Time-Dependent Fluctuations and Superdiffusivity in the Driven Lattice Lorentz Gas

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by  Sebastian Leitmann

Time-Dependent Fluctuations and Superdiffusivity in the Driven Lattice Lorentz Gas
Sebastian Leitmann and Thomas Franosch
Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020 Innsbruck, Austria
(Dated: January 4, 2017)
We consider a tracer particle on a lattice in the presence of immobile obstacles. Starting from
equilibrium, a force pulling on the particle is switched on, driving the system to a new stationary
state. We solve for the complete transient dynamics of the fluctuations of the tracer position
along the direction of the force. The analytic result, exact in first order of the obstacle density
and for arbitrarily strong driving, is compared to stochastic simulations. Upon strong driving, the
fluctuations grow superdiffusively for intermediate times; however, they always become diffusive in
the stationary state. The diffusion constant is nonanalytic for small driving and is enhanced by
orders of magnitude by increasing the force.
7
1
0
The material properties of complex fluids such as col- fluctuations as time progresses. The crowding is incor-
2
loidal dispersions [1, 2], solutions of biopolymers [3, 4], porated in the model by introducing hard and immo-

n
or biomaterials [5, 6] can be probed by pulling a meso- bile obstacles randomly distributed over the lattice. A
a
scopic tracer particle through the medium. In linear re- forceisswitchedonatacertaininstantoftimesuchthat
J
sponse by the fluctuation-dissipation theorem it is suf- thetracerperformsabiasedobstructeddiffusionthrough
3
ficient to monitor the force-free thermally agitated mo- the system. To first order in the obstacle density the

] tion of the tracer which is the principle of passive mi- moment-generating function for the displacements can
t
f crorheology[7–9]. ThenbyageneralizedStokes-Einstein be determined in principle exactly; so far only the time-
o
relation the dynamic mobility is connected to the lin- dependentvelocityresponsehasbeenelaborated[19]. In
s
. ear macroscopic frequency-dependent viscosity or elastic equilibrium, the fluctuations are also known for low ob-
t
a modulus. In contrast, in active microrheology the parti- stacle densities via the mean-square displacement [46–
m cle is manipulated by optical or magnetic tweezers and 48]. Within this model, we consider the fluctuations
- pulled through the environment in principle by arbitrar- along the direction of the force and provide for the first
d
ilystrongforces[10–12]. Here, thesystemisintrinsically time a complete time-dependent analytic solution for a
n
strongly out of equilibrium and a plethora of new phe- generic strongly interacting system driven far from equi-
o
c nomena have been found experimentally and in simula- librium.
[ tions,suchasforcethinning[13–15],(transient)superdif-

1 fusive behavior, and enhanced diffusivites [16, 17]. Model.— We consider a tracer particle performing
v To make theoretical progress in the nonlinear regime, a random walk with successive nearest-neighbor jumps
6 generic models have been investigated that focus on the N ={±aex,±aey} on a square lattice with lattice spac-
5
mutual exclusion originating from the strong repulsive ing a. The lattice consists of free sites accessible to the
7
interaction between the tracer and its environment as tracer, as well as sites with randomly placed immobile
0
0 the most important ingredient. The underlying dynam- hard obstacles of density n. If the tracer attempts to
. ics of the tracer is usually modeled as a random walk jumpontoanobstaclessite,itmerelyremainsatitsorig-
1
0 on a lattice or Brownian motion in continuum, while the inal position.
7 surroundingsrangefromdiluteandimmobileobstaclesto
1 dynamicandcrowdedenvironments. Forlatticesystems, For times t < 0 the tracer performs a symmetric ran-
:
v progress and even exact results have been achieved [18– dom walk and the system is in equilibrium, such that
i 25],predictinginteralia anomalousdiffusion[26]andsu- it is equally likely to find the tracer at any accessible
X
perdiffusivebehavior[27]inconfinedsystems. Incontin- site. For times t ≥ 0, we apply a force pulling the
r
a uum,theframeworkofmode-couplingtheoryoftheglass tracer along the x direction of the lattice. The dimen-
transition [28–34], Langevin equations [35, 36], kinetic sionless force F = force · a/k T introduces a bias in
B
theory [37], and continuous-time random walks [38–40] the nearest-neighbor transition probabilities W(d∈N),
successfully describe certain phenomena emerging in the and local detailed balance W(aex)/W(−aex) = exp(F)
nonlinear regime. Exact results in the stationary state and W(aey)/W(−aey) = 1 along both lattice direc-
infirstorderofthebathparticleshavebeenobtainedfor tionssuggestsW(±aex)=e±F/2/(eF/2+e−F/2+2)and
active microrheology in suspensions of hard spheres per- W(±aey)=1/(eF/2+e−F/2+2). Weperformcomputer
forming Brownian motion [41–45]; yet, an evaluation of simulations of this model and monitor the displacement
the transient dynamics and the approach to the steady alongtheforce∆xj =xj−x0indiscretetimecorrespond-
state has remained a challenge. ing to the number of (attempted) jumps j of the tracer
Here we rely on a lattice model for a driven tracer in particle. The trajectories are averaged over many ini-
a crowded environment to investigate the growth of the tial positions and obstacle realizations and transformed
2
(a) (b) (c)
FIG. 1. Fluctuations of the tracer along the force characterized by (a) the variance Var (t), (b) the local exponent α(t), and
x
(c)thetime-dependentdiffusioncoefficientD (t). Inallthreepanels,forceincreasesfrombottomtotop. Solidlinescorrespond
x
totheanalyticsolutionandsymbolsrepresentresultsfromcomputersimulations. Theblackdashedlinesrepresentsimulation
results for F =10 and 12 with mobile obstacles at the same density performing a symmetric random walk with mean waiting
time τ.
to continuous time t via a Poisson transform [49]: Stokes-EinsteinrelationtolinearorderinF. Aftersquar-
ing the mean displacement [Eq. (3)] and retaining only
X∞ (Γt/τ)j
h∆x(t)mi= h∆xmi e−Γt/τ, (1) terms up to first order in the obstacle density, we obtain
j j! the variance
j=0
with mean waiting time τ/Γ=2τ/[1+cosh(F/2)] of the h Z t i
tracer, valid for any order m of the moment. The choice Varx(t)=2Dx0t+n ∆Rx(t)−2v0t dt0∆v(t0) . (5)
0
for the mean waiting time corresponds to unnormalized
transition rates (Γ/τ)W(d); in particular, the transition We have calculated the terms in the square bracket ana-
rateperpendiculartotheforceisindependentofthedriv- lytically in the frequency domain by solving for the scat-
ing. Considering normalized rates as in Ref. [19] results tering matrix in the single obstacle case along the lines
only in a (force-dependent) multiplicative shift of the of Ref. [19]. The new term ∆Rx(t) is essentially ob-
time scale. tainedasasumovercertainmatrixelementsofthesingle-
Here, the quantity of interest is the time-dependent scattering matrix (see Supplemental Material [50]).
fluctuations of the tracer position along the direction of Forincreasingstrengthoftheforce,thevarianceshows
the force, encoded in the variance of the displacement: a significant increase of the fluctuations parallel to the
(cid:10) (cid:11) force[Fig.1(a)]. Inparticular,atintermediatetimes,the
Varx(t)= [∆x(t)−h∆x(t)i]2 =h∆x(t)2i−h∆x(t)i2. fluctuations are goverened by a marked increase faster
(2)
than diffusion ∼ t. Only at later times we recover diffu-
For the special case without driving, F = 0, the lattice sionalbehavior,howeverwithavastlyincreaseddiffusion
Lorentz gas is recovered and an analytic solution for the coefficient.
time-dependent fluctuations in first order of the obstacle The time-dependent behavior of the variance can
density was achieved years ago [46–48]. be quantified in more detail by considering the time-
Fluctuations of the tracer along the force.— We de- dependent diffusion coefficient [Fig. 1(c)]
composethemeandisplacementh∆x(t)icontainedinthe
1 d
vitayrivanc=e [E(aq/.2(τ2))]siinnhto(Fth/2e)baforre dthrieftevm0pttwyitlhatbtiacreeavneldoca- Dx(t):= 2dtVarx(t), (6)
0
correction:
and the local exponent α≡α(t) [Fig. 1(b)] defined by a
Z t
logarithmic time derivative
h∆x(t)i=v t+n dt0∆v(t0), (3)
0
where∆v(t)isthefirst-order-den0sityresponsefortheav- α(t):= dlnd(Vlna(rtx)(t)) = 2VDarxx((tt))t. (7)
erage velocity [19]. Similarly, the mean-square displace-
ment along the force Thus, ordinary diffusion corresponds to α = 1, whereas
subdiffusive and superdiffusive behavior is indicated by
h∆x(t)2i=2Dx0t+(v0t)2+n∆Rx(t), (4) α<1 and α>1, respectively. While there is still a sub-
contains a diffusive contribution with bare diffusion co- diffusiveregimeatsmalltimesoftheorderofthedensity
efficient D0 = (a2/4τ)cosh(F/2), drift (v t)2 and first- n,transportatstrongdrivingisdominatedbyasuperdif-
x 0
order-density response ∆Rx(t). The bare diffusion co- fusiveregimewhichgrowswithincreasingstrengthofthe
efficient D0 and the bare velocity v are connected by a driving [Fig. 1(b)].
x 0
3
The emergence of superdiffusion for large forces can
be rationalized by observing that up to times τ, the
time a tracer needs to go around an obstacle, the par-
ticle essentially moves only along the field until it hits
an obstacle. Thus, up to τ only the forward motion
needstobetakenintoaccountandthedynamicsisalong
one-dimensionallanes. Theprobabilitydistributionthen
reads P(∆x = a·j) = n(1−n)j +(1−n)j+1δjJ for J
jump attempts. Then, one can work out that asymptot-
ically the variance in continuous time is determined by
the fluctuations of the free path length and grows as
1na2 t3
Varx(t)= 3 64 exp(3F/2)τ3. (8)
This result suggests that α = 3 is the true asymp-
FIG. 2. Imaginary part of the frequency-dependent response
totic exponent of superdiffusion for the driven lattice
encoding the approach to the stationary state for different
Lorentz model (see Supplemental Material [50]). Match- strength F of the driving. Inset: Time-dependent diffusion
ing Eq. (8) to the short-time diffusion ≈a2eF/2t/4τ coefficientfordifferentforcesexemplifyingthenonmonotonic
√
yields as onset time of superdiffusion τ∗ ∼ τe−F/2/ n. behavior. Symbolscorrespondtosimulationresultsandlines
Therefore,thewindowofsuperdiffusionτ∗ (cid:46)t(cid:46)τ grows represent the theory.
with the force.
The instantaneous response Dx(t → 0) of the time- scattering events of the tracer with different obstacles
dependent diffusion coefficient Dx(t) [Eq.(6), Fig. 1(c)] and are not fully included in the first-order theory.
is determined by the first jump event only and one read-
It is interesting to ask how the superdiffusive behav-
ily obtains Dx(t → 0) = Dx0(1−n). The long-time be- ior emerges from the equilibrium reference system for
havior is obtained by evaluating the first-order-response
small forces. In equilibrium, the dynamics of a Brow-
terms ∆Rx(t) and ∆v(t) for long times leading to the nianparticlesatisfiesglobaldetailedbalanceandtheap-
asymptotic expansions
proach of the diffusion coefficient to the stationary state
t2 Dxeq(t)−Dxeq(t→∞) is described by a weighted sum of
∆Rx(t)=∆Rx,2 +∆Rx,1t+O(t0), (9) relaxingexponentials[52,53]i.e. acompletelymonotone
2
Z t function [54],
dt0∆v(t0)=∆v1t+∆v0+o(t0). (10) Z ∞
0 Deq(t)−Deq(t→∞)= e−γtm(dγ), (12)
x x
0
The expressions for the coefficients are lengthy and de-
with a non-negative measure m(dγ). In particular, in
pend only on the force and will not be shown here. The
analytic solution fulfills the relation ∆Rx,2 = 4v0∆v1, tghoevelrantetdicebyLaonreanltgzebgraasicindeecqauyil∼ibrti−u1mr,efltheectianpgprtohaecpheris-
such that the long-time diffusion coefficient is obtained
sistent memory in the system due to repeated interac-
as
tionwiththeobstacledisorder[46]. Takingtheone-sided
Dx(t→∞)=Dx0+nh∆R2x,1 −v0∆v0i, (11) tFaoiunrsi:er transform Dˆxeq(iω)=R0∞dt e−iωtDxeq(t), one ob-
exact in first order of the density of obstacles n and for Dˆeq(iω)− Dxeq(t→∞) =Z ∞ γ−iω m(dγ). (13)
arbitrary strong driving. Thus, in first order of the den- x iω γ2+ω2
0
sity, the long-time behavior is always diffusive. Yet, the
In particular, for ω > 0 the imaginary part of the
long-time diffusion coefficient increases by more than a
frequency-dependent approach to the stationary state in
factor of ten already at density n = 10−3 for the large
equilibriumisalwaysnegative(seealsoFig.2). Thenon-
forces in Fig. 1(c). The strong increase of the diffusion
vanishing contribution for ω →0 in equilibrium can be
coefficient at intermediate times is a fingerprint of the
traced back to a nonanalytic small-frequency behavior
superdiffusive behavior governing the transition to the of the diffusion coefficient Dˆeq(iω) ’ Deq(t→∞)/iω −
x x
stationary state.
(nπa2/8)ln(iωτ), corresponding to the algebraic tail
For fixed density and increasing force, deviations be- ∼t−1 in the temporal domain [46]. Thus, for the real
tween the analytic and the simulation results increase.
and imaginary part, we obtain
This is due to contributions higher order in the obsta-
cle density which become more and more important for 1 hDˆeq(iω)− Dxeq(t→∞)i’−π ln(ωτ)−iπ2, (14)
increasing force. Such higher-order terms arise due to na2 x iω 8 16
4
which rationalizes the small-frequency behavior in equi-
librium [Fig. 2].
Thisbehaviornolongerholdstrueinthepresenceofa
force on the tracer where positive contributions emerge
for any strength of the driving such that the approach
to the stationary diffusion coefficient is not necessarily
monotonically decreasing [Eq.(12)]. In fact, in the time
domain, this deviation from equilibrium becomes mani-
fest in a nonmonotonic behavior of the time-dependent
diffusion coefficient such that the point of least diffusiv-
ity is always attained at intermediate times (see inset of
Fig. 2).
It is convenient to characterize the transport behav-
ior in the stationary state in terms of the force-induced
diffusion coefficient Dind ≡Dind(F,n) [43] [Fig. 3]:
x x FIG. 3. Force-induced diffusion coefficient Dind (corrected
x
Dxind =Dx(t→∞)−Dxeq(t→∞), (15) bdyiffutshieonemcopetffiycliaenttticDe0Daxinsd(anfu=nct0io))n ionf uthneitsapopfliethdefobracree.
x
Symbols correspond to simulation results and lines represent
with the long-time diffusion coefficient in the absence of
(cid:2) (cid:3) the theory. The black dashed lines represent the asymp-
driving, Deq(t → ∞) = (a2/4τ) 1−(π−1)n [46]. For
x totic expansions for small and large forces. Inset bottom
small forces F →0, our explicit solution reveals that the
right: Same quantity as a function of the Péclet number
force-induced diffusion Dxind acquires a leading nonana- Pe := av0/Dx0(F = 0) = 2sinh(F/2) given by the ratio of
lytic term thebarevelocityv andthethermalfluctuationsD0(F =0).
0 x
Inset top left: Stationary diffusion coefficient measured in
Dind = nAa2F2[ln(1/|F|)+B]+O(cid:0)F2ln(1/|F|)(cid:1)2, units of Dx0 for different forces and densities.
x 4τ
(16)
Summary and conclusion.— We have solved for the
withprefactorA=(3π2+4π+8)/16π ≈0.998andasub- dynamics of a tracer particle on a lattice in response to
leadingcorrectiontermB ≡B(n)>0(seeSupplemental a step force in the presence of obstacles. The complete
Material[50]). Theoriginofthenonanalyticcontribution timedependenceofthefluctuationsofthetracerposition
can be understood by observing that the propagators in paralleltotheforcehavebeenevaluatedexactlyforarbi-
the presence of a force are essentially described by the trary strong driving in first order of the obstacle density.
equilibrium propagators up to a force-dependent shift in Our main result is the emergence of a superdiffusive
the frequency domain. In particular, they inherit the growthofthefluctuationsinanintermediatetimeregime
nonanalytic dependence for long times from the equilib- followed by ordinary diffusion with a stationary diffu-
riumpropagatorsleadingtotheemergenceofnonanalytic sion coefficient enhanced by orders of magnitude. These
contributions for small forces in the stationary state (see superdiffusively growing fluctuations have been discov-
Supplemental Material [50]). ered in simulations for crowded systems [16, 17, 38], but
ForlargeforcesF (cid:38)6,theforce-induceddiffusioncoef- have also been derived analytically in crowded lattices
ficient increases rapidly [Fig. 3] and assumes the asymp- in confined geometries [27] and one-dimensional kineti-
totic form cally constrained models [55]. Our results for the driven
latticeLorentzgasdemonstratethattheemergenceofsu-
na2 (cid:0) (cid:1)
Dind = exp(3F/2)+O exp(F) . (17) perdiffusion is generic and arises due to the competition
x 16τ
of exclusion interaction and nonlinear driving already at
Thisscalingbehaviorcanalsobeobtainedbyasymptotic low obstacle densities. In the lattice Lorentz model, the
matching of the superdiffusive behavior [Eq. (8)] to the superdiffusioncanbetracedbacktotherapidincreaseof
diffusive increase at time scale τ. the variance of the free path lengths as the particle per-
Remarkably, there exists a critical force F ≈ 1.45 formsapurelydirectedmotionalongthefielduntilithits
c
where in first order of the density the long-time diffu- an obstacle. The full time-dependent solution addition-
sion coefficient is identical to the bare diffusion coeffi- ally provides the first direct access to the intermediate
cient, Dx(t→∞,Fc)=Dx0(Fc). Thus, this critical force window of superdiffusive motion.
separatestworegimesofstrikinglydifferentbehavioralso In the lattice Lorentz gas in equilibrium, global de-
observed in other models [27, 35, 43]. For forces F <F , tailed balance holds and correlation functions purely re-
c
the intuitive picture holds where an increase in the dis- lax; i.e., they are completely monotone. Switching on
order suppresses the fluctuations, whereas in the regime the step force drives the system out of equilibrium and
F >F increasing disorder leads to an enhancement. theapproachofthediffusioncoefficienttothestationary
c
5
stateisnonmonotonic,instrikingcontrasttotheequilib-
riumparadigm. Whilethisbehaviorisobviousforstrong
forces where the stationary diffusion coefficient exceeds
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[44] J. W. Swan and R. N. Zia, Phys. Fluids 25, 083303 [56] B. J. Alder and T. E. Wainwright, Phys. Rev. A 1, 18
(2013). (1970).
[45] N.J.HohandR.N.Zia,J.FluidMech.795,739(2016). [57] M. H. Ernst and A. Weyland, Phys. Lett. A 34, 39
[46] Th.M.Nieuwenhuizen,P.F.J.vanVelthoven, andM.H. (1971).
Ernst, Phys. Rev. Lett. 57, 2477 (1986). [58] H. van Beijeren, Rev. Mod. Phys. 54, 195 (1982).
[47] Th.M.Nieuwenhuizen,P.F.J.vanVelthoven, andM.H. [59] T. Franosch, F. Höfling, T. Bauer, and E. Frey, Chem.
Ernst, J. Phys. A 20, 4001 (1987). Phys. 375, 540 (2010).
7
SUPPLEMENTAL MATERIAL
Analytic solution of the free dynamics
We define the lattice with lattice spacing a by the collection of all sites Λ = {r = (ax,ay) ∈ aZ×aZ : x,y ∈
[−L/2,L/2[}. We employ periodic boundary conditions and anticipate the limit of large lattices L → ∞. The
conditional probability hr|Uˆ (t)|0i for a tracer starting at the origin 0 and moving a distance r in lag time t is
0
determined by the time-evolution operator Uˆ0(t) which fulfills the master equation ∂tUˆ0(t) = Hˆ0Uˆ0(t) with free
’Hamiltonian’ Hˆ :
0
Hˆ0 = Γτ X(cid:2)−|rihr|+ X W(d)|rihr−d|(cid:3), N ={±aex,±aey}. (18)
r∈Λ d∈N
The invariance of the free dynamics under translations becomes manifest in the plane wave basis
1 X
|ki= √ exp(ik·r)|ri, (19)
N
r∈Λ
with number of lattice sites N = L2, wave vector k = (kx,ky) ∈ Λ∗ = {(2πx/aL,2πy/aL) : (x,y) ∈ Λ}, and
scalar product k·r=kxx+kyy. Then, the invariance of the free Hamiltonian under translations reads hk|Hˆ0|k0i=
(cid:15)(k)δ(k,k0) with eigenvalue:
Γ X(cid:2)(cid:0) (cid:1)(cid:1) (cid:3)
(cid:15)(k)=− 1−cos(k·d +isin(k·d) W(d) (20)
τ
d∈N
The moment-generating function F (k,t) of the tracer displacements ∆r = r − r0 with initial distribution
0
hr|p i=1/N is then obtained by the matrix elements of the time-evolution operator in the plane wave basis:
eq
F (k,t)= X e−ik·(r−r0)hr|Uˆ (t)|r0ihr0|p i=hk|Uˆ (t)|ki=hk|exp(Hˆ t)|ki=exp(cid:0)(cid:15)(k)t(cid:1), (21)
0 0 eq 0 0
r,r0∈Λ
where we used the formal solution Uˆ (t)=exp(Hˆ t) of the time-evolution operator. In particular, we obtain the first
0 0
two moments for the displacement along the field via
∂ (cid:12) a
h∆x(t)i=−i F (k,t)(cid:12) = sinh(F/2)t=v t, (22)
∂kx 0 (cid:12)k=0 2τ 0
∂2 (cid:12) a2
h∆x2(t)i=− F (k,t)(cid:12) =2D0t+(v t)2, D0 = cosh(F/2). (23)
∂kx2 0 (cid:12)k=0 x 0 x 4τ
Analytic solution to first order in the obstacle density
In the presence of obstacles, the moment-generating function F(k,t) of the displacements ∆r=r−r0 is defined in
terms of the disorder-averaged time-evolution operator [Uˆ(t)] :
av
X
F(k,t)= exp[−ik·(r−r0)]hr|[Uˆ(t)] |r0ihr0|p i, (24)
av eq
r,r0∈Λ
withinitialsite-occupationprobabilitydistributionhr0|p i=1/N. Inthefrequencydomain,themomentsareencoded
eq
in the Green function
1
[G] (k)= , (25)
av G (k)−1−Σ(k)
0
with the free propagator G (k) = hk|Gˆ |ki = R∞dt e−iωthk|Uˆ (t)|ki = [iω −(cid:15)(k)]−1 and self-energy Σ(k) which
0 0 0 0
accounts for all possible interactions of the tracer with the obstacle disorder. In first order of the obstacle density n,
the self-energy can be expressed by the single-scattering t-matrix which represents repeated collisions of the tracer
with the same obstacle: Σ(k)=nNt(k)+O(n2) [19, 46, 51].
8
The single-scattering t-matrix fulfills the relation
tˆ=vˆ+vˆGˆ tˆ=vˆ+tˆGˆ vˆ, (26)
0 0
where vˆ denotes the single obstacle potential which cancels transitions from and to the impurity site. The scattering
matrix is calculated in the real space basis by a 5×5 matrix inversion problem hr|tˆ|r0i = hr|vˆ(1−Gˆ vˆ)−1|r0i since
0
the obstacle potential has only nonvanishing contributions at the obstacle site and its four neighbors [19]. Then, the
forward-scattering amplitude Nt(k) = Nhk|tˆ|k0i = P exp[−ik·(r−r0)]hr|tˆ|r0i is derived by a transformation to
r,r0
the plane wave basis and the Green function is obtained as
[G] (k)=G (k)+nNt(k)G (k)2+O(n2). (27)
av 0 0
To make connection to the stochastic simulation we correct the Green function for the fraction n of immobile
random walkers starting at impurities by normalizing Eq. (27) with 1/(1−n)=1+n+O(n2):
[G] (k)=G (k)+n[G (k)+Nt(k)G (k)2]+O(n2). (28)
av 0 0 0
Then, we calculate the mean displacement
∂ (cid:12) ∂G (cid:12) h∂G ∂Nt i
[G] (cid:12) = 0(cid:12) +n 0 + G2 +O(n2), (29)
∂kx av(cid:12)k=0 ∂kx(cid:12)k=0 ∂kx ∂kx 0 k=0
and the mean-square displacement
∂2 (cid:12) ∂2G (cid:12) h∂2G ∂2Nt ∂Nt∂G i
[G] (cid:12) = 0(cid:12) +n 0 + G2+4G 0 +O(n2), (30)
∂kx2 av(cid:12)k=0 ∂kx2 (cid:12)k=0 ∂kx2 ∂kx2 0 0∂kx ∂kx k=0
along the force in the frequency domain in first order of the density n. The derivatives after the x-component of the
wave vector k are obtained as sum over the matrix elements hr|t|r0i:
∂Nt(cid:12) X
∂kx(cid:12)(cid:12)k=0 =−i ex·(r−r0)hr|tˆ|r0i, (31)
r,r0
∂2Nt(cid:12) X
∂kx2 (cid:12)(cid:12)k=0 =−r,r0[ex·(r−r0)]2hr|tˆ|r0i. (32)
Nonanalytic behavior
The conditional probability in real space can be calculated analytically [49] and is given by
hr|Uˆ (t)|0i=eFx/2ae−Γt/τI (t/2τ)I (t/2τ), r=(x,y), 0=(0,0), (33)
0 x/a y/a
where Im(·) denotes the modified Bessel function of integer order m. The free propagators Gˆ0(iω) are defined by a
one-sided Fourier transform hr|Gˆ (iω)|0i = R∞dt e−iωthr|Uˆ (t)|0i and encode the time-evolution of the system in
0 0 0
the frequency domain. Then, one observes that the propagator in the case of driving is essentially obtained by the
equilibrium propagator Gˆeq =Gˆ (F =0) via
0 0
Z ∞
hr|Gˆ (iω)|0i=eFx/2a dt e−[iω+(Γ−1)/τ]thr|Uˆ (t,F =0)|0i=eFx/2ahr|Gˆeq(iΩ)|0i. (34)
0 0 0
0
The only difference is a site-dependent prefactor and a shift in the frequency of the equilibrium propagator iΩ =
iω+(Γ−1)/τ. For example, the propagator h0|Gˆeq(iω)|0i = (2τ/π)K[(1+iωτ)−2]/(1+iωτ) can be expressed by
0
the complete elliptic integral of the first kind K[x] = Rπ/2dθ [1−xsin2(θ)]−1/2 and has the following nonanalytic
0
expansion for iω →0 and −π/2<arg(iω)<π/2:
h0|Gˆeq(iω)|0i= τ ln(8/iωτ)+ τ iωτ[1−ln(8/iωτ)]+O(cid:0)ω2ln(ω)(cid:1). (35)
0 π 2π
9
Thus, for long times iω →0, and small forces Γ−1=F2/16+F4/768+O(F6), the propagators in the presence of a
force inherit the nonanalytic dependence from the propagators in equilibrium:
h0|Gˆ (iω)|0i= τ ln(128/F2)+ τ F2[1−3ln(128/F2)]+O(cid:0)F4ln(1/F)(cid:1), F ↓0. (36)
0 π 2π 48
Sincethestationarydiffusioncoefficientessentiallyresultsfromsolvinga5×5matrixproblemwiththefreepropagators
as entries, the stationary diffusion coefficient displays nonanalytic contributions of the same type:
Dx(t→∞)=Dxeq(t→∞)+ nA4τa2F2[ln(1/|F|)+B]+O(cid:0)F2ln(1/|F|)(cid:1)2, (37)
with equilibrium diffusion coefficient Deq(t→∞)=(a2/4τ)[1−(π−1)n]. The subleading correction B ≡B(n) can
x
be explicitly evaluated to
7 4π4−π3−6π2−24π 2π/n
B = ln(2)− + . (38)
2 2(π−2)(3π2+4π+8) 3π2+4π+8
Asymmetric simple exclusion process for mobile obstacles
Wehavesimulatedthedynamicsofthetracerinthepresenceofmobileobstaclesperforminganasymmetricsimple
exclusion process [Fig. 4]. The velocity and the diffusion coefficient behave rather similarly to the case of unbiased
mobile obstacles. In particular, the effect of giant diffusion accompanied by a crossover regime persists also for the
asymmetric case.
1 4
symmetric Density n=10−3
ASEP p =0.5
right Force F =10
0.99 ASEP pleft =0.5 p =p =0.25
up down
3
0x
0 D
vt/v() 0.98 Density n=10−3 Dt/()x 2
0.97 Force F =10 symmetric
p =p =0.25 ASEP p =0.5
up down right
ASEP p =0.5
left
0.96 1
10−2 10−1 100 101 102 10−1 100 101 102
Time t/τ Time t/τ
FIG. 4. Time-dependent velocity v(t) and diffusion coefficient D (t) for the case that the obstacles perform an asymmetric
x
simpleexclusionprocess. Thejumpprobabilityperpendiculartothefieldisalwayssymmetricandgivenbyp =p =1/4.
up down
The curves shown correspond to p =p =1/4 (symmetric), bias in direction of the field p =1/2, p =0, and bias
right left right left
in the opposite direction p =1/2, p =0.
left right
Asymptotic model
For large forces, the transition rate along field dominates the transport behavior and the motion of the tracer
perpendicular and against the field can be ignored. Hence, in every jump the tracer hits an obstacle with probability
p=n and the probability for a displacement ∆x after J jumps is given by
P(∆x=a·j)=qjp+qj(1−p)δjJ =p+δjJ[1−(J +1)p]+O(p2), j =0,...,J, q =1−p. (39)
10
Then, one can readily calculate the mean and mean-square displacement of the tracer:
J
X 1
h∆xJi= ajP(∆x=a·j)= naJ(J +1)+aJ[1−(J +1)n]+O(n2), (40)
2
j=0
J
X 1
h∆x2i= (aj)2P(∆x=a·j)= na2J(J +1)(2J +1)+a2J2[1−(J +1)n]+O(n2). (41)
J 6
j=0
After performing the transformation to continuous time via the Poisson transform with a mean waiting time of
τ/γ =4τ/eF/2,
(cid:20)X∞ (γt/τ)J (cid:21)2 (cid:16)γt(cid:17)3 (cid:16)γt(cid:17)2 (cid:16)γt(cid:17)2
h∆x(t)i2 = h∆xJi J! e−γt/τ =−na2 τ −2na2 τ +a2 τ +O(n2), (42)
J=0
X∞ (γt/τ)J 2 (cid:16)γt(cid:17)3 5 (cid:16)γt(cid:17)2 (cid:16)γt(cid:17) (cid:16)γt(cid:17)2
h∆x2(t)i= h∆x2i e−γt/τ =− na2 − na2 +a2(1−n) +a2 +O(n2), (43)
J J! 3 τ 2 τ τ τ
J=0
we obtain the variance of the displacements in first order of the density n:
1na2 t3 1na2 t2 a2 t
h∆x2(t)i−h∆x(t)i2 = exp(3F/2) − exp(F) + (1−n)exp(F/2) +O(n2). (44)
3 64 τ3 2 16 τ2 4 τ
For large forces, the asymptotic model captures the dynamics of the first-order solution quantitatively until the
diffusive motion perpendicular to the force becomes relevant t (cid:38) τ [Fig. 5]. Hence, for intermediate times, the
dynamics becomes asymptotically superdiffusive ∼ tα with exponent α = 3. We also performed simulations at high
forces for different densities where the exponent α=3 can be observed in simulations.
3
106 Force F =28
Force
104 D1e0n−s3ity n t()x 2.5 F =28
τ 10−4 ar
0D2x 102 1100−−65 t/)V 2
/) 10−7 t( 1.5
t(x 100 10−8 Dx
r 2
a 1
V =
10−2 Density n
t) 10−3 10−6
α( 0.5 10−4 10−7
10−4 10−5 10−8
0
10−5 10−4 10−3 10−2 10−1 100 101 10−5 10−4 10−3 10−2 10−1 100 101
Time t/τ Time t/τ
FIG. 5. Time-dependent variance Var (t) and local exponent α(t) for fixed force F = 28 and different densities. Solid lines
x
correspond to the full time-dependent solution in first order and symbols represent simulation results. The black dashed lines
correspond to the asymptotic model [Eq. (44)]. Density decreases from left to right.