Persistent memory for a Brownian walker in a random array of obstacles
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Persistent memory for a Brownian walker in a random array of obstacles
Thomas Franosch a,b , Felix H¨ofling c , Teresa Bauer b , Erwin Frey b a Institut f¨ur Theoretische Physik, Universit¨at Erlangen-N¨urnberg, Staudtstraße 7, 91058 Erlangen, Germany b Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience (CeNS), Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨atM¨unchen, Theresienstraße 37, 80333 M¨unchen, Germany c Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, England, United Kingdom
Abstract
We show that for particles performing Brownian motion in a frozen array of scatterers long-time correlations emerge inthe mean-square displacement. Defining the velocity autocorrelation function (VACF) via the second time-derivativeof the mean-square displacement, power-law tails govern the long-time dynamics similar to the case of ballistic mo-tion. The physical origin of the persistent memory is due to repeated encounters with the same obstacle which occursnaturally in Brownian dynamics without involving other scattering centers. This observation suggests that in this casethe VACF exhibits these anomalies already at first order in the scattering density. Here we provide an analytic solutionfor the dynamics of a tracer for a dilute planar Lorentz gas and compare our results to computer simulations. Ourresult support the idea that quenched disorder provides a generic mechanism for persistent correlations irrespective ofthe microdynamics of the tracer particle.
Keywords:
Brownian motion, disordered solids, computer simulation
PACS:
1. Introduction
The essence of the Brownian motion of mesoscopic par-ticles suspended in a solvent has been understood sincethe pioneering works of Einstein [1] and von Smolu-chowski [2]. Today, Brownian motion constitutes a ba-sic paradigm of stochastic processes [3–5] with appli-cations in numerous fields, which may even utilize thenoise as by stochastic resonance [6] and Brownian mo-tors [7–9], not forgetting important generalizations be-yond classical statistical physics to quantum Browniansystem [10] and relativistic Brownian motion [11].Following Einstein and von Smoluchowski, thestatistics of the particle’s trajectories is described interms of a probability distribution for the displacementas was measured shortly after by Perrin [12] relyingon the newly developed dark field microscope. Thesedisplacements are cumulative, i.e., they are the sumof many small steps, and provided the steps are un-correlated, the central limit theorem applies. Then foran unbiased random walk the probability cloud is de-scribed by a Gaussian which broadens as time proceeds.The corresponding trajectories are continuous but al-most nowhere di ff erentiable curves, self-similar in a sta-tistical sense. This picture obviously cannot apply to the very small time and length scales where collisionswith individual solvent molecules are resolved. Never-theless it seems plausible that on a coarse-grained timeand length scale all correlations due to these molecu-lar processes are rapidly suppressed and Brownian mo-tion emerges as mathematically described by a stochas-tic di ff erential equation by Langevin [13] with a Wienerprocess as noise term.This established picture had to be reconsidered whenAlder and Wainwright [14, 15] showed in molecular dy-namics simulations that the basic assumption of weaklycorrelated displacements is not fulfilled for the singleparticle motion in a fluid. Rather they found a long-timeanomaly Bt − d / , B > d is the spatial dimension of the sys-tem. For a mesoscopic colloidal particle the direct ex-perimental observation of these anomalies `a la Perrinwas achieved only 100 years after Einstein’s work us-ing high-precision photonic force microscopy [16]. Thepower law correlations persist even in the presence of abounding wall where momentum can be transferred to,although they decay more rapidly [17–19].In this paper we argue that quenched disorder consti- Preprint submitted to Chemical Physics September 26, 2018 utes a second generic mechanism for persistent corre-lations in the velocity autocorrelation function. It wasshown in the late 60s that in the Lorentz model, wherea single tracer scatters ballistically in a random array offrozen hard obstacles, ring collisions lead to a nonana-lytic dependence of the di ff usion constant on the den-sity of scatterers [20]. The same mechanism implies apower-law tail in the VACF [21] of the form − At ( d + / , A >
0. First, in contrast to Alder’s discoveries, the ori-gin of this tail is not connected to momentum conser-vation, since the tracer exchanges momentum with thefrozen scatterers. Second, the persistent correlation isnegative reflecting the caging due to the obstacles. Herethe correlations in the velocity are inherited from theconfiguration space since, loosely speaking, the particleremembers forever which paths are blocked by obsta-cles [22]. The predicted anomalies were soon partiallyconfirmed by molecular dynamics simulations [23, 24]yet the prefactor of the tail was by an order of magni-tude larger than expected. At higher densities the relax-ation appears to become slower suggesting a density-dependent exponent [25, 26]. Later it was suggestedthat a second universal power law takes over as the thepercolation threshold is approached [27] in qualitativeagreement with a theoretical prediction obtained in amode-coupling approach [28, 29]. The scenario of acrossover behavior explaining the enhanced prefactor ofthe power law was fully confirmed only recently [30].Since the kinetic theory developed by Weijland andvan Leeuwen [20] is rather involved, one is temptedto assume that their result constitutes a peculiarity ofthe Lorentz model with few general consequences. Inthis paper we show that the long-time anomalies in theVACF persist in a system where particles move di ff u-sively rather than ballistically through the course of ob-stacles. We calculate analytically the first systematiccorrection for the complete scattering function and spe-cialize to low wavenumbers to obtain the VACF. Thelong-time tail emerges again due to repeated encoun-ters with the same scatterer, yet the derivation of this re-sult drastically simplifies compared to the ballistic case,since a di ff usive tracer finds the same obstacle manytimes without requiring the series of Boltzmann scat-tering events as the ballistic particle. We corroborateour analytic result by Brownian dynamics simulationsin the dilute regime and discuss the range of validityof the low-density expansion. Last we conclude thatfrozen disorder generically entails long-time algebraicdecay in the VACF irrespective of the microscopic dy-namics implying that the basic assumption of quicklydecaying correlations in Brownian motion is generallynot fulfilled in heterogeneous media. The notion of universal long-time anomalies also ap-plies to hopping transport in disordered lattice models[31, 32] where again repeated encounters with the samescatterer lead to persistently correlated motion. Therethe VACF has been calculated up to second order in theobstacle density and the predictions have been nicelyconfirmed by computer simulation [33].The paper is organized as follows. In Sec. 2, webriefly recall the main results of the anomalies in thelow-density expansion of the ballistic Lorentz model.For the Brownian particle exploring a course of obsta-cles introduced in Sec. 3, a multiple scattering expan-sion is derived in Sec. 4 revealing that for the first-orderdensity expansion it is su ffi cient to solve the single ob-stacle scattering problem. Section 5 calculates the cor-responding forward scattering amplitude which is usedin Sec. 6 to discuss the intermediate scattering functionand, in particular, the velocity autocorrelation functionin the dilute case. The analytic predictions are testedagainst computer simulation in Sec. 7. A summary ofthe results and conclusions are given in Sec. 8.
2. Ballistic Lorentz model and kinetic theory
In the Lorentz model randomly distributed, possiblyoverlapping obstacles of radius σ represent a random,frozen environment in which a single, point-like tracerparticle moves. The dynamics of the tracer is consid-ered as ballistic with elastic scattering whenever an ob-stacle is encountered. In particular, energy is conservedand the particle’s velocity remains constant in magni-tude for all times. Di ff usion emerges after many col-lisions with the scatterer as the direction of the veloc-ity is randomized. Since the obstacles are distributedrandomly and independently, the structures are charac-terized solely via the obstacle density n . The only di-mensionless control parameter is the reduced obstacledensity n ∗ : = n σ d .To lowest order in the obstacle density the motionof a ballistic particle in the course of obstacles is de-scribed by the Lorentz-Boltzmann equation [34]. Thereuncorrelated scattering events lead to di ff usion on scaleslarger than the mean-free path ℓ ∼ σ/ n ∗ . In this geo-metric problem the time scale is set by the mean col-lision time τ = ℓ/ v by means of the velocity v ofthe particle. The corresponding di ff usion coe ffi cient D ∼ v ℓ ∼ v σ/ n ∗ diverges for small densities since itis due to the rare scattering events only that the ballis-tic motion becomes di ff usive after all. Via a series ofthese uncorrelated scattering events the particle may re-turn to an already visited obstacle, giving rise to non-trivial correlations. To account for these repeated col-2ision events, one has to go systematically beyond theLorentz-Boltzmann equation resulting in a non-analyticcorrection to the di ff usion coe ffi cient D . In two dimen-sions, d =
2, to which we restrict the discussion in thefollowing, one obtains [20, 23], D D = − n ∗ n ∗ − . n ∗ + . n ∗ ln n ∗ ) + . . . (1)where the Lorentz-Boltzmann di ff usion coe ffi cient isgiven by D = v σ/ n ∗ . Uncorrelated scatter-ing events predict an exponential decay in the veloc-ity autocorrelation function Z ( t ) : = h v ( t ) · v (0) i / = ( v /
2) exp( − t / τ ) with the mean collision rate τ − = n ∗ v /σ . The repeated scattering from the same obstaclethen introduces an algebraic tail Z ( t ) ≃ − σ π n ∗ t , for t → ∞ , n ∗ → . (2)reflecting the infinite memory of the motion in the disor-dered system. The amplitude diverges as the system be-comes more and more dilute, yet since the initial valueof the velocity autocorrelation function is density inde-pendent, Z ( t = = v , the time required to attain thetail is set by the growing collision time τ . Provided timeis measured in terms of τ , the tail is a small O ( n ∗ ) cor-rection to the Lorentz-Boltzmann theory.In summary, in the framework of the ballistic Lorentzmodel the non-analytic dependence of D on n ∗ and thelong-time anomaly emerge in a higher order correctionbeyond the Lorentz-Boltzmann theory.
3. Brownian particle in a disordered array
The Lorentz model can be extended to particles thatperform Brownian motion in the void space, i.e the do-main not excluded by the frozen scatterers. In the planarcase, this might be a useful minimal model for the com-plex transport found in cellular membranes or modelsthereof [35–38]. The configuration of the environmentis described by the centers of the obstacles x , . . . , x N in a finite hypercubic box of length L . The density ofthe scatterers is kept fixed, n = N / L d , as the thermody-namic limit is performed. Due to exclusion, the distanceof the tracer to any of the obstacles always exceeds theradius of the scatterers, | r − x i | ≥ σ , i = , . . . , N . Pe-riodic boundary conditions are assumed throughout forconvenience.A complete statistical description is given in terms ofthe conditional probability density P ( r , t | R t ′ ) to find theparticle at time t at position r provided it was knownto be at R at an earlier time t ′ ≤ t . Since the stochastic process is stationary, the conditional probability is time-translationally invariant and we choose t ′ = P ( r , t | R
0) isthen simply the di ff usion equation restricted to the voidspace ∂ t P ( r , t | R = D ∇ P ( r , t | R
0) (3)where the nabla operator ∇ acts on the current positionof the particle r and now D refers to the short-time dif-fusion coe ffi cient. Since the tracer particle cannot pen-etrate the obstacles, this equation of motion has to besupplemented by the von Neumann boundary condition( r − x i ) · ∇ P = , for | r − x i | = σ, i = , . . . , N (4)i.e., the flux through the boundary vanishes.To avoid di ffi culties with hard core repulsion, we con-sider for the moment a random potential consisting offinite spherical barriers, U ( r ) = N X i = u ( r − x i ) (5)where, e.g., u ( r ) = U ϑ ( σ − | r | ) with the Heaviside stepfunction ϑ ( · ) , and the hard core limit U → ∞ is antic-ipated. Then the propagator obeys the Smoluchowskiequation ∂ t P = D k B T ∇ · ( P ∇ U ) + D ∇ P (6)where k B T is the thermal energy.Analytic progress is made by studying the one-sidedtemporal Fourier transform G ( ω ; r , R ) = Z ∞ e i ω t P ( r t | R t (7)for complex frequencies ω in the upper half-plane,rather than studying the time dependence of the prop-agator P ( r t | R
0) directly. Then the equation of motiontranslates to( − i ω − D ∇ ) G − D k B T ∇ · ( G ∇ U ) = δ ( r − R ) . (8)This form of the Smoluchowski equation identifies G asan inverse of the Smoluchowski operator and constitutesthe starting point for the elaborated framework of thescattering theory.
4. Formal scattering theory
The Smoluchowski equation in the form of Eq. (8)has a mathematical analogy to the time-independent3chr¨odinger equation, allowing us to employ the tech-niques developed for the scattering problem of a quan-tum particle. First it is convenient to adopt an ab-stract bra-ket notation. The Hilbert space is spanned bythe (generalized) ket states | r i normalized by h r | r ′ i = δ ( r − r ′ ). Then we introduce an operator G ( ω ) witha positional representation h r | G ( ω ) | R i = G ( ω ; r , R ).The dependence on the complex frequency ω will besuppressed in the following. Similarly, we introducethe unperturbed Smoluchowski operator Ω via its ma-trix elements h r | Ω | ψ i = D ∇ h r | ψ i and perturbation V by h r | V | ψ i = ( D / k B T ) ∇ · ( h r | ψ i ∇ U ). The Smolu-chowski equation (8) is equivalent to the following op-erator equation ( − i ω − Ω ) G = I , (9)where Ω = Ω + V and I is the identity operator. Theproblem is to evaluate the full Green function G in thepresence of the scatterers. The case of no obstaclesleads to the unperturbed Green function G which satis-fies ( − i ω − Ω ) G = I . (10)Simple operator algebra reveals that the full Green func-tion obeys the Lippmann-Schwinger equation G = G + G VG . (11)which yields upon iteration the Born series with the firstorder approximation G = G + G VG . The scatteringmatrix defined by S : = ( I − G V ) − advances solutionsof the unperturbed problem to the full one G = SG .It is convenient to single out the event of no scatteringand to define the T-matrix as G T = S − I . Then thefollowing identities are easily derived G = G + G TG , (12) T = V + VG T . (13)The first relation states that the T-matrix acts like an ef-fective potential such that the full multi-scattering pro-cess is obtained in first Born approximation. The secondequation states how the T-matrix is obtained by itera-tion, T = V + VG V + VG VG V + . . . , (14)implying that the multiple scattering events are catego-rized as single-, double-, . . . scattering events.In our case the scattering potential V = P i v ( i ) is asum of single obstacle potentials di ff ering merely in theposition of the scatterer, and thus a multi-scattering ex-pansion is appropriate. Let us define the correspondingsingle obstacle t-matrices via t ( i ) = v ( i ) + v ( i ) G t ( i ) , (15) describing multiple scattering form a single obstacle i .Then the direct scattering expansion of Eq. (14) can bereorganized into T = X i t i + X i , j t ( i ) G t ( j ) + X i , jj , k t ( i ) G t ( j ) G t ( k ) + . . . , (16)where the first term accounts for the repeated scatteringwith the same scatterer, the second one repeated scatter-ing with one scatterer followed by a series of scatteringevents with a di ff erent scatterer. The third term con-tinues this series including scattering from a third scat-terer, which may coincide with the first one. It is nowclear how the term involving n t-matrices is constructed:pick all sequences consisting of n obstacles such that allneighboring pairs are distinct. These sequences are rep-resented by multiplying the single obstacle t-matricessandwiched with an unperturbed propagator. The mul-tiple scattering expansion, Eq. (16), remains valid in thecase of hard obstacles provided the single obstacle t-matrix is calculated via the full propagator of the singleimpurity problem analogous to Eq. (12).The dynamics of the tracer depends on the details ofits local environment. Performing an ensemble averageover di ff erent initial positions of the tracer and mea-suring only the relative displacements reduces the taskto computing the disorder-averaged propagator G . ByEqs. (12,13), this is achieved by equivalently evaluat-ing the average many obstacle T-matrix T . The multiplescattering expansion shows that to first order it is su ffi -cient to determine the average single-obstacle t matrix T = nL d ¯ t + O ( n ) (17)since the next terms involve at least two obstacles.Furthermore the disorder average restores translationalsymmetry, implying T is diagonal in a plane wave basis.The correlations induced by the interaction are con-ventionally represented in terms of a self-energy Σ , de-fined via Dyson’s equation G = G + G Σ G . (18)Comparison with Eqs. (12), (17) shows that to first orderin the obstacle density Σ = nL d ¯ t + O ( n ) . (19)For the evaluation of the first-order correction in thedensity, it is thus su ffi cient to solve the dynamics of thetrace for a single scatterer.4 . A single scatterer The motion of a single particle di ff using in the presenceof a fixed spherical hard obstacle is exactly solvablewhich is the main result of this section. Here we usean approach which uses a mixed real-space momentum-space representation, making the calculation much moretransparent. Second we choose a formulation reminis-cent of the scattering problem of a quantum particle ofa hard disk using the Hilbert space of the free motionas reference. The solution of the quantum scatteringproblem can be found in many textbooks, see for ex-ample Ref. 39. Then to account for the exclusion bythe obstacle a fictitious dynamics of ghost particles isintroduced, such that the full propagator is a superposi-tion of the obstructed motion and a free di ff usion pole.The hard-sphere scattering also arises in the context ofthe low-density expansion of hard-sphere suspensions,solved by Ackerson and Fleishman [40] in real spaceand later by Felderhof and Jones [41–43].Here we calculate the t-matrix for the single obstacleproblem via a partial wave decomposition of the solu-tion of the Helmholtz equation. To indicate that the op-erators evaluated here refer to the single impurity prob-lem, we use small letters for the operators. First, it isuseful to consider the unperturbed Green function g (which is of course identical to G ) satisfying( − i ω − D ∇ ) h r | g | R i = δ ( r − R ) , (20)without restrictions on the values of the initial and ter-minal positions r and R . The propagator g for theBrownian particle moving in a two-dimensional planein the presence of a single hard core obstacle of radius σ placed at the origin satisfies the equation of motion( − i ω − D ∇ ) h r | g | R i = δ ( r − R ) ϑ ( | R | − σ ) . (21)Here the Heaviside step function ϑ ( | R | − σ ) accounts forthe fact that the tracer cannot start within the obstacleimplying that the matrix elements vanish h r | g | R i = | R | < σ . The hard-core repulsion imposes a no-fluxboundary condition on the propagator, r · ∇ h r | g | R i = | r | = σ , (22)at the surface of the obstacle. To make contact withthe formal scattering theory, the propagator g is to beinterpreted as the resolvent of some time-evolution op-erator. In the case of a hard disk, the dynamics has beendefined only for particles initially outside of the obsta-cle. Completing the description, we assume that parti-cles positioned inside an obstacle at the beginning be-have as ghost particles di ff using freely without feeling the exclusion. The resolvent associated to this operatorthen consists of the propagator we seek plus a di ff usivepole with a residue corresponding to the packing frac-tion of the single obstacle on the plane.The solution will be obtained by switching to a mixedcoordinate-momentum representation h r | g | q i = √ A Z d R h r | g | R i e i q · R , (23)where the plane wave states | q i are defined by their over-lap with the positional representation h r | q i = e i q · r / √ A .The wavenumbers q form a discrete set such that the po-sitional representation obeys periodic boundary condi-tions in the plane A = L ; furthermore they are properlynormalized h k | q i = δ k , q . Then the two Green functionssatisfy ( − i ω − D ∇ ) h r | g | q i = √ A e i q · r , (24)( − i ω − D ∇ ) h r | g | q i = √ A e i q · r , (25)where for h r | g | q i satisfies the no-flux boundary condi-tion, Eq. (22), for | r | > σ and describes the ghost parti-cle for | r | < σ . The free motion allows for a plane wavesolution h r | g | q i = e i q · r / √ A − i ω + D q , (26)which also is the solution for h r | g | q i for terminal posi-tion inside the obstacle. From the observation that thedi ff erence of Eqs. (24) and (25),(i ω + D ∇ ) h r | g − g | q i = , (27)vanishes for terminal positions outside the obstacle, theremaining case | r | > σ follows immediately. The gen-eral solution decaying rapidly at infinity of the two-dimensional source-free Helmholtz equation (27) in po-lar coordinates is given in terms of modified Besselfunctions of the second kind K ν ( · ) as h r | g − g | q i = ∞ X m = −∞ a m K m ( µ r )e i m ϕ , (28)where we abbreviated µ = √− i ω/ D = (1 − i ) √ ω/ D and ϕ = ∠ ( r , q ). The coe ffi cients a m are determined bythe boundary condition r · ∇ h r | g | q i =
0. Noting that h r | g | q i can be represented using the Jacobi-Anger ex-pansion [44, Eq. 9.1.41]e i z cos ϕ = ∞ X n = −∞ i n J n ( z )e i n ϕ , (29)5ne obtains a m = − / √ A − i ω + D q i m qJ ′ m ( q σ ) µ K ′ m ( µσ ) . (30)The propagator can now be evaluated in the momen-tum representation via Fourier transform. Here we shallneed only the forward scattering matrix element h q | g − g | q i = √ A Z | r | >σ d r e − i q · r h r | g − g | q i = − π/ A − i ω + D q ∞ X m = −∞ qJ ′ m ( q σ ) µ K ′ m ( µσ ) × Z ∞ σ r d r J m ( qr ) K m ( µ r ) , (31)where we used the fact that for terminal position in-side the obstacle, the free and full propagator are identi-cal. The second line is obtained using again the Jacobi-Anger expansion and integrating over the relative angle.With the help of the indefinite integral Z r d rJ m ( qr ) K m ( µ r ) = − rq + µ (cid:2) qK m ( µ r ) J ′ m ( qr ) − µ K ′ m ( µ r ) J m ( qr ) (cid:3) , (32)and the relation [44, Eq. 9.1.76]) ∞ X m = −∞ J m ( z ) = ⇒ ∞ X m = −∞ J m ( z ) J ′ m ( z ) = , (33)one obtains the forward scattering amplitude in closedform h q | g − g | q i = − πσ / A (cid:0) − i ω + D q (cid:1) D q ∞ X m = −∞ J ′ m ( q σ ) K m ( µσ ) µσ K ′ m ( µσ ) . (34)The t-matrix for the current case is again defined via theoperator relation g = g + g tg , and since the unper-turbed propagator is diagonal in the wave number rep-resentation h k | g | q i = − i ω + D q δ k , q , (35)the t-matrix for forward scattering is readily obtained h q | t | q i = − πσ A D q ∞ X m = −∞ J ′ m ( q σ ) K m ( µσ ) µσ K ′ m ( µσ ) . (36) Note that the forward scattering amplitude is indepen-dent of the position of the scatterer. The self-energy Σ is to first order in the density, n ∗ = σ N / A , equal tothe average T-matrix of the single scattering t-matrices,Eq. (19), h q | Σ | q i = − π n ∗ D q ∞ X m = −∞ J ′ m ( q σ ) K m ( µσ ) µσ K ′ m ( µσ ) + O ( n ∗ ) , (37)which includes the motion of the ghost particles withresidue n ∗ π . Since averaging over the disorder restorestranslational symmetry the self-energy is diagonal in thewave number representation. The average propagatorthen reads h q | G | q i = " (cid:16) − i ω + D q (cid:17) + π n ∗ D q ∞ X m = −∞ J ′ m ( q σ ) K m ( µσ ) µσ K ′ m ( µσ ) − (38)up to order O ( n ∗ ). This form still contains the di ff usivemotion of a the ghost particle. Yet subtracting the cor-responding pole yields up to first order in the packingfraction only (1 − n ∗ π ) as common prefactor, which re-flects the total probability for a randomly placed tracernot overlap with an obstacle. Dropping this factor again,one can take the preceding result as the conditionalpropagator for particles initially in the void space.For reference we give also the corresponding result inthree dimensions, n ∗ d = N σ / L : h q | G | q i = h ( − i ω + D q ) + π n ∗ d D q ∞ X ℓ = (2 ℓ +
1) [ j ′ ℓ ( q σ )] µσ k ′ ℓ ( µσ ) k ℓ ( µσ ) i − . (39)
6. Velocity autocorrelation function
The simplest quantity characterizing deviations fromsimple di ff usion is the averaged mean-square displace-ment δ r ( t ) or the velocity autocorrelation function de-fined via Z ( t ) = d d d t δ r ( t ) , t > . (40)The corresponding one-sided Fourier transform Z ( ω ) = R ∞ e i ω t Z ( t )d t is related to the long-wavelength limit ofthe Green function G ( q , ω ) : = h q | G | q i , see e.g. [45].For completeness the derivation is repeated here. Since6he mean-square displacement is the second moment ofthe averaged real-space propagator, the VACF can berepresented as Z ( t ) = d d d t Z r P ( r t | d r . (41)The averaged intermediate scattering function F ( q , t ) : = Z e i q · r P ( r t | d r , (42)exhibits a long wavelength expansion F ( q , t ) = − q δ r ( t ) / d + O ( q ), implying Z ( t ) = − lim q → q ∂ t F ( q , t ) . (43)Since G ( q , ω ) corresponds to the one-sided Fouriertransform of F ( q , t ), this last relation can be translatedto the frequency domain asRe Z ( ω ) = lim q → ω q Re G ( q , ω ) , (44)and to the self-energy Σ ( q , ω ) = h q | Σ | q i via Z ( ω ) = D − lim q → q Σ ( q , ω ) . (45)Specializing again to the planar case, Eq. (38), only theterms m = ± Z ( ω ) = D + π n ∗ D K ( µσ ) µσ K ′ ( µσ ) , (46)where the frequency dependence is encoded in µ = (1 − i ) √ ω/ D . For reference let us provide also thecorresponding result for the three-dimensional case Z ( ω ) = D + π n ∗ d D k ( µσ ) µσ k ′ ( µσ ) . (47)The velocity autocorrelation inherits the dynamic cor-relations due to the frozen landscape of obstacles, inparticular, it displays non-analytic behavior for low-frequencies Z ( ω ) = D + π n ∗ D " − − i ωσ D ln D − i ωσ ! + i ωσ γ D + O ( ω ln ω ) , (48)where γ = . . . . denotes the Euler-Mascheroni con-stant. By a Green-Kubo relation, the zero-frequency limit yields the long-time di ff usion constant D in thepresence of obstacles, D = Z ( ω = = D (1 − π n ∗ ) + O ( n ∗ ) , (49)highlighting the suppression of transport due to the ex-cluded volume. A Fourier backtransform Z ( t ) = π Z ∞ [Re Z ( ω )] cos( ω t )d ω , (50)shows that the non-analytic low-frequency expansioncorresponds to a long-time anomaly, Z ( t ) ≃ − π n ∗ σ t as t → ∞ , (51)with the same power-law as for the ballistic planarLorentz model. Here the prefactor is first order inthe density reflecting the fact that the tracer can en-counter the same obstacle by di ff usion many times with-out going through a series of scattering events with otherfrozen obstacles.The di ff usive motion at high frequencies is singulartoo, Z ( ω ) = D − π n ∗ r D − i ωσ + O ( ω − ) , (52)which is reminiscent of the skin e ff ect for electromag-netic waves on a metal. The corresponding skin pene-tration length δ : = √ D /ω is the characteristic lengthscale for di ff usive transport, and the reduction reflectsthe probability to encounter an obstacle within thatlength. A Fourier back-transform yields the singularshort-time behavior Z ( t ) = − n ∗ D r π D σ t , t → , (53)characteristic for Brownian motion in an environmentof hard obstacles.
7. Simulation results and Discussion
We have performed Brownian dynamics simulationwhich test our analytic result for the VACF and explorethe range of validity of the first order approximation tothe low-density approximation.The array of immobilized obstacles is generated byplacing randomly N hard disks of radius σ in a plane ofsize A = L . The positions of the scatterers are inde-pendently drawn from a uniform distribution and scat-terers may overlap which in principle occurs at any den-sity n = N / A . To minimize finite-size e ff ects periodic7 -3 -2 -1 -2 -1 -3 -2 -1 -2 -1 PSfrag replacements t / t t / t − Z ( t )( t / σ ) − Z ( t )( t /σ ) / n ∗ Density n ∗ Figure 1: Dimensionless negative VACF − Z ( t ) for the dilute planarLorentz gas for Brownian dynamics. Symbols correspond to simula-tion results, full lines to the first-order density approximation. Densityincreases from bottom to top. Inset: Rescaling of − Z ( t ) over n ∗ boundary conditions are employed, with typical systemsizes of L /σ = .A single tracer explores the void space of the frozenlandscape by Brownian motion. Special care has to betaken to account for the hard-core exclusion at the ob-stacles. Here we have relied on an event-driven algo-rithm developed for hard sphere liquids [46] recentlyemployed also for the three-dimensional Lorentz modelclose to the percolation transition [47]. The basic idea isto compute ballistic trajectories including the collisionswith obstacles, which are interrupted by fictitious colli-sions with a solvent acting as a heat bath. In its simplestversion, these kicks from the solvent are instantaneousat regular time intervals with period τ B where a newvelocity is drawn at random from a two-dimensionalMaxwell distribution of variance v . At time scaleslarge with respect to the algorithmic time τ B , the mo-tion is Brownian and solves the di ff usion equation witha short-time di ff usion constant D = v τ B /
4. By con-struction, the tracer never leaves the void space and thehard-core repulsion is manifested merely in the usualspecular scattering at the surface of the obstacles.The simulations for Brownian tracer particles in thelow-density range include 5 trajectories for each of the155 di ff erent obstacle realizations drawn for each den-sity, except for n ∗ = .
01 where 500 di ff erent obstaclerealizations where examined. We measure time in termsof microscopic scale t : = σ / D , i.e., the time neededfor the particle to di ff use one obstacle radius withoutobstruction. The algorithmic time τ B should be muchsmaller than t and here we used τ B = . t . Thenegative velocity autocorrelation function − Z ( t ) is dis- -2 -1 PSfrag replacements t / t − Z ( t ) t / t / / σ Density n ∗ Figure 2: Negative VACF − Z ( t ) rectified by the short-time prediction( t / t ) − / . Symbols correspond to our simulation results, full lines tothe first-order-density approximation. Density increases from bottomto top. played in Fig. 1 on double-logarithmic scales coveringfour non-trivial decades in time and more than three or-ders of magnitude in signal. For the smallest density n ∗ = .
01 the data coincide with the theoretical first-order density approximation for all times. The time axisin the figure starts at t = τ B and the small increase in thefirst data points visible is still a ff ected by the algorithmicresolution. The curves corresponding to moderate den-sities still follow the first-order low-density prediction atshort times but start to deviate at long times. The long-time decay is slower than the expected one quantifiedby an apparent exponent which becomes smaller uponincreasing the obstacle density. The crossover regimeshows only a slight flattening of the curvature consistentwith the observed increase of the exponent. The inset inFig. 1 corroborates very nicely the direct density depen-dence of the VACF following from Eq. (48). All simu-lation results overlap in the rectification with the theo-retical curve besides the exponent variations discussedabove. Let us mention that for the ballistic case a nu-merical confirmation of the long-time tail with univer-sal exponent has been achieved only recently [30], yetthe amplitude for the power law still deviated by 25%from the theoretical value for n ∗ = . t − / behavior of Eq. (53).For the smallest density again a perfect agreement is ob-served. The moderate densities are still well described8 -3 -2 -1 -2 -1 PSfrag replacements t / t − Z ( t )( t / σ ) Density n ∗ Figure 3: Long-time behavior of Z ( t ) rectified by the long-time tail( t / t ) − . Symbols correspond to the simulation results, full lines arethe first-order-density approximation. Density increases from bottomto top. by the first-order-density theory reflecting the fact thatthe particle did not have time to undergo collisions withmore than one obstacle. At short times the e ff ects aris-ing from a finite τ B amplify as the system becomesdenser, because the number of collisions per τ B is aug-mented by up to a factor of 15. For the highest density n = .
15 we measure already 0.026 collisions on aver-age between the algorithmic assignment of new randomvelocities. Assuming a Poissonian distribution for thescattering events the probability that two collisions takeplace in this time interval is roughly about 0 .
1% andtherefore not completely negligible.A rectification plot for the long-time behavior is de-picted in Fig. 3. The VACF reaches the predicted scal-ing t − for all densities, yet the observed plateau val-ues at long times increase stronger than expected. Forthe ballistic Lorentz gas it has been shown that suchan increase arises due to a competition of the long-time tails and the critical relaxation at the percolationtransition [30]. There, the underlying fractal structureinduces anomalous transport for the mean-square dis-placement [48] resulting in a fractal power-law decay ofthe VACF. The dynamics of the two-dimensional Brow-nian motion close to the percolation transition shall bediscussed elsewhere [49].The long-time di ff usion coe ffi cients were extractedvia D = lim t →∞ (1 / δ r ( t ) / d t from the simulatedmean-square displacements. The suppression of thelong-time di ff usion coe ffi cient predicted in Eq. (49) isshown in Fig. 4. Up to densities of about n ∗ = . PSfrag replacements D i ff u s i on c on s t a n t D / D − π n ∗ Density n ∗ Figure 4: Long-time di ff usion coe ffi cient D as a function of reducedobstacle density n ∗ = n σ . Symbols correspond the simulation re-sults, the straight line D / D = − π n ∗ is the first-order-density ap-proximation. description of the data for all densities. Extrapolationsuggests that the di ff usion coe ffi cient should vanish at n ∗ = /π = . .. which is surprisingly close to themeasured critical density of the percolation transition n ∗ c = . .. [49].
8. Conclusion and Outlook
The notion that correlations quickly die out at timescales beyond some characteristic relaxation time of theproblem has been shown to be incorrect in general. Be-sides the meanwhile established long-time anomaliesdue to momentum conservation in fluids [14, 15], a sec-ond paradigm leading to persistent correlations is iden-tified. Quenched disorder implies repeated encounterswith the same obstacle, and the information encoded inthe exclusion of the configuration space manifests itselfin measurable quantities such as the mean-square dis-placement or the velocity autocorrelation function.Previous studies focused on the ballistic motion inquenched disorder [20, 21] where the theoretical analy-sis is quite involved, and the long-time anomalies couldbe considered as merely a peculiarity of the Lorentzmodel. The identification of similar persistent correla-tions for hopping transport in disordered lattices [31–33] suggests that memory e ff ects may apply to a muchlarger class of systems. Here we have calculated thememory e ff ects for a Brownian particle in a random en-vironment of hard scatterers to first order in the obstacledensity. We conclude that the only ingredient necessaryfor the long-time tails is the frozen disorder. Since dis-order is ubiquitous in nature and the e ff ects arise at all9bstacle densities, we conclude that the power laws inthe VACF are present for all real systems.We have confirmed our analytical results by computersimulation for Brownian motion in a disordered arrayof obstacles. The Brownian particle follows the first-order theory quantitatively at low densities and qualita-tively for moderate ones confirming that the long-timeanomaly persists for all densities where di ff usion occursin the long-time limit. Furthermore, we have shown thatthe first-order-density expansion gives a reliable picturefor Brownian dynamics, in contrast to the ballistic case.The Brownian dynamics in the presence of hard ob-stacles displays a second power law in the VACF at shorttimes which is due to single scattering events from anobstacle. These e ff ects have no analog in the ballisticcase nor in hopping transport in disordered lattices.In the present study the unobstructed dynamics isconsidered to be a random walk where momentum doesnot play a role. For a colloidal particle suspended in afluid and moving through a dilute course of obstacles,one may expect that both the vortex di ff usion of mo-mentum in the fluid as well as the repeated encounterswith static obstacles leads to an algebraic long-time de-cay of the VACF. Since the decay t − d / due to vortex dif-fusion for unconfined motion is slower than the one forthe obstructed motion t − ( d + / one may anticipate thatmomentum conservation is more important than sterichindrance at least for long times. Yet, the obstacles canalso carry away momentum and the vortex di ff usion inthe fluid should be cut o ff at time scales where the agi-tated fluid encounters an obstacle. For the case of a wallin three dimensions it has been shown that the long-time anomalies are suppressed [17–19] to t − / ratherthan t − / . Hence, for a disordered system in a fluid acompetition between both scenarios appears to be rele-vant, yet no theory is available. Computer simulationsfor dense hard sphere liquids have indicated a crossoverfrom a positive tail due to vortex di ff usion as well asa negative tail in the densely packed regime, where thecages act as a quasi-frozen disordered environment [50].This phenomenology has been recently corroborated forLennard-Jones particles [51].Hydrodynamic fluctuations in two-dimensional fluidslead to a series of divergences in the correlation func-tions. The di ff usion coe ffi cient in an infinite system isexpected to be (logarithmically) divergent since the fluidcannot carry momentum away fast enough. For a dilutedensity of static obstacles one may hope that some ofthese e ff ects are regularized in a natural way, giving anintriguing interplay of non-analytic behaviors; a theoryfor such a scenario, however, remains an open question.
9. Acknowledgments
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