x + a 1 k 2 x / 2 + a 2 k 2 y / 2 ∞ � Ω n ( k ) z n Ω z ( k ) = sinh( β F / 2) n =0 a 0 = (2 π − 3) cosh( β F / 2) + 1 , 1 Φ ( k ) ) ∼ cosh( β F / 2) (1 − z ) 1 − z + Φ ( k ) / χ z a 1 = (2 π − 3) cosh( β F / 2) + 1 , 1 a 2 = cosh( β F / 2) + 2 π − 3 . with π χ z ∼ − (1 − z ) ln(1 − z ) the leading asymptotic term of the generating function of the mean number of “new” (virgin) sites visited on the n -th step BD Hughes, (2005) Random walks in random environments

Velocity and variance 0.09 0.03 0.003 0.002 v 0.001 0.06 0.02 0 0 0.005 0.01 y /n ρ 0 v σ 2 0.03 0.01 0 0 0 0.2 0 0.2 0.1 0.1 ρ 0 ρ 0 ρ 0 n ρ 0 sinh( β F / 2) σ 2 v ∼ y ∼ cosh( β F / 2) + 2 π − 3 (2 π − 3) cosh( β F / 2) + 1 The intruder moves at constant velocity along the field direction and di ff uses along the transversal direction O B´ enichou, C M-M, G Oshanin, PRE 87 020103 (2013)

Velocity and variance 0.09 0.03 0.003 0.002 v 0.001 0.06 0.02 0 0 0.005 0.01 y /n ρ 0 v σ 2 0.03 0.01 0 0 0 0.1 0.2 0 0.1 0.2 ρ 0 ρ 0  βρ 0 4( π − 1) F , β F � 1  ρ 0 sinh( β F / 2)  (2 π − 3) cosh( β F / 2) + 1 = v ∼ ρ 0 v ∞ = 2 π − 3 , β F � 1   O B´ enichou, et al, PRL (2000) 84 , 511; PRB (2001) 63 , 235413 O B´ enichou, C M-M, G Oshanin, PRE 87 020103 (2013)

Weak superdiffusion 0.4 0.3 φ ( n ) 0.2 0.1 0 0 2 3 4 1 10 10 10 10 10 n a 1 + 2 a 2 π ( γ − 1) + 2 a 2 � � σ 2 0 0 ln( n ) x ∼ ρ 0 n π � 1 � lim n →∞ H n +1 = ln( n ) + γ + O with γ ≈ 0 . 577 n O B´ enichou, C M-M, G Oshanin, PRE 87 020103 (2013)

Anomalous fluctuations broadening 0.15 0.1 P n (x) 0.05 In the limit ρ 0 → 0 0 0 10 20 30 40 50 60 − ( x − vn )2 x x ) − 1 / 2 e 2 σ 2 P n ( x ) = (2 πσ 2 (1 + A/n + . . . ) , x 0.2 − y 2 2 σ 2 y ) − 1 / 2 e y (1 + B ln n/n + . . . ) , P n ( y ) = (2 πσ 2 P n (y) 0.1 0 -10 -5 0 5 10 v ∼ ρ 0 a 0 , y a 1 + 2 a 2 ( γ − 1) + 2 a 2 � � σ 2 0 0 ln( n ) n , x ∼ ρ 0 π π σ 2 y ∼ ρ 0 a 2 n , a i ≡ a i ( β F )

Confined geometries

Confined geometries The variance of the intruder’s displacement can be represented as n σ 2 x ∼ ρ 0 a 1 n + ρ 0 a 2 0 , χ n χ n : mean # of new sites visited on the n -th step by any vacancy. In terms of S n , the mean # of distinct lattice sites visited by any of the vacancies up to time n χ n = S n − S n − 1 S n is a fundamental characteristic property of a lattice discrete-time RW . O B´ enichou, P Illien, C M-M, G Oshanin (2013)

Confined geometries In general, for infinite systems (at least in one direction) S n ∼ n α α is and indicator of the mixing of the lattice gas and depends on the e ff ective dimensionality of the lattice. ◮ for larger α , a vacancy mostly moves to new sites ◮ for smaller α , a vacancy predominantly revisits already visited sites In general α < 1 for systems in which the RW is recurrent , while α = 1 for non-recurrent RW’s.

Confined geometries We have χ n ∼ n α − 1 σ 2 x ∼ ρ 0 a 1 n + ρ 0 a 2 0 n 2 − α ⇒ ◮ For non-recurrent random walk ( α = 1), the behaviour is di ff usive ◮ For recurrent random walks ( α < 1) σ 2 x ∼ ρ 0 a 2 0 n 2 − α The less e ffi cient the mixing of the lattice gas is the faster the variance of the intruder’s displacement grows