Anomalous diffusion and response in branched systems: a simple analysis
Giuseppe Forte, Raffaella Burioni, Fabio Cecconi, Angelo Vulpiani
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Anomalous diffusion and response in branched systems: a simpleanalysis
Giuseppe Forte
Dipartimento di Fisica Universit`a di Roma “Sapienza”,P.le A. Moro 2, I-00185 Roma, Italy.
Raffaella Burioni
Dipartimento di Fisica and INFN Universit`a di Parma,Viale G.P.Usberti 7/A, I-43100 Parma, Italy.
Fabio Cecconi
CNR-Istituto dei Sistemi Complessi (ISC),Via dei Taurini 19, I-00185 Roma, Italy.
Angelo Vulpiani
Dipartimento di Fisica Universit`a di Roma “Sapienza”and CNR-Istituto dei Sistemi Complessi (ISC),P.le A. Moro 2, I-00185 Roma, Italy. (Dated: September 29, 2018)
Abstract
We revisit the diffusion properties and the mean drift induced by an external field of a randomwalk process in a class of branched structures, as the comb lattice and the linear chains of plaquettes.A simple treatment based on scaling arguments is able to predict the correct anomalous regimefor different topologies. In addition, we show that even in the presence of anomalous diffusion, theEinstein’s relation still holds, implying a proportionality between the mean-square displacement ofthe unperturbed systems and the drift induced by an external forcing. NTRODUCTION
The Einstein’s work on Brownian motion represents one of brightest example of howStatistical Mechanics [1] operates by providing the first-principle foundation to phenomeno-logical laws. In his paper, the celebrated relationship between the diffusion coefficient andthe Avogadro’s number N A was the first theoretical evidence on the validity of the atomistichypothesis. In addition, he derived the first example of a fluctuation dissipation relation(FDR) [2, 3].Let x t be the position of a colloidal particle at time t undergoing collisions from smalland fast moving solvent particles, in the absence of an external forcing. At large times wehave: h x t i = 0 , h x t i ≃ D t , (1)where D is the diffusion coefficient and the average h· · · i is over an ensemble of independentrealizations of the process. The presence of an external constant force-field F induces a lineardrift h δx t i F = h x t i F − h x t i = µF t (2)where h· · · i F denotes the average over the perturbed system trajectories and µ indicates themobility. Einstein was able to prove that the following remarkable relation holds: h x t i h x t i F − h x t i = 2 k B TF . (3)The above equation is an example of a class of general relations known as FluctuationDissipation Relations, whose important physical meaning is the following: the effects ofsmall perturbations on a system can be understood from the spontaneous fluctuations ofthe unperturbed system [2, 3].Anomalous diffusion is a well known phenomenon ubiquitous in Nature [4–6] characterizedby an asymptotic mean square displacement behaving as h x t i ≃ t ν with ν = 12 . (4)The case ν > / ν < / x t . This happens in the presence of strong time cor-relations and can be found in chaotic dynamics [7, 8], amorphous materials [9] and porousmedia [10, 11] as well.Anomalous diffusion is not an exception also in biological contexts, where it can be ob-served, for instance, in the transport of water in organic tissues [12, 13] or migration ofmolecules in cellular cytoplasm [14, 15]. Biological environments which are crowded withobstacles, compartments and binding sites are examples of media strongly deviating from theusual Einstein’s scenario. Similar situations occur when the random walk (RW) is restrictedon peculiar topological structures [16–18], where subdiffusive behaviours spontaneously arise.In such conditions, it is rather natural to wonder whether the fluctuation-response relation-ship (3) holds true and, if it fails, what are its possible generalizations.2he goal of this paper is to present a derivation based on a simple physical reasoning,i.e. without sophisticated mathematical formalism, of both the anomalous exponent ν andEq. (3) for RWs on a class of comb-like and branched structures [19] consisting of a mainbackbone decorated by an array of sidebranches as in Fig. 1. Such branched topology istypical of percolation clusters at criticality, which can be viewed as finitely ramified fractals[20, 21]. Comb-like structures moreover are frequently observed in condensed matter andbiological frameworks: they describe the topology of polymers [22, 23], in particular ofamphiphilic molecules, and can be also engineered at the nano and microscale. Moreover,they are studied as a simple models for channels in porous media and a general account forthese systems can be found in Ref. [16]. FIG. 1. Cartoon of a one dimensional lattice (backbone) decorated by identical arbitrary-shapedsidebranches or dead-ends depicted as lateral irregular objects. Such sidebranches act as temporarytraps for the random walk along the backbone.
The diffusion along the backbone, longitudinal diffusion , can be strongly influenced bythe shape and the size of such branches and anomalous regimes arise by simply tuning theirgeometrical importance over the backbone. In other words, the dangling lateral structures,dead-ends, introduce a delay mechanism in the hopping to neighbour backbone-sites thateasily leads to non Gaussian behaviour, as it was observed for instance in flows across porousmedia [24, 25].The simple analysis of the RW on such lattices is based on the homogenization time ,meant as the shortest timescale after which the longitudinal diffusion becomes standard.The homogenization time t ∗ ( L ) can be identified with the typical time taken by the walker tovisit most of the M sb ( L ) sites in a single sidebranch of linear size L . Such a time is expectedto be a growing function of M sb ( L ) and thus of L : t ∗ ( L ) = g [ M sb ( L )]. In the following,we shall see how the scaling properties of t ∗ ( L ) = g [ M sb ( L )] can be easily extracted fromgraph-theoretical considerations, in simple and complex structures as well.Once such a scaling is known, we can apply a “matching argument” to derive the exponent ν in the relation (4). For finite-size sidebranches indeed, the anomalous regime in thelongitudinal diffusion is transient and soon or later it will be replaced by the standard3iffusion, h x t i ∼ ( t ν if t ≪ t ∗ ( L ) D ( L ) t if t ≫ t ∗ ( L ) (5)where D ( L ) is the effective diffusion coefficients depending on L . The power-law and thelinear behaviors have to match at time t ∼ t ∗ ( L ), thus we can write the matching condition t ∗ ( L ) ν ∼ D ( L ) t ∗ ( L ) or equivalently t ∗ ( L ) ν − ∼ D ( L ) , (6)accordingly, both the scaling D ( L ) ∼ L − u and t ∗ ( L ) ∼ L v provide a direct access to theexponent ν via the expression (1 − ν ) v = u . We shall see in the following, how the valuesof u and v are determined by two relevant dimensions of RW problem: the spectral ( d S ) andthe fractal ( d ) dimensions. The former is related to return probability to a given point ofthe RW and the latter defines the scaling M sb ( L ) ∼ L d .Moreover, we will show that the anomalous regimes observed in branched graphs satisfythe FDR (3) supporting the view that FDR has a larger realm of applicability than Gaussiandiffusion, as already pointed out by other authors in similar and different contexts [26–29].In the branched systems considered in this work, the generalization of FDR is due to aperfect compensation in the anomalous behaviour of the numerator and the denominator ofthe ratio (3).The paper is organized as follows, in sect.2, we discuss the diffusion and the responseby starting from the simplest branched structure: the classical comb-lattice (Fig. 2), i.e. astraight line (backbone) intersected by a series of sidebranches. The generalization to moresophisticated ”branched structures” made of complex and fractal sidebranches is reported insect.3. Sect.4 contains conclusions, where, possible links of the FDRs here derived to otherframeworks are briefly discussed. THE SIMPLEST BRANCHED STRUCTURE
At first, we consider the basic model: the simplest comb lattice is a discrete structureconsisting of a periodic and parallel arrangement of the “teeth” of length L along a “back-bone” line (B), see Fig. 2. This model was proposed by Goldhirsch et al. [30] as a elementarystructure able to describe some properties of transport in disordered networks and can bewell adapted to all physical cases where particles diffuse freely along a main direction butcan be temporarily trapped by lateral dead-ends. The walker occupying a site can jump toone of the nearest neighbour sites. Denoting by r t = ( x t , y t ) the position of the walker attime t , we can define for the longitudinal displacement from the initial position: x t − x = t X j =1 δ j (7)where { δ j } are non independent random variables such that δ j = ( δ k j if r j ∈ B0 if r j / ∈ B4 /2 Backbone BL/2 FIG. 2. Sketch of the simplest comb-lattice structure made of a “backbone” (horizontal array) and“tooth” (lateral arrays) of size L . where δ k j = {− , , } with probability { / , / , / } respectively and B denotes the set ofpoints with y = 0, i.e. forming the backbone B (Fig. 2). A simple algebra yields h ( x t − x ) i = t X j =1 h δ j i + 2 t X j =1 t X i>j h δ j δ i i where terms h δ j i = 0 if r j / ∈ B, whereas h δ j i = 1 / r j ∈ B. On the other hand h δ j δ i i = 0for all j = i . Therefore we have h ( x t − x ) i = 12 tf t (8)where f t is the mean percentage of time (frequency) the walker spends in the backboneB during the time interval [0 , t ]. To evaluate f t , we begin from the case t > t ∗ ( L ), t ∗ ( L )being the homogenization time , meant as the time taken by the walker to span a wholetooth, visiting at least once all the sites [31]. Since along the y -direction the one-dimensionaldiffusion h y t i ≃ D t is fast enough to explore exhaustively the size L and, more importantly,it is recurrent, t ∗ ( L ) can be taken as the time such that h y t i ∼ L and thus t ∗ ( L ) ∼ L .Since, after the time of the order t ∗ ( L ) ∼ L , the probability for the walker to be in a siteof the tooth can be considered to be almost uniform, we have f t = 11 + L ≃ L − , hence for t ≥ t ∗ ( L ), the mean square displacement behaves as h ( x t − x ) i ≃ L ) t (9)with an effective diffusion coefficient D ( L ) = 1 / [4( L + 1)]. In the above derivation, wehave assumed that the lateral teeth are equally spaced at distance 1. When the spacing5s ℓ > D ( L ) = 1 / [4( L + ℓ )]. This formula can be interpreted asthe ratio between the free D = 1 and the effective diffusivity D ( L ). In the literature ontransport processes, this ratio is sometimes referred to as tortuosity and it describes thehindrance posed to the diffusion process by a geometrically complex medium in comparisonto an environment free of obstacles [13, 32].The diffusion on a simple comb lattice for L = ∞ is known to be anomalous [4, 17, 33].For finite L the diffusion remains anomalous as long as the RW does not feel the finite sizeof the sidebranches. Therefore for times t < t ∗ ( L ), we expect an anomalous behaviour h ( x t − x ) i ∼ t ν (10)where the exponent ν can be computed by the matching condition (6), with t ∗ ( L ) ∼ L and D ( L ) ∼ L − , yielding L ν ∼ L − × L , from which ν = 1 / h ( x t − x ) i ∼ t / . (11)This result can be rigorously derived from standard random walks techniques [17]. It is inter-esting to note that, as the homogenization time t ∗ ( L ) diverges with the size L , upon choosingthe appropriate L , the anomalous regime can be made arbitrarily long till it becomes thedominant feature of the process.The longitudinal diffusion is a process determined by the return statistics of the walkersto the backbone. The walker indeed becomes “active” only after a return time T r = T r ( t )(operational time) which is actually a stochastic variable of the original discrete clock t = nt . This is an example of subordination: the longitudinal diffusion is subordinated to asimple discrete-time RW through the operational time T r . In a more familiar language,we are observing a Continuous Time Random Walk (CTRW) where waiting times are thereturn times to backbone sites [33] during the motion along the teeth. CTRW on a lattice,proposed by Montroll and Weiss [34], is a generalization of the simple RW where jumpsamong neighbour sites do not occur at regular intervals ( t k = kt ) but the waiting timesbetween consecutive jumps are distributed according to a probability density ψ ( t ). Shlesinger[35] showed that anomalous diffusion arises if ψ ( t ) is long tailed.The equation governing the CTRW is P ( x, t ) = ∞ X n =0 G ( x, n ) P ( n, t ) (12)where G ( x, n ) is the probability distribution of the variable x after n -steps along the back-bone from the origin x = 0 and P ( n, t ) indicates the probability to make exactly n -steps inthe time interval [0 , t ]. The probability P ( n, t ) is related to the waiting-time distribution ψ ( t ). On the comb lattice, the waiting-time distribution ψ ( t ) coincides with the distribu-tion of first-return time to the backbone sites, which for infinite sidebranches is long-tailedand asymptotically decays as ψ ( t ) ∼ t − / (see [17]). For finite sidebranches of size L , thedistribution is truncated to t ∗ ( L ) by the finite-size effect, thus ψ ( t ) ∼ t − / exp[ − t/t ∗ ( L )],Refs. [4] and [33]. 6e now consider the problem of the response of a driven RW on a comb lattice in thepresence of an infinitesimal longitudinal (i.e. parallel to the backbone line) external field ǫ [26, 36]. In that case, the displacement on the backbone is x t − x = t X j =1 ∆ ( ǫ ) j where ∆ ( ǫ ) j = ( δ ( ǫ ) j if r j ∈ B0 if r j / ∈ B δ ( ǫ ) j = {− , , } with probabilities, { (1 / δp ) , / , (1 / − δp ) } , so that h δ ( ǫ ) j i = ǫ . Thusa biased RW with jumping probabilities 1 / − δp and 1 / δp to the left and to the rightrespectively is used to model the effect of a static external field. The average jump is h δ ( ǫ ) j i = 1 × (1 / δp ) − × (1 / − δp ) = 2 δp , thus ǫ = 2 δp . Notice that ǫ plays the role ofthe external field F . By the same argument used for the free RW on the comb, we obtain h δx t i ǫ = h ( x t − x ) i ǫ − h ( x t − x ) i = ǫtf t . (13)The comparison of Eq. (8) and Eq.(13) provides the general result h ( x t − x ) i h δx t i ǫ = 12 ǫ . (14)We stress that this expression holds at any time: for both t & t ∗ ( L ) and t . t ∗ ( L ) [26], thusit works even when the averages are not taken over the realizations of a Gaussian process.In this respect, Eq. (14) represents a generalization of the Einstein’s relation (14) to theRW over comb lattices in agreement with analogous results found in different systems andcontexts [27–29].This property is a simple consequence of the subordination condition expressed byEq. (12). In fact, the small bias in the left/right jump ( ǫ = 2 δp ) along the backboneintroduces a shift in the distribution of steps G ǫ ( x, n ) = 1 √ πD s n exp (cid:20) − ( x − ǫn ) D s n (cid:21) where D s = 1 / D s = lim n →∞ h (˜ x n − ˜ x ) i / (2 n ) and ˜ x n indicates the position after n jumps on the backbone; for ǫ = 0 thedistribution is a unbiased Gaussian (in the limit of large t also n is large and the Binomial iswell approximated by Gaussian G ( x, n )). Actually the precise shape of G ǫ ( x, n ) is not veryrelevant. Since h ˜ x n i = ǫn we can compute the biased displacement in the perturbed system h x t i ǫ = ǫ ∞ X n =0 P ( n, t ) n . Considering that h n ( t ) i = P n P ( n, t ) n , we can re-write h x t i ǫ = ǫ h n ( t ) i . h x t i = h n ( t ) i / P n ( t ) remains unaltered with respect to that of theunperturbed system.To verify the above results, we generated N p = 7 × independent RW trajectories for t = 2 × time steps over a regular comb-lattice with different sidebranch sizes L . Panel A ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ -4 -2 t/L -2 〈 x 〉 / L L = 2 L = 2 L = 2 L = 2 ✩ L = 2 ✜ L = 2 ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ 〈 x 〉 〈 x 〉 ε − 〈 x 〉 L = 2 L = 2 L = 2 L = 2 ✩ L = 2 ✜ L = 2 t t tt A) B) FIG. 3. A) rescaled MSD, h x i /L , for the comb lattice (Fig. 2) of tooth length L , as a function ofthe rescaled time t/L . There is a crossover, at t/L ∼ t ≃ t ∗ ∼ L ) between a subdiffusive, t / , to a standard regime t . Inset : plot of h x i vs. t without rescaling for different L . B) plotshowing the generalized fluctuation-dissipation relation (14). The slope of the dashed straight lineis 2 ǫ , where ǫ = 2 δp = 0 .
02 as prescribed by Eq. (14).
Inset : separate plot of MSD and fluctuation h δx t i ǫ vs. time to appreciate their common behaviour in both anomalous and standard regime. of Fig. 3 refers to the mean square displacement (MSD) for an ensemble of walkers on thetraditional comb-lattice (Fig. 2) at different teeth length to probe the homogenization effectscharacterized by the time t ∗ ( L ) ∼ L . The rescaled data ( t/L , h x i /L ) collapse onto amaster curve showing a clear crossover, at the rescaled crossover time, from a subdiffusive, t / , to a standard regime, t . The response (panel B of Fig. 3) for the same lattice fulfillsthe generalized fluctuation-dissipation relation (14), thus a plot of the response h x t i ǫ − h x t i vs. the fluctuation h ( x t − x ) i shows that the data for different values of L align alonga straight line with slope ǫ = 2 δp . The perfect alignment is a consequence of the exactcompensation at every time between fluctuations and response (inset of Fig. 3B). In thesimulations of Fig. 3B, the drift is implemented by an unbalance δp = 0 .
01 in the jumpprobability along the backbone giving ǫ = 2 δp = 0 . GENERALIZED BRANCHED STRUCTURES
Interestingly, the previous analysis can be easily extended to the cases where each toothof the comb is replaced by a more complicated structure, e.g. a two dimensional plaquette,a cube or a even graph with fractal dimension d and spectral dimension d S . The spectral8imension is defined by the decay of the return probability P ( t ) to a generic site in t steps P ( t ) ∼ t − d S / [16, 37], while the ratio between d S and d is known to control the mean-squaredisplacement behaviour [16] h x ( t ) i ∼ t d S /d . (15)Of course, formula (7) still applies to fractal-like graphs and Eqs. (8,14) hold true, providedan appropriate change in the “geometrical” prefactor is introduced, as we explain in thefollowing. FIG. 4. Sketch of the comb structures used in the simulations and obtained as an infinite periodicalarrangement of the same geometrical element: A) comb of plaquettes (dubbed “kebab”) and B)two-nested comb lattices (“antenna”).
In the general case where the “teeth” are fractal structures with spectral and fractaldimensions d S and d respectively, the lateral diffusion satisfies h y t i ∼ t d S /d . Here, and in the following, y t indicates the transversal process with respect to the backbone.The previous argument for the homogenization time stems straightforwardly by noting that awalker on an infinite sidebranch, in an interval t , visits a number of different sites [16, 17, 37] M sb ( t ) ∼ (cid:26) t d S / if d S ≤ t if d S > L , the homogenization time t ∗ ( L ) isobtained by the condition M sb [ t ∗ ( L )] ∼ L d of an almost exhaustive exploration of the sites.9hen when the sidebranch has spectral dimension d S ≤
2, the first condition of (16) yieldsan homogenization time t ∗ ( L ) ∼ L d/d S Whereas, if the sidebranch has d S >
2, the secondcondition of (16) must be used to obtain t ∗ ( L ) ∼ L d . The physical reason of a differentexpression of t ∗ ( L ) above and below d S = 2 is due to the non-recurrence of the RW for d S > d S < h ( x t − x ) i ∼ t ν , ν = 1 − d S . (17)These results coincide with the exact relations obtained by a direct calculation of the spec-tral dimension on branched structures, based on the asymptotic behavior of the returnprobability on the graph, or on renormalization techniques [19, 38, 39].The case d S = 2 deserves a specific treatment thus, as an example, we consider the ”kebablattice” (Fig. 4) where each plaquette is a regular two dimensional square lattice, for which d S = d = 2. Indeed d S = 2 is the critical dimension separating recurrent ( d S <
2) and notrecurrent ( d S >
2) RWs. Thus d S = 2 is the marginal dimension [4] which reflects into thelogarithmic scaling of the transversal MSD h y t i ∼ t/ ln( t ), hence the homogenization timeis now t ∗ ( L ) ∼ L ln( L ). Applying once again the matching argument, we obtain the scaling h ( x t − x ) i ∼ ln( t ) (18)indicating a logarithmic pre-asymptotic diffusion along the backbone. The time evolution ofMSD from initial positions of the simulated random walkers on the “kebab” lattice verifiesthe transient behaviour (18) at different sizes L , Fig. 5A.Notice that, in our matching arguments, we only make use of the spectral and fractaldimension of the sidebranches. Interestingly, these two parameters are left unchanged if oneperforms a set of small scale transformations on the graph [40], without altering their largescale structure. Our results hold therefore true also for different and disordered sidebranches,provided the two dimensions are unchanged.Following the same steps as those described for the comb lattice, the generalizedfluctuation-dissipation relation also holds for all branched structures. To check the re-sult we study the “kebab” lattice (Fig. 4A), where the two-dimensional plaquette is aregular square lattice of side L y and unitary spacing [38]. It follows that: h ( x t − x ) i h δx t i ǫ = 13 ǫ , (19)the prefactor 1 / /
6. Panel B of Fig. 5 reports the verification ofthe fluctuation-dissipation relation: independently of the lattice size, the plot response vs.MSD is a straight line with slope 1 / (3 ǫ ).To show the effect of d S on the homogenization time and on the diffusion process, weconsider a structure composed by two-nested comb lattices that we dub “antenna” (Fig. 4B),10 t 〈 x 〉 L = 2 L = 2 L = 2 L = 2 L = 2 L = 2 ✩ ✩ ✩✩✩✩✩✩✩✩✩✩✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✜ ✜ ✜✜✜✜ ✜ ✜ ✜ ✜ ✜✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ 〈 x 〉 -2 〈 x 〉 ε − 〈 x 〉 L = 2 L = 2 L = 2 L = 2 ✩ L = 2 ✜ L = 2 l n ( t ) A) B)
FIG. 5. Panel A: linear-log plot of MSD, h x t i vs. time for plaquette-comb lattice at different L , (Fig. 4A). Data show an initial collapse onto the common baseline ln( t ), in perfect agreementwith the scaling result (18). Panel B: plot of the response h δx t i ǫ vs. the fluctuations h ( x t − x ) i showing the generalized Einstein’s relation (19). The slope of the dashed line is 3 ǫ , with ǫ = 2 δp and δp = 0 .
01 (unbalance in the left-right jump probability along the backbone). i.e. a comb lattice, where the teeth are comb lattices themselves on the y, z plane. Thisstructure is then characterized by two length-scales, the vertical, L y , and transversal, L z ,teeth length; only for sake of simplicity we assume L y ∼ L z ∼ L .Also in this case there exists a crossover time t ∗ ( L ) ∼ L depending on the length ofthe teeth along- z , such that: for t & t ∗ ( L ), the diffusion becomes standard, whereas for t . t ∗ ( L ), an anomalous diffusive regime takes place. Since for a simple comb lattice, d S = 3 /
2, see [17], we obtain from Eq. (4) h ( x t − x ) i ∼ t / . For finite L , the MSD in Fig. 6A exhibits an initial regime t / followed by a t / -behaviourwith a final crossover to the standard one. Such a particular scaling, t / , is certainly dueto the ”double structure” of the sidebranches. t 〈 x 〉 ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✜ ✜ ✜ ✜✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ 〈 x 〉 -2 〈 x 〉 ε − 〈 x 〉 L z = 2 L z = 2 L z = 2 L z = 2 ✩ L z = 2 ✜ L z = 2 t t t L y = 2L z A) B)
FIG. 6. A-panel: time behaviour of MSD, h x t i for the “antenna” (Fig. 4B) with L y = 2 L z = L at different values of L . B-panel: generalized fluctuation-dissipation relation (19): response h δx t i ǫ vs. h x t i . The slope of the dashed straight-line is the proportionality factor 3 ǫ . ✩ ✩ ✩ ✩ ✩✩✩✩✩✩ ✩✩✩ ✩✩ ✩✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩✩ ✩ ✩✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✜ ✜ ✜ ✜ ✜ ✜✜✜✜✜✜✜ ✜✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜✜✜ ✜✜✜ ✜✜✜ ✜✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ -8 -6 -4 -2 t/L -1 〈 x 〉 L = 12L = 36L = 72L = 128 ✩ L = 512 ✜ t L=12L=36L=72L=128L=256L=512 ✩ ✩✩✩✩✩✩✩✩✩✩✩ ✩✩ ✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✩ ✜ ✜✜✜✜✜✜✜✜✜✜✜✜ ✜✜✜✜✜✜✜✜✜✜✜✜ ✜✜✜ ✜✜✜ ✜✜✜✜✜✜✜✜ -2 -1 〈 x 〉 ε - 〈 x 〉 〈 x 〉 L = 12L = 36L = 72L = 128 ✩ L = 512 ✜ t -2 -1 〈 x 〉 (L=72) 〈 x 〉 ε (L=72) t t 〈 x 〉 A) B)
FIG. 7. Comb lattice of compenetrating cubes. A) collapse of the MSD h x t i at different cubesides L vs. the rescaled time t/L . The data show the plateau which is a precursor of the standarddiffusion. Inset : same data not rescaled. B) Plot showing the generalized fluctuation-dissipationrelation: response h δx t i ǫ vs. h x t i . Inset : plots of h δx t i ǫ and h x t i showing the parallel behaviourof response and MSD independently of the regime. Also in this case, the generalized Einstein’s relation is verified (Fig. 6A) which coincideswith Eq. (19) for the “kebab”. Indeed, the walkers on both antenna and kebab lattices havethe same probability 1 / d S > d S = d = 3, so we consider a comb-like structure where the lateralteeth are compenetrating but non-communicating cubes. For computational simplicity thecubes are arranged with centers at a unitary distance from one another along the backbone.Actually, the minimal distance among the centers of non-compenetrating cubes with edge L , is L/ L/ L which is of course larger than 1 as soon as L >
1, but in our modelthe cubes, despite their large overlap, are still considered as distinct sidebranches connectedonly through the backbone. The homogenization time will be t ∗ ( L ) ∼ L d and D ( L ) ∼ L − d .Therefore, for t ≫ t ∗ ( L ), we expect the standard diffusive growth h ( x − x ) i ∼ t/L d , whilebelow t ∗ ( L ), h ( x − x ) i ∼ t ν and the matching condition at t ∗ ( L ) predicts the existence ofa plateau h ( x t − x ) i ∼ const, as derived by exact relations based on return probabilities[19]. The simulation data are in agreement with the above results, see Fig. 7, and also theproportionality between fluctuation and response is again perfectly verified. CONCLUSION
In this paper we have analyzed the random walk (RW) and the Einstein’s response-fluctuation relation on a class of branched lattices generalizing the standard comb-lattice.For any sidebranch of finite-size, a transient regime of anomalous diffusion is observed whoseexponents can be derived by an heuristic argument based on the notion of homogenitaziontime and on the geometrical properties of the lateral structures.Our analysis has been here restricted to branched lattices where the distance betweentwo consecutive sidebranches is unitary, but it can be straightforwardly extended to cases12ith arbitrary spacing.We can conclude by noting that a random walk on generic branched lattice satisfies ageneralized Einstein’s relation for different shapes and sizes L of the sidebranches. This isclearly apparent in figures: 5B, 6B and 7B, where data perfectly collapse onto a straight linewhen plotting the free mean square displacements against the response.Since this is a straightforward consequence of Eqs. (8) and (13), including their analoguesin more complex comb-structures, the result that R ( t ) = h ( x t − x ) i h δx t i ǫ = constis exact and valid for any comb-like structure both in the transient and asymptotic regimes.It stems from the perfect compensation, at any time, between the response of the biasedRW and the mean square displacement of the unbiased RW.Our results may add other elements to the general issue [41–43] about the validity of thefluctuation-dissipation relations (FDR) in far from equilibrium systems and non Gaussiantransport regimes.There are by now sufficient theoretical [26, 27, 44] and experimental [45, 46] evidencesto claim that FDR can be often generalized well beyond its realm of applicability. Thistraditional issue of Statistical Mechanics received a renewed interest also thanks to theamazing progresses in single-molecule manipulation techniques. Experiments whereby acolloidal particle is dragged by optical tweezers well approximate the ideal system of asingle Brownian particle driven out of equilibrium. This offers the opportunity to test in alaboratory the FDR on a minimal non equilibrium system. To some extent, invoking thesimilarity between RW and Brownian motion, the issues addressed in this work involve thatclass of behaviours encountered in mesoscopic systems [47], where either particles or genericdegrees of freedom move diffusively on a complex support.13
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