On anomalous diffusion and the out of equilibrium response function in one-dimensional models
D Villamaina, A Sarracino, G Gradenigo, A Puglisi, A Vulpiani
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n On anomalous diffusion and the out of equilibriumresponse function in one-dimensional models
D Villamaina, A Sarracino, G Gradenigo, A Puglisi and AVulpiani
CNR-ISC and Dipartimento di Fisica, Universit`a Sapienza - p.le A. Moro 2, 00185,Roma, ItalyE-mail: [email protected],[email protected]@gmail.com,[email protected]@roma1.infn.it
PACS numbers: 05.40.-a,05.60.-k,05.70.Ln
Abstract.
We study how the Einstein relation between spontaneous fluctuationsand the response to an external perturbation holds in the absence of currents, for thecomb model and the elastic single-file, which are examples of systems with subdiffusivetransport properties. The relevance of nonequilibrium conditions is investigated: whena stationary current (in the form of a drift or an energy flux) is present, the Einsteinrelation breaks down, as it is known to happen in systems with standard diffusion. Inthe case of the comb model, a general relation - appeared in the recent literature -between response function and an unperturbed suitable correlation function, allows usto explain the observed results. This suggests that a relevant ingredient in breakingthe Einstein formula, for stationary regimes, is not the anomalous diffusion but thepresence of currents driving the system out of equilibrium. n anomalous diffusion and the out of equilibrium response function...
1. Introduction
In his seminal paper on the Brownian Motion, Einstein, beyond the celebrated relationbetween the diffusion coefficient D and the Avogadro number, found the first exampleof fluctuation-dissipation relation (FDR). In the absence of external forcing one has, forlarge times t → ∞ , h x ( t ) i = 0 , h x ( t ) i ≃ Dt , (1)where x is the position of the Brownian particle and the average is taken over theunperturbed dynamic. Once a small constant external force F is applied one has alinear drift δx ( t ) = h x ( t ) i F − h x ( t ) i ≃ µF t (2)where h . . . i F indicates the average on the perturbed system, and µ is the mobility ofthe colloidal particle. It is remarkable that h x ( t ) i is proportional to δx ( t ) at any time: h x ( t ) i δx ( t ) = 2 βF , (3)and the Einstein relation (a special case of the fluctuation-dissipation theorem [1]) holds: µ = βD , with β = 1 /k B T the inverse temperature and k B the Boltzmann constant.On the other hand it is now well established that beyond the standard diffusion, asin (1), one can have systems with anomalous diffusion (see for instance [2, 3, 4, 5, 6]),i.e. h x ( t ) i ∼ t ν with ν = 1 / . (4)Formally this corresponds to have D = ∞ if ν > / D = 0 if ν < / ν < /
2. Itis quite natural to wonder if (and how) the FDR changes in the presence of anomalousdiffusion, i.e. if instead of (1), Eq. (4) holds. In some systems it has been showedthat (3) holds even in the subdiffusive case. This has been explicitly proved in systemsdescribed by a fractional-Fokker-Planck equation [7], see also [8, 9]. In addition there isclear analytical [10] and numerical [11] evidences that (3) is valid for the elastic singlefile, i.e. a gas of hard rods on a ring with elastic collisions, driven by an externalthermostat, which exhibits subdiffusive behavior, h x i ∼ t / [12].The aim of this paper is to discuss the validity of the fluctuation-dissipation relationin the form (3) for systems with anomalous diffusion which are not fully described bya fractional Fokker-Planck equation. In particular we will investigate the relevance ofthe anomalous diffusion, the presence of non equilibrium conditions and the (possible)role of finite size. Since we are also interested in the study of transient regimes, wewill consider models with microscopic dynamics described in terms of transition ratesor microscopic interactions.First, we focus on the study of a particle moving on a “finite comb” lattice with teethof size L [13]. In the limit L = ∞ an anomalous subdiffusive behavior, h x i ∼ t / , holdsand the system can be mapped, for large times, onto a continuous time random walk [13]. n anomalous diffusion and the out of equilibrium response function... L the subdiffusion is only transient and at very large time t > t ∗ ( L ) ∼ L one has a standard diffusion: h x i ∼ t . We will see that Eq. (3), where in this case theperturbed average is obtained with unbalanced transition rates driving the particle alongthe backbone of the comb, holds both for t > t ∗ ( L ) and t < t ∗ ( L ) with the same constant.This in spite of the fact that the probability densities P ( x, t ) in the two regimes are verydifferent. The scenario changes in the presence of “non equilibrium” conditions, i.e. witha drift, which induces a current, in the unperturbed state: the relation (3) does not holdanymore. On the other hand, in this case it is possible to use a generalized fluctuation-dissipation relation, derived by Lippiello et al. in [14], which gives the response functionin terms of unperturbed correlation functions and is an example of non equilibrium FDRvalid under rather general conditions [15, 16, 14, 17, 18, 19, 20, 21]. A generalization ofthe Einstein formula was also proved in the framework of continuous time random walksin [22]. So we can say that the Einstein relation (3) also holds in cases with anomalousdiffusion when no current is present, but it is necessary to introduce suitable correctionswhen a perturbation is applied to a system with non zero drift.In addition we compare the results found in comb models, with those obtained forsingle-file diffusion with a finite number of particles. There we will also consider a nonequilibrium case, with the introduction of inelastic collisions which induce an energy fluxcrossing the system. Our results suggest that the presence of non equilibrium currentsplays a relevant role in modifying Eq. (3) in stationary states.
2. Comb: diffusion and response function
The comb lattice is a discrete structure consisting of an infinite linear chain (backbone),the sites of which are connected with other linear chains (teeth) of length L [13]. Wedenote by x ∈ ( −∞ , ∞ ) the position of the particle performing the random walk alongthe backbone and with y ∈ [ − L, L ] that along a tooth. The transition probabilitiesfrom ( x, y ) to ( x ′ , y ′ ) are: W d [( x, → ( x ± , / ± dW d [( x, → ( x, ± / W d [( x, y ) → ( x, y ± / y = 0 , ± L. (5)On the boundaries of each tooth, y = ± L , the particle is reflected with probability1. The case L = ∞ is obtained in numerical simulations by letting the y coordinateincrease without boundaries. Here we consider a discrete time process and, of course,the normalization P ( x ′ ,y ′ ) W d [( x, y ) → ( x ′ , y ′ )] = 1 holds. The parameter d ∈ [0 , / x axis, producing a non zero drift of the particle. A state with a non zero drift can beconsidered as a perturbed state (in that case we denote the perturbing field by ε ), orit can be itself the starting state where a further perturbation can be added changing d → d + ε . n anomalous diffusion and the out of equilibrium response function... t/L < x ( t ) > / L L=16L=32L=64L=128L=256L=512~t ~tt * /L
100 1000 10000
Left panel: h x ( t ) i /L vs t/L is plotted for several values of L in the combmodel. Right panel: h x ( t ) i and the response function δx ( t ) for L = 512. In the insetthe parametric plot δx ( t ) vs h x ( t ) i is shown. Let us start by considering the case d = 0. For finite teeth length L < ∞ , we havenumerical evidence of a dynamical crossover from a subdiffusive to a simple diffusiveasymptotic behaviour (see Fig. 1) h x ( t ) i ≃ ( Ct / t < t ∗ ( L )2 D ( L ) t t > t ∗ ( L ) , (6)where C is a constant and D ( L ) is an effective diffusion coefficient depending on L . Thesymbol h . . . i denotes an average over different realizations of the dynamics (5) with d = 0 and initial condition x (0) = y (0) = 0. We find t ∗ ( L ) ∼ L and D ( L ) ∼ /L andin the left panel of Fig. 1 we plot h x ( t ) i /L as function of t/L for several values of L ,showing an excellent data collapse.In the limit of infinite teeth, L → ∞ , D → t ∗ → ∞ and the system shows apure subdiffusive behaviour [23] h x ( t ) i ∼ t / . (7)In this case, the probability distribution function behaves as P ( x, t ) ∼ t − / e − c (cid:16) | x | t / (cid:17) / , (8)where c is a constant, in agreement with an argument `a la Flory [2]. The behaviour (8)also holds in the case of finite L , provided that t < t ∗ . For larger times a simpleGaussian distribution is observed. Note that, in general, the scaling exponent ν , inthis case ν = 1 /
4, does not determine univocally the shape of the pdf. Indeed, for thesingle-file model, discussed below, we have the same ν but the pdf is Gaussian [24].In the comb model with infinite teeth, the FDR in its standard form is fulfilled,namely if we apply a constant perturbation ε pulling the particles along the 1-d latticeone has numerical evidence that h x ( t ) i ≃ Cδx ( t ) ∼ t / . (9) n anomalous diffusion and the out of equilibrium response function... h x ( t ) i and δx ( t ) is fulfilled also with L < ∞ , where boththe mean square displacement (m.s.d.) and the drift with an applied force exhibit thesame crossover from subdiffusive, ∼ t / , to diffusive, ∼ t (see Fig. 1, right panel).Therefore what we can say is that the FDR is somehow “blind” to the dynamicalcrossover experienced by the system. When the perturbation is applied to a statewithout any current, the proportionality between response and correlation holds despiteanomalous transport phenomena.Our aim here is to show that, differently from what depicted above about the zerocurrent situation, within a state with a non zero drift [25] the emergence of a dynamicalcrossover is connected to the breaking of the FDR. Indeed, the m.s.d. in the presenceof a non zero current, even with L = ∞ , shows a dynamical crossover h x ( t ) i d ∼ a t / + b t, (10)where a and b are two constants, whereas δx d ( t ) ∼ t / , (11)with δx d ( t ) = h x ( t ) i d + ε − h x ( t ) i d : at large times the Einstein relation breaks down (seeFig. 2). The proportionality between response and fluctuations cannot be recovered bysimply replacing h x ( t ) i d with h x ( t ) i d − h x ( t ) i d , as it happens for Gaussian processes(see discussion below), namely we find numerically h [ x ( t ) − h x ( t ) i d ] i d ∼ a ′ t / + b ′ t, (12)where a ′ and b ′ are two constants, as reported in Fig. 2.
3. Comb: application of a generalized FDR
The discussion of the previous section shows that the first moment of the probabilitydistribution function with drift P d ( x, t ) and the second moment of P ( x, t ) are alwaysproportional. Note that in the presence of a drift the pdf is strongly asymmetric withrespect to the mean value, as shown in Fig. 3 for a system with L = ∞ . Differently, thefirst moment of P d + ε ( x, t ) is not proportional to the second moment of P d ( x, t ), namely h x ( t ) i d + ε ≁ h x ( t ) i d − h x ( t ) i d . In order to find out a relation between such quantities,we need a generalized fluctuation-dissipation relation.According to the definition (5), one has for the backbone W d + ε [( x, y ) → ( x ′ , y ′ )] = W d [( x, y ) → ( x ′ , y ′ )] (cid:18) ε ( x ′ − x ) W + d ( x ′ − x ) (cid:19) ≃ W d e εW ( x ′ − x ) , (13)where W = 1 /
4, and the last expression holds under the condition d/W ≪ local detailed balance condition for our Markovprocess we can rewrite it, for ( x, y ) = ( x ′ , y ′ ), as W d + ε [( x, y ) → ( x ′ , y ′ )] = W d [( x, y ) → ( x ′ , y ′ )] e h ( ε )2 ( x ′ − x ) , (14) n anomalous diffusion and the out of equilibrium response function... t δ x d (t)/h( ε)
Response function (black line), m.s.d. (red dotted line) and secondcumulant (black dotted line) measured in the the comb model with L = ∞ , field d = 0 .
01 and perturbation ε = 0 . where h ( ε ) = 2 ε/W . For general models where the perturbation enters the transitionprobabilities according to Eq. (14), the following formula for the integrated linearresponse function has been derived [14, 19, 21] δ O d h ( ε ) = hO ( t ) i d + ε − hO ( t ) i d h ( ε ) = 12 [ hO ( t ) x ( t ) i d − hO ( t ) x (0) i d − hO ( t ) A ( t, i d ] , (15)where O is a generic observable, and A ( t,
0) = P tt ′ =0 B ( t ′ ), with B [( x, y )] = X ( x ′ ,y ′ ) ( x ′ − x ) W d [( x, y ) → ( x ′ , y ′ )] . (16)The above observable yields an effective measure of the propensity of the system to leavea certain state ( x, y ) and, in some contexts, it is referred to as activity [26]. Recallingthe definitions (5), from the above equation we have B [( x, y )] = 2 dδ y, and therefore thesum on B has an intuitive meaning: it counts the time spent by the particle on the x axis. The results described in the previous section can be then read in the light of thefluctuation-dissipation relation (15):i) Putting O ( t ) = x ( t ), in the case without drift, i.e. d = 0, one has B = 0 and,recalling the choice of the initial condition x (0) = 0, δxh ( ε ) = h x ( t ) i ε − h x ( t ) i h ( ε ) = 12 h x ( t ) i . (17)This explains the observed behaviour (9) even in the anomalous regime and predicts thecorrect proportionality factor, δx ( t ) = ε/W h x ( t ) i .ii) Putting O ( t ) = x ( t ), in the case with d = 0, one has δx d h ( ε ) = 12 (cid:2) h x ( t ) i d − h x ( t ) A ( t, i d (cid:3) . (18) n anomalous diffusion and the out of equilibrium response function... h x ( t ) i d ∼ t , turns out to be exactly canceled by the term h x ( t ) A ( t, i d , sothat the relation between response and unperturbed correlation functions is recovered(see Fig. 2).iii) As discussed above, it is not enough to substitute h x ( t ) i d with h x ( t ) i d − h x ( t ) i d to recover the proportionality with δx d ( t ) when the process is not Gaussian. This can beexplained in the following manner. By making use of the second order out of equilibriumFDR derived by Lippiello et al. in [27, 28, 29], which is needed due to the vanishing ofthe first order term for symmetry, we can explicitly evaluate h x ( t ) i d = h x ( t ) i + h ( d ) 12 (cid:20) h x ( t ) i + 14 h x ( t ) A (2) ( t, i (cid:21) , (19)where A (2) ( t,
0) = P tt ′ =0 B (2) ( t ′ ) with B (2) = − P x ′ ( x ′ − x ) W [( x, y ) → ( x ′ , y ′ )] = − / δ y, . Then, recalling Eq. (17), we obtain h x ( t ) i d − h x ( t ) i d = h x ( t ) i + h ( d ) (cid:20) h x ( t ) i + 18 h x ( t ) A (2) ( t, i − h x ( t ) i (cid:21) . (20)Numerical simulations show that the term in the square brackets grows like t yieldinga scaling behaviour with time consistent with Eq. (12). On the other hand, in the caseof the simple random walk, one has B (2) = − A (2) ( t,
0) = − t and then h x ( t ) i d − h x ( t ) i d = h x ( t ) i + h ( d ) (cid:20) h x ( t ) i − t h x ( t ) i − h x ( t ) i (cid:21) . (21)Since in the Gaussian case h x ( t ) i = 3 h x ( t ) i and h x ( t ) i = t , the term in the squarebrackets vanishes identically and that explains why, in the presence of a drift, the secondcumulant grows exactly as the second moment with no drift. -50 0 50 100 150 200 250 300 x P d ( x , t ) t=10 t=10 t=5 . d=0.01 Figure 3. P d ( x, t ) in the comb model with L = ∞ and d = 0 .
01 at different times.Notice that the mean value increases with time mostly due to the spreading, while themost probable value remains always close to zero. n anomalous diffusion and the out of equilibrium response function...
4. Conclusions and perspectives
In order to evaluate the generality of the above results, let us conclude by discussinganother system. Indeed, subdiffusion is present in many different problems wheregeometrical constraints play a central role. In this framework, a well studiedphenomenon is the so-called single-file diffusion. Namely, we have N Brownian rodson a ring of length L interacting with elastic collisions and coupled with a thermal bath.The equation of motion for the single particle velocity between collisions is m ˙ v ( t ) = − γv ( t ) + η ( t ) , (22)where m is the mass, γ is the friction coefficient, and η is a white noise with variance h η ( t ) η ( t ′ ) i = 2 T γδ ( t − t ′ ). The combined effect of collisions, noise and geometry (sincethe system is one-dimensional the particles cannot overcome each other) produces anon-trivial behaviour. In the thermodynamic limit, i.e. L , N → ∞ with N/L → ρ , asubdiffusive behaviour occurs [12].Analogously to the comb model, the case of N and L finite presents some interestingaspects. In order to avoid trivial results due to the periodic boundary conditions onthe ring, it is suitable to define the position of a tagged particle as s ( t ) = R t v ( t ′ ) dt ′ ,where v ( t ) is its velocity. For the m.s.d. h s ( t ) i , averaged over the thermalized initialconditions and over the noise, we find, after a transient ballistic behaviour for shorttimes, a dynamical crossover between two different regimes: h s ( t ) i ≃ ( − σρ ) ρ q Dπ t / t < τ ∗ ( N ) DN t t > τ ∗ ( N ) , (23)where σ is the length of the rods and D is the diffusion coefficient of the single Brown-ian particle [12]. Note that the asymptotic behaviour is completely determined by themotion of the center of mass, which is not affected by the collisions and simply diffuses.Moreover, as evident from numerical simulations, τ ∗ ∼ N and in the limit of infinitenumber of particles the behaviour becomes subdiffusive, in perfect analogy with whatobserved for the comb model, where the role of L is here played by N . The main differ-ence is that, in this case, the probability distribution is Gaussian in both regimes. As aconsequence of the Gaussian nature of the problem, applying a perturbation as a smallforce F in Eq. (22), one finds that the Einstein relation is always fulfilled [11, 10, 18, 30],also for finite N and L (see Fig. 4). Strong violations of the Einstein relation, can beobtained, in dense cases, when the collisions between the rods are inelastic so that ahomogeneous energy current crosses the system [11].In this note we have considered systems with subdiffusive behaviour, showing thatthe proportionality between response function and correlation breaks down when “nonequilibrium” conditions are introduced. In the comb model, non equilibrium effects areinduced by unbalanced transition probabilities driving the particle along the backbone,while the single-file model is driven away from equilibrium by inelastic collisions. Inthe first case, the generalized FDR of Eq. (15), developed in the framework of aging n anomalous diffusion and the out of equilibrium response function... t δ s(t) δ s ( t ) ~t~t Figure 4. h s ( t ) i and the response function δs ( t ) for the single-file model withparameters: N = 10, L = 10, σ = 0 . m = 1, γ = 2, T = 1 and perturbation F = 0 .
1. In the inset the parametric plot δs ( t ) vs h s ( t ) i is shown. systems [14], can be explicitly written, providing the off equilibrium corrections to theEinstein relation. In the second case, the transition rates are not known and anotherformalism must be exploited [18], which requires the knowledge of the probabilitydistribution for the relevant dynamical variables of the model. For instance, followingthe ideas of [11], a distribution which couples the velocities of neighbouring particlescould be a reasonable guess. Still, the indentification of the relevant variables and theircoupling in the single-file and other granular systems is a central issue, requiring furtherinvestigations. Acknowledgments
We thank R. Burioni and A. Vezzani for interesting discussions on random walkson graphs. We also thank F. Corberi and E. Lippiello for a careful reading of themanuscript. The work of GG, AS, DV and AP is supported by the “Granular-Chaos”project, funded by Italian MIUR under the grant number RBID08Z9JE.
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