Geometry-Induced Superdiffusion in Driven Crowded Systems Carlos - - PowerPoint PPT Presentation
Geometry-Induced Superdiffusion in Driven Crowded Systems Carlos Meja-Monasterio Technical University of Madrid Galileo Galilei Institute, Arcetri Florence May 2014 Active micro-rheology Active manipulation of small probe particles by
Technical University of Madrid
Galileo Galilei Institute, Arcetri Florence May 2014
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
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
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
10 10
1
10
2
10
10
10
10
−x
In the wake
∼ x−3/2 5 10 15 10
10
10
10
10
2
x
In front
The medium remembers the passage of the intruder on large temporal and spatial scales
C M-M, G Oshanin Soft Matter (2011), 7 993
◮ We consider a square lattice of Lx × Ly sites, of unit spacing, with P.B.C and populated with hard-core particles. ◮ Each site can be either empty or
◮ The system evolves in discrete time n and particles move randomly. ◮ One particle, the intruder, is subject to a constant force F
probability 1/4.
pν = Z −1e
β 2 F·eν ,
where Z = 2(1 + cosh (βσF/2)) and β is the inverse temperature.
Simple Exclusion Process
2 4 6 8
F
0.1 0.2 0.3
|V|
βF = 0.5 βF = 10
Stokesian regime V = F ξ , with friction coefficient ξ = ξmf + ξcoop where
ξmf = 4τ βσ2(1 − ρ) and ξcoop = 4τ βσ2(1 − ρ) (π − 2)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
2 4 6 8
F
0.1 0.2 0.3
|V|
We are interested in the limit of very dense lattices or very strong pulling forces.
O B´ enichou, G Oshanin, PRE (2001) 64. 020103 MJAM Brummelhuis, HJ Hilhorst, Physica A (1989) 156, 575
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
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 rn at time n conditioned to {Zj
n}
P(rn|{Zj
n}) =
n
· · ·
n
δ(rn, r1
n + · · · + rM n )P(r1 n, . . . , rM n |{Zj n}) P(r1 n, . . . , rM n |{Zj n}) is the conditional probability that within the
time interval n the intruder moved to r1
n due to its interaction with
vacancy 1, to r2
n due to its interaction with vacancy 2, etc. In the lowest order in ρ0 the vacancies contributions are
independent and P(r1
n, . . . , rM n |{Zj n}) ≃ M
P(rn|Z j
n)
The problem reduces to M single vacancies, correct to O(ρ0).
Averaging P(rn|Z j n) over the initial distribution of vacancies
P(rn) ≃
n
· · ·
n
δ(rn, r1
n + · · · + rM n ) M
P(rn|Z j
n) Defining the Fourier transformed distribution
Pn(k) =
exp (−ik · rn) P(rn|{Z j
n})
and summing over rn one obtains that it factorizes into Pn(k) =
exp (−ik · rn) P(rn|Z j
n)
M
Taking the thermodynamic limit Lx, Ly → ∞ with ρ0 fixed we
Pn(k) ≃ exp(−ρ0Ωn(k)) Ωn(k) is implicitly defined by Ωn(k) =
n
∆n−l(k|eν)
F ∗
l (0|eν|Z) ,
F ∗
l (0|eν|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 + eν at time l − 1 and ∆l(k|eν) = 1 − pl(k) exp (i(k · eν)) P(Rn) ' 1 4π2
Z π
π
dk exp (i (k · Rn) ρ0 Ωn(k))
Ωn(k) can be solved explicitly in terms of its generating function Ωz(k) =
∞
Ωn(k) zn In the large n (and ρ0 1) limit z → 1− Ωz(k) ∼ 1 (1 − z) Φ(k) 1 − z + Φ(k)/χz with χz ∼ − π (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
Then Ωz(k) ∼ Φ(k) (1 − z)2
π Φ(k) −1 , with Φ(k) = −ia0kx + a1k2
x /2 + a2k2 y /2
a0 = sinh(βF/2) (2π − 3) cosh(βF/2) + 1 , a1 = cosh(βF/2) (2π − 3) cosh(βF/2) + 1 , a2 = 1 cosh(βF/2) + 2π − 3 .
P(Rn) ' 1 4π2
Z π
π
dk exp (i (k · Rn) ρ0 Ωn(k))
In the large n (and ρ0 1)
Ωz(k) =
∞
Ωn(k) zn ) ∼ 1 (1 − z) Φ(k) 1 − z + Φ(k)/χz
with χz ∼ − π (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
Φ(k) = −ia0kx + a1k2
x /2 + a2k2 y /2
a0 = sinh(βF/2) (2π − 3) cosh(βF/2) + 1 , a1 = cosh(βF/2) (2π − 3) cosh(βF/2) + 1 , a2 = 1 cosh(βF/2) + 2π − 3 .
0.1 0.2 0.03 0.06 0.09 0.1 0.2 0.01 0.02 0.03
0.005 0.01 0.001 0.002 0.003
v σ2
y/n
ρ0 ρ0
v ρ0
v ∼ ρ0 sinh(βF/2) (2π − 3) cosh(βF/2) + 1 σ2
y ∼
ρ0n cosh(βF/2) + 2π − 3 The intruder moves at constant velocity along the field direction and diffuses along the transversal direction
O B´ enichou, C M-M, G Oshanin, PRE 87 020103 (2013)
0.1 0.2 0.03 0.06 0.09 0.1 0.2 0.01 0.02 0.03
0.005 0.01 0.001 0.002 0.003
v σ2
y/n
ρ0 ρ0
v ρ0
v ∼ ρ0 sinh(βF/2) (2π − 3) cosh(βF/2) + 1 =
βρ0 4(π−1) F ,
βF 1 v∞ =
ρ0 2π−3 ,
βF 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)
10 10
1
10
2
10
3
10
4
0.1 0.2 0.3 0.4 φ(n) n
σ2
x ∼ ρ0
π (γ − 1) + 2a2 π ln(n)
limn→∞ Hn+1 = ln(n) + γ + O 1
n
O B´ enichou, C M-M, G Oshanin, PRE 87 020103 (2013)
In the limit ρ0 → 0
Pn(x) = (2πσ2
x)−1/2e − (x−vn)2
2σ2 x
(1 + A/n + . . .) , Pn(y) = (2πσ2
y)−1/2e − y2
2σ2 y (1 + B ln n/n + . . .) ,
10 20 30 40 50 60
x
0.05 0.1 0.15
Pn(x)
5 10
y
0.1 0.2
Pn(y)
v ∼ ρ0 a0 , σ2
x ∼ ρ0
π (γ − 1) + 2a2 π ln(n)
σ2
y ∼ ρ0 a2n ,
ai ≡ ai(βF)
The variance of the intruder’s displacement can be represented as σ2
x ∼ ρ0a1n + ρ0a2
n χn , χn: mean # of new sites visited on the n-th step by any vacancy. In terms of Sn, the mean # of distinct lattice sites visited by any
χn = Sn − Sn−1 Sn is a fundamental characteristic property of a lattice discrete-time RW.
O B´ enichou, P Illien, C M-M, G Oshanin (2013)
In general, for infinite systems (at least in one direction) Sn ∼ nα α is and indicator of the mixing of the lattice gas and depends on the effective 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.
We have χn ∼ nα−1 ⇒ σ2
x ∼ ρ0a1n + ρ0a2 0n2−α ◮ For non-recurrent random walk (α = 1), the behaviour is
diffusive
◮ For recurrent random walks (α < 1)
σ2
x ∼ ρ0a2 0n2−α
The less efficient the mixing of the lattice gas is the faster the variance of the intruder’s displacement grows
symbols, ðtÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffi 3=2 p L2=ð4a2
00tÞ2 xðtÞ]
ð Þ ¼ ffiffiffiffi p ð
2
Þ
2ð Þ
ð Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffi p ð
0 Þ
symbols, ðtÞ ¼ 3 ffiffiffiffi
00tÞ2 xðtÞ]
capillaries: stripes:
1/2 1/2
1/2 1/2
X
0.2 0.4 0.6 0.8 1
2 4 6 8 10 Pn(X) X n = 104 n = 105 n = 106
0.8 1 1.2 1.4 1.6 1.8 1 10 100 ˜ κeven(n), ˜ κodd(n) n
(a)
1 10 100 0.85 0.9 0.95 1 n
(b)
. Pn(X) '
⇢0→0 exp (κeven(n))
✓κeven(n) + κodd(n) κeven(n) κodd(n) ◆X/2 ✓q ◆
✓
✓q κ2
even(n) κ2
◆
lim
⇢0→0
κn
(2j+1)
ρ0 = (p1 p−1) r 2n π 2p1p−1(p1 p−1)+o(1) (13) lim
⇢0→0
κn
(2j)
ρ0 = r 2n π + o(1) , j = 0, 1, 2, . . . .
We need to determine the long time limit of the variance at fixed vacancy density For confined geometries (limt!1lim0!02
x lim0!0limt!12
x).
between two consecutive visits to the intruder, a given vacancy experiences an effective bias due to the motion
O Bénichou, et al, PRL 111, 260601 (2013)
The long time behaviour is always diffusive In quasi-1D the longitudinal diffusivity is enhanced
lim
t!1
2
x
t
8 > > > < > > > : B quasi-1D; 4a2
010 lnð1 0 Þ
2D lattice; 2a2
0½A þ cothðf=2Þ=ð2a0Þ0
3D lattice;
In 2D
Dk D? ∼ 1 ρ0
?
Dk D? ∼ ln(ρ1
0 )
No enhancement is observed in 3D
O Bénichou, et al, PRL 111, 260601 (2013)
In the intermediate regime we find
2
x
8 > > > < > > > : tgð2
0tÞ
quasi-1D;
0t lnðð0a0Þ2 þ 1=tÞ
2D lattice; 2a2
0½A þ cothðf=2Þ=ð2a0Þ0t
3D lattice;
O Bénichou, et al, PRL 111, 260601 (2013)
Glass-forming Yukawa fluid Winter, et al., PRL 108, 028303 (2012) Binary mixture of Lennard-Jones particles Schroer, Heuer, PRL 110, 067801 (2013)
O Bénichou, et al, PRL 111, 260601 (2013)
1d single-file
2d 3d
2d stripes
2d slit pores
3d capillaries
X
x
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
∞ ∼ n1/2
πn ln(n)
π ln(n)
ρ0a2 π n ln(n)
ln1/2
l n1/2 ρ0a2 l
n3/2 l2n1/2
l2 n1/2 ρ0a2 l2 n3/2 ln ln(n) l ln(n) ρ0a2 l
n ln(n)
∞ ∼ n
New phenomena field-induced broadening of fluctuations in overcrowded environments
McGuffee and Elcock, PLoS Computational Biology (2010)