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arXiv:1003.2918v1 [cond-mat.soft] 15 Mar 2010

EPJ manuscript No.

(will be inserted by the editor)

The Localization Transition of the Two-Dimensional

Lorentz Model

Teresa Bauer1, Felix H¨ ofling2,3, Tobias Munk1, Erwin Frey1, and Thomas Franosch4,1

1Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS),

Fakult¨ at f¨ ur Physik, Ludwig-Maximilians-Universit¨ at M¨ unchen, Theresienstraße 37, 80333 M¨ unchen,

Germany

2Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, England, United

Kingdom

3Max Planck Institute for Metals Research, Heisenbergstraße 3, 70569 Stuttgart, Germany

4Institut f¨ ur Theoretische Physik, Universit¨ at Erlangen-N¨ urnberg, Staudtstraße 7, 91058 Erlangen,

Germany

Abstract. We investigate the dynamics of a single tracer particle performing

Brownian motion in a two-dimensional course of randomly distributed hard ob-

stacles. At a certain critical obstacle density, the motion of the tracer becomes

anomalous over many decades in time, which is rationalized in terms of an under-

lying percolation transition of the void space. In the vicinity of this critical density

the dynamics follows the anomalous one up to a crossover time scale where the

motion becomes either diffusive or localized. We analyze the scaling behavior

of the time-dependent diffusion coefficient D(t) including corrections to scaling.

Away from the critical density, D(t) exhibits universal hydrodynamic long-time

tails both in the diffusive as well as in the localized phase.

1 Introduction

Understanding transport in heterogeneous media is fundamental for a variety of applications

ranging from material sciences, porous catalysts, oil recovery, and even biological system. Often,

the motion of particles inside such materials is strongly hindered due to the presence of slowly

rearranging or immobilized obstacles of many different length scales. Many heterogeneous ma-

terials, e.g. rocks, soils, cements, foams and ceramics, consist of solid frames permeated by a

network of pores [1, 2], and a mobile agent can meander through this static course of obstacles

and display long-range transport. Likewise, transport in densely packed systems is strongly ob-

structed by the presence of surrounding particles via their excluded volume effect. In many cases

a separation of time scales naturally occurs, for example in strongly heterogeneous mixtures

such as sodium ions in silicates [3–5] or size-disparate soft or Yukawa spheres [6–8], leading to

a much slower diffusion of one component. Similarly, the dense packing of differently sized pro-

teins, lipids and sugars in the cell cytoplasm leads to strongly suppressed transport known as

molecular crowding [9–12]. Again the motion of a smaller sized molecule is much faster than of

surrounding macromolecules and the small molecule explores a quasi-static array of obstacles.

Molecular crowding is also relevant in quasi two-dimensional systems such as protein diffu-

sion in lipid bilayers as studied by single molecule fluorescence microscopy [13] or fluorescence

correlation spectroscopy [14–16].

The motion of a tracer in these materials often displays anomalous transport as manifested

in a subdiffusive increase of the mean-square displacement (MSD). This behavior is displayed

in a finite window of time and a crossover to ordinary diffusion occurs at sufficiently long times.

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The exponent characterizing the subdiffusive behavior often appears to depend on the details

of the system and even changes as the experimental parameters are varied.

The Lorentz model consisting of a single tracer exploring a course of randomly distributed

frozen obstacles constitutes a minimal model for transport in heterogeneous media [17]. In its

simplest variant the obstacles are spheres or disks and the positions are Poisson distributed.

Hence, they may overlap and form clusters which restrict the motion of a pointlike tracer

particle to the remaining void space. Above a certain obstacle density, the void space no longer

permeates the entire system and a percolation transition occurs [18]. It has been shown that

this transition is accompanied with a divergence of a characteristic length scale ξ, known as

correlation length. The dynamics close to the transition displays critical behavior and the mean-

square displacement exhibits behavior similar to experimental observations for heterogeneous

media. It has been shown that the Lorentz model generically leads to large crossover windows

explaining the apparent drift in characteristic exponents [19–21]. Recently, the two-dimensional

Lorentz model has been introduced in the context of lateral diffusion of proteins in the plasma

membrane [22–24].

Systematic studies on the two-dimensional Lorentz model were mostly restricted to low

densities focusing on the algebraic decay of the velocity autocorrelation functions (VACF) and

the non-analytic dependence of the diffusion coefficient on the obstacle density [25–30]. The

recurrent collisions with the obstacles lead to infinite memory resulting in a negative long-time

tail ∼ −t−2of the VACF [17, 31, 32]; yet close to the transition, the critical behavior shifts

the onset of the hydrodynamic tail to longer and longer times [33]. The critical behavior of

the Lorentz model in two-dimensions is expected to be qualitatively different from the three-

dimensional case since in the latter the conductances through narrow channels determines the

dynamic exponent from pure geometric reasons [34, 35]. In the former the narrow gaps are

expected to be less relevant and the universality of transport on a percolating lattice should

be recovered. Beyond universality one would like to know the range of validity of the universal

behavior, the size of the crossover region, and the importance of corrections to scaling.

In this work we present simulation results for the two-dimensional Lorentz model for Brown-

ian tracer particles, in particular for densities close to the percolation transition. We have mea-

sured the mean-square displacement, the time-dependent diffusion coefficient, and the VACF,

and analyze their respective critical behavior. Then we compare the subdiffusive behavior as

well as the diffusion coefficient with the predicted power-law behavior. The non-algebraic de-

cay of the VACF at long times emerges also for the case of a Brownian tracer corroborating

the notion that the frozen configuration space alone gives rise to persistent correlations in the

dynamics. A scaling theory that includes the leading corrections to scaling is developed and

tested against the simulation data by suitable rectification plots.

2 Lorentz Model

The Lorentz model constitutes the minimal model for particle transport through a disordered

material. In its simplest variant, a single classical tracer particle traverses a d-dimensional array

of frozen hard-core obstacles of density n. Each obstacle acts as a scattering center of radius σ

restricting the motion of the tracer to the void space. For independently distributed scatterers

the only control parameter characterizingthe structure is then the dimensionless number density

n∗= nσd. Equivalently, one may employ the porosity ϕ, i.e., the volume fraction accessible to

the tracer due to the possibly overlapping obstacles. In the planar problem (d = 2) which we

address in this work, one easily calculates

ϕ = exp(−πnσ2). (1)

Already at intermediate obstacle density, the void space decomposes into many pockets of

different sizes, and long-range particle transport occurs only through the void space that is

percolating through the entire system. At a certain obstacle density n∗

component ceases to exist and all particles are trapped in finite pockets. The problem of con-

tinuum percolation constitutes a critical phenomenon of purely geometric origin [18], and a

c≈ 0.359, the infinite

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series of predictions has been made for the characteristic behavior in the close vicinity of the

critical density n∗

c. The linear dimension of the largest finite cluster (of the void space) defines

the correlation length ξ, which is expected to diverge as ξ ∼ |n∗−n∗

is approached. Below the length scale ξ the geometric structures appear, in a statistical sense,

indistinguishable to the ones at n∗

cforming the basis for the notion of self-similarity. Simulta-

neously, the root-mean-square size ℓ of all finite clusters diverges, yet with a smaller exponent

ℓ ∼ |nc− n∗

infinite component as ncis increased towards the percolation threshold, P∞∼ (n∗

respect to geometric properties, continuum percolation shares the same universality class as

lattice percolation, and in two dimensions the exact values of the exponents are known from a

mapping to the Baxter line of the eight-vertex model ν = 4/3,β = 5/36 [35–37].

Transport of a single particle is expected to become anomalous and universal close to the

percolation threshold independent of the details of the dynamics at the microscale. Here we

consider a particle undergoing Brownian motion confined to the void space. Then the short-

time diffusion coefficient D0 fixes the microscopic time scale t0 := σ2/D0, i.e., the typical

time needed for the particle to diffuse the distance of one obstacle radius without obstruction.

The simplest quantity characterizing the motion of the tracer is the mean-square displacement

δr2(t) = ?[R(t) − R(0)]2?, where the brackets indicate averaging both over all initial positions

of the particle as well as different realizations of the disorder. In particular, particles that

are initially in a finite pocket will remain there forever and do not contribute to long-range

transport.

Directly at the percolation threshold (n∗= n∗

ics of a walker becomes subdiffusive δr2(t) ∼ t2/zfor long times t ≫ t0. The dynamic exponent

z is independent of the geometric exponents of the percolation problem, but is determined from

the universality class of random resistor networks. For obstacle densities above n∗

are trapped in finite clusters and correspondingly the mean-square displacement is expected to

saturate at the mean-square cluster size δr2(t → ∞) = ℓ2. However, close to the transition the

subdiffusive behavior should be visible in a finite time window t0≪ t ≪ txwhere txdenotes

the crossover time to localization. These arguments suggest that for small separation parameter

ǫ := (n∗− n∗

cthe mean-square displacement should obey the scaling law

c|−νas the critical density

c|−ν+β/2. The same exponent β governs the vanishing of the relative weight of the

c−nc)β. With

c) the void space is self-similar and the dynam-

c, all particles

c)/n∗

δr2(t;ǫ) ≃ t2/zδˆ r2

±(ˆt),

ˆt = t/tx

(2)

for ǫ↓0 and t ≫ t0and a scaling function δˆ r2

the diffusive side (ǫ ↑ 0) with a corresponding scaling function δˆ r2

time scale tx. To describe the crossover from critical dynamics to localization/diffusion, the

scaling functions should exhibit the following asymptotics: δˆ r2

∞) ∼ˆt−2/z, δˆ r2

localized side one infers for the crossover scaling time tx∼ ℓz∼ |ǫ|z(−ν+β/2). Interestingly, the

relevant length scale that determines the divergence of the inherited time is given by the mean

cluster size ℓ rather than the correlation length ξ. Since tx also marks the crossover to the

diffusive regime for ǫ < 0, one immediately concludes that the long-time diffusion coefficient

should vanish as D ∼ (−ǫ)µfor ǫ ↑ 0 with the conductivity exponent µ = (z − 2)(ν − β/2).

We use the value determined by Grassberger [38] in high-precision computer simulations for

the electrical conductivity of a percolating lattice, µ = 1.310 ± 0.001, as reference value and

calculate the anomalous dimension to z = 3.036.

+(·). We anticipate that scaling is also obeyed on

−(·) and the same crossover

±(ˆt → 0) = const. and δˆ r2

+(ˆt →

−(ˆt → ∞) ∼ˆt1−2/z, respectively. From the known long-time behavior on the

3 Simulation Results

We have performed Brownian dynamics simulations for a single particle moving in a fixed array

of hard disks of radius σ. The obstacles are distributed independently with a fixed average

density n. We have employed periodic boundary conditions for system sizes of L/σ = 10,000

and generated Brownian trajectories for very long times following an algorithm discussed in

[39] which was also employed recently for the three-dimensional Lorentz model [20]. In essence,

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100

102

104

106

10-2

100

102

104

106

108

0.01

0.05

0.1

0.2

0.3

0.33

0.34

0.35

0.355

0.359

0.37

0.38

0.40

0.41

10-2

100

102

104

106

108

10-1

100

t/t0

δr2(t)/σ2

D(t)/D0

t/t0

∼ t2/z

Density n∗

Fig. 1. Mean-square displacement

δr2(t) of the Lorentz model for Brow-

nian particles. The obstacle density

n∗increases from top to bottom; the

thick black line indicates the long-

time asymptote for anomalous trans-

port at criticality δr2(t) ∼ t2/zwith

z = 3.036. The inset displays the time-

dependent diffusion coefficient D(t) at

obstacle density n∗= 0.30 for varying

algorithmic parameter τB/t0 = 0.25,

0.015625, 0.0025, and 0.0005625 .

particles are propagated along a deterministic, straight trajectory with specular scattering every

time the tracer hits an obstacle, yet this dynamics is interrupted at regular time intervals τB,

where new velocities are drawn from a Maxwell distribution of variance v2. Then on time scales

longer compared to τB and length scales larger than vτB, a free particle undergoes Brownian

motion with diffusion coefficient D0= v2τB/4. In the presence of the obstacles the particle can

still be considered as Brownian walker at the microscale with short-time diffusion coefficient

D0, provided τB is small relative to the inverse collision rate τc= 1/2πnσv. We shall use the

characteristic time t0 = σ2/D0 as our basic unit of time, i.e., the time a free particle needs

to traverse an obstacles radius. The algorithmic condition to mimic Brownian dynamics at

the microscale is thus given by τB/t0 ≪ 1/(2n∗)2. We have verified that τB/t0 = 0.0025 is

sufficiently small in order that the long-time behavior is independent of the microparameters

v2and τBas exemplified in the inset of Fig. 1 for a moderate obstacle density n∗= 0.3. There

the time-dependent diffusion coefficient is displayed for different τB and the curves neatly

superimpose for τB < 0.015625t0. For the production runs we have fixed τB = 0.0025t0and

have calculated mean-square displacements as running-time averages over several trajectories

for at least 100 different realizations of the disorder resulting in more than 775 trajectories in

total for each density n∗to generate high-accuracy data. The trajectories extend over huge

time windows of typically 2.5 × 106t0, yet close to the percolation threshold and for densities

n∗≤ 0.15 15 times longer trajectories have used. With current computing resources, a single

trajectory at the critical density and for the longest simulation times runs approximately 40

hours on one core of a Quad Core Intel(R) Xeon(R) CPU X5365 (3.00GHz).

Results for the mean-square displacement δr2(t) for all obstacle densities are exhibited in

Fig. 1 on double logarithmic scales. First one should notice, that the data display almost no noise

even for the longest times. For short times, all data start from the short-time diffusive motion,

δr2(t) = 4D0t, and only at times t ≃ t0the presence of the obstacles suppresses the motion and

the curves fan out. For low obstacle densities, the long-time behavior is again diffusive yet with

a suppressed diffusion coeffficient D(n∗). On the other hand, the mean-square displacements

for high n∗saturate at long times, reflecting the fact that all particles are localized.

The localized and diffusive curves are nicely discriminated by a critical density n∗

where the MSD behaves subdiffusively over at least six decades in time, i.e., it extends to

our longest observation times. This critical density coincides with the numerical estimate

n∗

c= 0.359072(4) for continuum percolation [40, 41]. The value z = 3.036, as inferred from

the exponent µ determined by finite-size scaling of the conductivity at the critical point [38]

provides an excellent description of the long-time behavior of the critical MSD δr2(t) ∼ t2/z.

Our simulations provide the first quantitative test that the 2d Lorentz model shares the uni-

versality class of two-dimensional random resistor networks.

c= 0.359

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106

108

0.01

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0.355

0.359

0.37

0.38

0.40

0.41

D(t)/D0

Density n∗

Time t/t0= tD0/σ2

∼ t2/z−1

∼ t−2

Fig. 2. The time-dependent diffusion

coefficient D(t) := (1/4)dδr2(t)/dt.

The obstacle density n∗increases from

top to bottom; the arrows indicate

the long-time diffusion coefficient D.

For the critical density, the time-

dependent diffusion coefficient van-

ishes as a power law, D(t) ∼ t2/z−1.

The thick line indicates the power law

t−2expected as long-time asymptote

in the localized regime due to the cul-

de-sacs.

For densities close to n∗

cross over to either diffusive or localized behavior. By naked eye one infers already that this

crossover time increases as the critical density is gradually approached. Let us mention that in

the three-dimensional Lorentz model, the curves off the critical point deviate much more from

the critical one, than in the planar Lorentz model. Nevertheless they still display subdiffusion

in a finite time window, yet with apparent density-dependent exponents [19, 20].

c, the data follow the critical one up to some finite time where they

3.1 Time-dependent diffusion coefficients

A quantity more sensitive to the anomalous transport behavior is given in terms of the time-

dependent diffusion coefficient

1

2d

where the dimension is d = 2 for the planar problem. We have taken numerical derivatives

of the MSDs taking advantage of the fact that δr2(t) varies significantly only on logarithmic

scales. Since the MSDs are calculated using our standard blocking scheme [42], the numerical

derivatives essentially do not introduce new noise to the data. The time-dependent diffusion

coefficient D(t) is displayed in Fig. 2 for the same densities considered above. First, one notices

that all curves start from the short-time diffusion constant D0, corroborating that our numerical

algorithm reproduces Brownian motion at small time and length scales. For increasing time D(t)

is gradually suppressed reflecting that obstacles can only slow down the overdamped dynamics.

For densities below n∗

cthe time-dependent diffusion coefficient approaches a nonzero limit D

for long-times. The values of the long-time diffusion constant D decrease rapidly as the critical

density is approached from below. Directly at the critical point, D(t) reaches a power-law

long-time asymptote D(t) ∼ t2/z−1corresponding to a subdiffusive mean-square displacement.

For densities above the critical one, the time-dependent diffusion coefficient vanishes even more

rapidly. Following the argument of persistent correlations due to power-lawdistributed exit rates

of the cul-de-sacs, one should expect a universal long-time tail D(t) ∼ t−2in the entire localized

phase [34]. Such a behavior has indeed been observed recently in molecular dynamics simulations

for the two-dimensional Lorentz model for ballistic particles [33], though they considered the

velocity autocorrelation function rather than D(t). Our simulations exhibit clear evidence that

this tail remains present for Brownian particles too as we shall argue below.

The diffusion constants D extracted as long-time limits of D(t) are displayed in Fig. 3

for varying obstacle density. Over the investigated range of densities, the diffusion constant

is suppressed by a factor of 100. It vanishes as the critical density is approached and follows

amazingly well the scaling prediction D ∼ (−ǫ)µ. Even for the lowest density considered, where

the motion is practically unobstructed by the obstacles, D(n∗= 0.01) = 0.97D0deviates by only

D(t) :=

d

dtδr2(t),(3)

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10-2

10-1

100

-10-1

-100

0

0 0.10.20.30.4

Diffusion constant D/D0

ε = (n∗−n∗

c)/n∗

c

Density n∗

D ∼ |ε|µ

D1/µ

n∗

c

Fig. 3. Scaling behavior of the long-

time diffusion coefficient D with sep-

aration parameter ǫ = (n∗− n∗

The straight line in the rectification

plot (inset) confirms the value of the

conductivity exponent µ = 1.309.

c)/n∗

c.

8.4% from the scaling asymptote. It appears as a coincidence that the critical regime connects

down to the low-density asymptote without an intermediate region of moderate obstructed

motion. For the corresponding three-dimensional system the convergence towards the scaling

behavior is approached slowlier, however, since the corresponding conductivity exponent is much

higher µ3d= 2.88, the diffusion vanishes much more rapidly and a suppression by five orders

of magnitude can be observed [20]. The rectification plot in the inset of Fig. 3 corroborates

that µ = 1.310 obtained by measuring the conductivity on a lattice close to percolation is

indeed the correct value. Our simulations for Brownian particles provide an independent test

that two-dimensional random resistor networks and the planar Lorentz model indeed share the

same universality class. The critical density has been determined by extrapolating the straight

line in the rectification plot to zero diffusivity, yielding n∗

value of n∗

cwas used throughout this work to simulate the critical dynamics.

c= 0.359±0.001. The thus determined

3.2 Velocity autocorrelation functions

Let us also discuss the velocity autocorrelation function (VACF), Z(t) = ?v(t) · v(0)?/d, for

the Brownian particle. Although the notion of velocity for Brownian particles is conceptually

questionable, their correlation function is well defined for times t > 0. Here we rely again on

numerical derivatives, i.e., we employ

Z(t) :=

1

2d

d2

dt2δr2(t) (4)

as definition. Then the relation to the time-dependent diffusion coefficient is provided by

D(t) = D0+

?t

0+Z(t′)dt′, (5)

where the integral is evaluated excluding the time t = 0. This form constitutes the analog of

the Green-Kubo relation, alternatively one can include a δ-distribution in the VACF to account

for the Brownian motion at the microscale.

Figure 4 displays the VACF, and one first observes that it is negative for all times, except

on time scales associated with our algorithmic microparameter τB. This fact is consistent with

the notion that obstacles can only slow down the diffusion, Eq. (5), and in the case of Brownian

motion one can show that the VACF is a completely monotone function, see Appendix A. The

long-time behavior for the diffusive regime (n < nc) is characterized by persistent correlations

that slowly decay as a power-law. The low densities display a tail Z(t) ∼ −t−2consistent

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10-6

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100

10-2

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100

101

Time t/t0

102

103

104

105

0.359

0.3

0.25

0.2

0.15

0.1

0.05

0.01

10-2

100

102

100

102

104

Time t/t0

−Z(t)(t0/σ)2

−(Z(t)/σ2)t2/n∗

Density n∗

∼ t2/z−2

∼ t−2

Fig. 4. Velocity autocorrelation func-

tion Z(t)= (1/4)d2δr2(t)/dt2for

Brownianparticles

Lorentz model. The thick lines indi-

cate the hydrodynamic tail t−2and

the critical behavior t2/z−2, respec-

tively. The inset displays a rectifica-

tion plot −t2Z(t)/σ2/n∗as a function

of time.

in theplanar

100

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102

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106

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0.355

10-810-610-410-2100102104106

t/tx

100

101

102

D(t)/ADt2/z−1(1+CDt−y)

t/tx∼ tε(2ν−β+µ)

D(t)/ADt2/z−1

Density n∗

Fig. 5. Scaling behavior of the time-

dependent diffusion coefficient for the

densities below n∗

able reads tx = t0|ǫ|β−2ν−µThe in-

set includes the leading correction to

scaling with a correction amplitude

CD = −0.14ty

critical density.

c. The scaling vari-

0consistent with the

with the theoretical prediction for kinetic theory for ballistic particles [32]. It has been antici-

pated earlier [17] that also Brownian particles exhibit the same behavior, since the long-time

correlations originate from repeated encounters of the same frozen heterogeneities. Indeed the

Lorentz model for Brownian tracers can be solved analytically to lowest order in the scattering

density n∗and the time-dependence of the VACF including its long-time tail can be worked

out exactly [43]. Nevertheless, to the best of our knowledge, our simulation results provide the

first direct evidence for this universality at all densities. As the density is gradually increased,

the overall signal in the VACF becomes larger and the exponent of the power-law tail appears

to drift. A rectification plot shows that the t−2behavior is assumed for all densities as the

late-time relaxation. The critical asymptote appears in an intermediate window which extends

to longer and longer times as the critical density is approached. Our data also show that the

amplitude of the hydrodynamic tail diverges close to n∗

same arguments as in the ballistic case [33].

cwhich can be rationalized using the

4 Dynamic scaling analysis

The power-law behavior in both the mean-square displacement or the time-dependent diffusion

coefficient at criticality and the vanishing of the diffusion constant as a power law upon ap-

proaching the critical density is merely one aspect of the critical behavior. Yet, the universality

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0.8

1

10-2

100

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106

10-2

10-1

100

10-2

100

102

D(t)/ADt2/z−1

1−D(t)/ADt2/z−1

t/t0

t/t0

∼ t−y

Fig. 6. Rectification of the time-

dependent diffusion coefficient D(t)

at the critical density n∗

cal amplitude is determined to AD =

0.508D0t1−2/z

0

. The inset displays the

approach of the MSD towards the crit-

ical law. The thick line indicates a

power laws with the universal correc-

tion exponent y = 0.49.

c. The criti-

hypothesis suggests a much more sensitive test in terms of scaling. For example, the mean-

square displacement δr2(t;ǫ) for small separation parameters |ǫ| ≪ 1 and long times t ≫ t0is

expected to fulfill Eq. (2). Here we show that the time-dependent diffusion coefficient D(t;ǫ)

can be used equivalently to test the scaling prediction. Taking derivatives, we suggest

D(t;ǫ) ≃ t2/z−1ˆD±(ˆt),

ˆt = t/tx

(6)

with the scaling time tx := t0|ǫ|−(2ν−β+µ). The connection with the scaling function for the

mean-square displacement is easily established,

ˆD±(ˆt) =

1

2d

?2

zδˆ r2

±(ˆt) +ˆtd

dˆtδˆ r2(ˆt)

?

.(7)

For short rescaled times,ˆD±(ˆt → 0) = const. =: ADand the critical behavior is recovered. For

long timesˆD−(ˆt) ∼ˆt1−2/zsuch that diffusion is reached for ǫ < 0. On the localized side, the

mean-square displacements saturate and the leading behavior δr2

Eq. 7, thusˆD+(ˆt) = o(ˆt−2/z).

Figure 5 displays a rectification plot for D(t) for obstacle densities below the critical ones.

For large rescaled times the curves nicely superimpose, though this reflects merely the fact that

the diffusion regime is reached for all cases and that the long-time diffusion coefficient obeys the

scaling prediction D ∼ (−ǫ)µ. For short rescaled times the curves converge to a constant which

is given by the long-time behavior of the time-dependent diffusion coefficient at the critical

point. The fanning out of the curves arises due to corrections to scaling and eventually because

of the crossover to the microscopic regime.

To gain further insight into the scaling behavior we extend our scaling hypothesis by a

generic irrelevant scaling variable. Then it has been shown recently within a cluster-resolved

scaling theory [44] that the mean-square displacement should obey

+(ˆt) ∼ˆt−2/zcancels exactly in

δr2(t;ǫ) = t2/zδˆ r2

±(ˆt)?1 + t−y∆±(ˆt)?,(8)

where y is another universal exponent characterizing the approach of the critical dynamical

behavior. It is connected to a correction-to-scaling exponent Ω for the cluster-size distribution

via the scaling relation y = Ω(νd − β)/[z(ν − β/2)]. For the two-dimensional case the value

y = 0.49(3) [44] was determined for random walks on a lattice, which we shall use in the

following. Taking derivatives with respect to time, the corresponding prediction for the time-

dependent diffusion coefficient is

D(t;ǫ) ≃ t2/z−1ˆD±(ˆt)?1 + t−y∆D

±(ˆt)?,(9)

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where the new scaling function ∆D

via

±(ˆt) is connected to the one of the mean-square displacement

2dˆD±(ˆt)∆D

±(ˆt) =

?2

+ˆtd

z− y

?

δˆ r2

±(ˆt)∆±(ˆt)

dˆt

?δˆ r2

±(ˆt)∆±(ˆt)?.(10)

For small rescaled times the correction-to-scaling function reduces to a constant ∆D

const. =: CD. and one easily infers the relation to the correction-to-scaling constant C :=

∆±(ˆt → 0) for the mean-square displacement: CD= C(2 − yz)/2. In particular at criticality,

the time-dependent diffusion coefficient displays a power-law correction for long times

±(ˆt → 0) =

D(t;ǫ = 0) ≃ ADt2/z−1?1 + CDt−y?. (11)

For long rescaled timesˆt → ∞, the correction to scaling function behaves as a power-law again

∆D

constant

D(ǫ) ∼ (−ǫ)µ?

The time-dependent diffusion coefficient D(t) at the critical density is displayed in Fig. 6 in

a rectification plot. Within the statistical errors of our simulation one observes a saturation at

long times provided the established value for z is used, indicating that the asymptotic behavior

is reached. At very long times the curve starts to deviate again due to statistical fluctuations,

possibly finite-size effects, and the uncertainty of the value for the critical obstacle density.

The amplitude of the critical relaxation AD= limt→∞t1−2/zD(t;ǫ = 0) has been determined

to AD = 0.508D0t1−2/z

0

from our numerical data to optimize data collapse for the scaling in

the diffusive and localized regime, see below. The approach towards this power-law behavior

is consistent with a power law according to Eq. (11) with a correction to scaling amplitude

CD= −0.14ty

It appears that for a deeper analysis of the correction to scaling behavior as suggested by

Eqs. (8) and (9), the precise form of the scaling function ∆D

arguments, we know that the correction function for the MSD, ∆±(ˆt), smoothly interpolates

between a constant ∆±(ˆt → 0) =: C and power laws ∆±(ˆt → ∞) ≃˜C±ˆty. Then the correction

term for the time-dependent diffusion coefficient behaves for large rescaled timesˆt → ∞ as

ˆD(ˆt)∆D

Upon closer inspection of the correction behavior, Eq. (12), and the simulation results, Fig. 3,

the corrections are very small for large rescaled times. Then one may approximate ∆D

its short-time asymptote ∆D

diffusive/localized time regime. The result of this procedure is displayed in the inset of Fig. 5 for

the diffusive side with a quite impressive improvement of the scaling behavior. Let us emphasize

that the only new parameter CDis in principle fixed by the critical behavior, such that no free

parameters enter the scaling plot.

The scaling behavior for the time-dependent diffusion coefficient D(t) is tested for the lo-

calized regime, n∗> n∗

c, in Fig. 7. Again the curves approach a constant at short rescaled times

as the critical density is approached. Accounting for the correction by the same procedure

as for the diffusive side yields an almost perfect data collapse without any new parameters.

The scaling behavior at large times reflects that also the approach of the saturation in the

MSDs is universal. From our general discussion, the scaling function is known to vanish rapidly

ˆD+(ˆt) = o(ˆt−2/z). Following Machta and Moore [34] there should universal power-law correla-

tions in the entire localized phase, D(t) ∼ t−2for t → ∞, due to the meandering of the particle

in the self-similar cul-de-sacs. Assuming that the crossover from the critical law to these univer-

sal hydrodynamic tails is again given by txas we have argued earlier for ballistic particles [33],

leads to a scaling prediction ofˆD+(ˆt) ∼ˆt−2/z−1. In particular, one expects a divergence of the

prefactor of the tail in the localized phase according to t2/z+1

Fig. 7 and provides a nice description of the data.

−(ˆt → ∞) ≃˜CD

±ˆty, yielding corrections for the asymptotic behavior of the long-time diffusion

1 +˜CD

−t−y

0(−ǫ)yµz/(z−2)?

.(12)

0, see inset of Fig. 6.

±(ˆt) has to be known. From general

−(ˆt) ∼ˆty−2/zon the diffusive side andˆD(ˆt)∆D

−(ˆt) = o(ˆty−2/z) in the localized regime.

±(ˆt) by

±(ˆt) = CDfor all rescaled times, i.e., ignore the corrections on the

x

. This prediction is indicated in

Page 10

10 Will be inserted by the editor

10-4

10-3

10-2

10-1

100

10-8

10-6

10-4

10-2

100

102

104

0.37

0.38

0.40

0.41

10-8

10-6

10-4

10-2

t/tx

100

102

10-4

10-3

10-2

10-1

100

101

D(t)/ADt2/z−1(1+CDt−y)

t/tx∼ tε(2ν−β+µ)

D(t)/ADt2/z−1

Density n∗

∼ t−2/z−1

Fig. 7. Scaling behavior of the time-

dependent diffusion coefficient for the

densities above n∗

time tx

= t0|ǫ|β−2ν−µ. The thick

line indicates the power law t−2/z−1

which is expected to hold in the en-

tire localized phase. The inset includes

the leading correction to scaling with

the same correction amplitude CD =

−0.14ty

cwith the crossover

0as in the diffusive regime.

5 Conclusion

The dynamics of a tracer particle in a densely packed planar course of obstacles has been

investigated by Brownian dynamics simulation. The slowing down of the dynamics close to the

percolation threshold is accompanied by critical behavior observed over more that 6 decades in

time. We corroborate that the planar Lorentz model shares a universality class with the random

resistor network where the critical exponents are known from earlier simulations. We have

shown that the time-dependent diffusion coefficient constitutes a suitable quantity to analyze

the scaling behavior close to the transition. The corresponding scaling relations have been

derived and extended by the leading correction. We find that scaling behavior is in general well

obeyed and the corrections to scaling appear much less important than for the three-dimensional

case [20].

The Lorentz model exhibits power-law long-time anomalies away from the critical density

due to repeated encounters with the same scatterer. These tails have been derived originally for

ballistic particles, yet they turn out to be universal irrespective of the dynamics at microscopic

scales. Then the velocity autocorrelation, defined via a second derivative of the mean-square

displacement exhibits the tails even for Brownian tracers. On the localized side we also find

long-time tails due to the self-similar distribution of exit times of the cul-de-sacs [34], again

irrespective of the microscopic dynamics. Interestingly, these long-time tails are part of the

scaling function for the time-dependent diffusion coefficient.

The assumption that the obstacles are distributed independently is certainly an oversimpli-

fication in real systems. Then one would like to extend the Lorentz model where the matrix

consists of some frozen-in configuration of a strongly interacting liquid or a snapshot of a slowly

rearranging matrix of obstacles. Second, experiments are usually for a finite concentration of

particles meandering in the array of obstacles and one may ask at what time and length scales

these interaction of the tracers modifies the dynamics in the labyrinth. In three dimensions an

intriguing interplay of the physics of the glassy dynamics and the localization transition has

been discovered recently [3–8], and since the glass transition in two-dimensions is qualitatively

similar [45, 46] one may hope that the physics of the planar Lorentz model is applicable in

size-disparate two-dimensional mixtures.

Financial support from the Deutsche Forschungsgemeinschaft via contract No. FR 850/6-1 and from

the Konrad-Adenauer-Stiftung (T.B.) is gratefully acknowledged. This project is supported by the

German Excellence Initiative via the program “Nanosystems Initiative Munich (NIM).”

Page 11

Will be inserted by the editor11

A Appendix: Completely monotone functions

In this Appendix we develop a spectral representation for the mean-square displacement and

the velocity-autocorrelation function for arbitrary dimension d.

For a Brownian particle in an external potential U(r) the time-evolution of the conditional

probability distribution Ψ(r,t) to find the particle at r at time t provided it has been at r′and

some earlier time t′is governed by the Smoluchowski equation

∂tΨ(r,t|r′t′) =

∂

∂r

?D0

kBT

∂U

∂rΨ

?

+ D0∂2Ψ

∂r2≡ˆΩ(r)Ψ ,(13)

whereˆΩ(r) denotes the Smoluchowski operator acting on the position r. At the very end, we

are interested in hard potentials with infinite barriers, however we anticipate that this case is

assumed as limit of smooth potentials becoming increasingly steep.

One can also consider the evolution of Ψ with respect to the conditional time and one can

show that

− ∂t′Ψ(r,t|r′t′) = −D0

kBT∂r′

where the adjointˆΩ+(r′) is with respect to the standard scalar product. Furthermore,ˆΩ+(r′)

is identified with the backward Smoluchowski operator and now acts on r′.

The mean-square displacement is obtained as an average

∂U

∂Ψ

∂r′+ D0∂2Ψ

∂r′2≡ˆΩ+(r′)Ψ ,(14)

δr2(t − t′) ≡

?

drdr′(r − r′)2Ψ(rt|r′t′)Ψeq(r′),(15)

where Ψeq(r) = Z−1exp(−U(r)/kBT) denotes the equilibrium distribution. In principle one

may also introduce a disorder average for different realizations of the potential U(r), yet we

anticipate that for large enough systems the quantities of interest are self-averaging. Since in

equilibrium the MSD is stationary, it depends only on the time difference and one derives

d2

dt2δr2(t − t′) = −d

?

?

?

?

dt

d

dt′δr2(t − t′)

=drdr′(r − r′)2?ˆΩ(r)ˆΩ+(r′)Ψ(rt|r′t′)

drdr′Ψ(rt|r′t′)

?ˆΩ(r′)ˆΩ+(r)(r − r′)2Ψeq(r′)

drdr′Ψ(rt|r′t′)

?

drdr′?ˆΩ+(r)r

?

Ψeq(r′)

=

?

=

ˆΩ+(r′)ˆΩ+(r)(r − r′)2?

?

Ψeq(r′)

= −2·

?ˆΩ+(r′)r′?

Ψ(rt|r′t′)Ψeq(r′),(16)

where in the second to last line the property of the Smoluchowski operatorˆΩ(r)[A(r)Ψeq(r)] =

?ˆΩ+(r)A(r)

result shows that the second derivative of the MSD can be interpreted essentially as the negative

of the autocorrelation function ofˆΩ+(r)r. To make connection with the ballistic case it is helpful

to introduce v = iˆΩ+(r)r as a formal velocity, and one recovers the usual relation to the velocity

autocorrelation function

?v(t) · v(t′)? =1

2

Next we recall that autocorrelation functions C(t) = ?A(t)∗A(0)? for overdamped dynamics

are completely monotone, i.e., their derivatives exhibit fixed sign

?

Ψeq(r) valid for any well-behaved function A(r) has been employed. The preceding

d2

dt2δr2(t − t′).(17)

(−1)ndn

dtnC(t) ≥ 0for all n ∈ N0,t ≥ 0.(18)

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12Will be inserted by the editor

A sketch of a non-rigorous proof is as follows. First, consider the complex scalar product

?A|B? =

?

drA(r)∗B(r)Ψeq(r).(19)

Then one easily verifies thatˆΩ+is hermitian with respect to this scalar product ?A|Ω+B? =

?Ω+A|B?. Then with the formal solution of Eq. (13), Ψ(rt|r′t′) = exp[(t−t′)ˆΩ(r)]δ(r−r′), one

finds a representation of the autocorrelation function as

C(t) = ?eΩ+tA|A?.(20)

Yet sinceˆΩ+is hermitian all eigenvalues are real, and by a ‘variational principle’,

?ˆΩ+A|A? = −D0

?

dr

????

∂A(r)

∂r

????

2

Ψeq(r) ≤ 0,(21)

also negative semi-definite. Zero constitutes the non-degenerate eigenvalue with constant eigen-

function |0?. A spectral decomposition of the backwardsSmoluchowski operatorˆΩ+= −?

in terms of eigenfunctionsˆΩ+|λ? = λ|λ? shows that an autocorrelation function can be repre-

sented as

C(t) =

?

λ

λλ|λ??λ|

|?A|λ?|2exp(−λt), for t > 0.(22)

From this representation one immediately infers that C(t) is completely monotone. By the

famous Bernstein theorem [47] the converse is also true, i.e., any completely monotone function

allows for a representation as a superposition of relaxing exponentials with positive weights.

For the VACF one concludes

?v(t) · v(t′)? = −

?

λ

????λ|ˆ Ω+r?

???

2

exp(−λt).(23)

Since the equilibrium state |0? is annihilated by the backward Smoluchowski operatorˆΩ+|0? =

0, the sum extends in fact only over positive eigenvalues λ > 0.

Integration yields the time dependent diffusion coefficient

D(t) = D0−

?

λ>0

????λ|ˆΩ+r?

???

21 − exp(−λt)

λd

,(24)

and one immediately infers that D(t) is monotonically decreasing to the long-time diffusion

coefficient

D = D0−

?

λ>0

????λ|ˆΩ+r?

???

21

λd.

(25)

For the mean-square displacement one obtains the representation

δr2(t) = 2dD0t − 2

?

λ>0

????λ|ˆΩ+r?

???

2λt − 1 + exp(−λt)

λ2

,(26)

valid for t ≥ 0.

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