Anomalous Collective Diffusion in One-Dimensional Driven Granular Media

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1 Typeset with jpsj2.cls <ver.1.2> Anomalous Collective Diffusion in One-Dimensional Driven Granular Media Yasuaki Kobayashi and Masaki Sano Department of Physics, University of Tokyo, Hongo, Tokyo We numerically investigate the nonequilibrium steady state of a one-dimensional rarefied granular system in which each particle is driven by Gaussian white noise and subject to inelastic mutual collisions. We find that the dynamic structure factor S(q, t) decays as exp( Dq 2 t 2 ), which is in marked contrast to the usual Fickian behavior exp( Dq 2 t). Employing newly obtained S(q, t), we demonstrate that the static spatial correlation function and the one-point time correlation function obey the same scaling behavior. KEYWORDS: granular media, clustering, dissipation, power law, dynamic structure factor, collective diffusion 1. Introduction Simple driven dissipative systems have been widely investigated to gain insights into a variety of phenomena found in systems far from equilibrium. For example, systems that are subjected to a driving field such as a shear flow or an electric field often exhibit various self-organized structures, which never occur in equilibrium states. Above all, granular media have been the subject of much interest. Among many intriguing properties of a granular gas, one of the most striking features is the clustering; Goldhirsch and Zanetti 1) showed that a gas composed of inelastically colliding particles loses its stability and forms high density clusters. Another example is dynamic properties of granular media passing in a narrow pipe. In the experiment of granular pipe flow, the power law decay in density power spectra was obtained. 2 7) Lattice Gas Automata have also confirmed such power law behavior. 8, 9) Granular media with continuous energy input evolve into a nonequilibrium steady state in which much interesting properties have been studied such as an equation of state, 10) static spatial correlations, 10 12) velocity correlations, 11, 12) and a large scale spatial structure. 13) It is worth noting that in such a nonequilibrium system clusters are always unstable and the system repeats ceaseless creation and destruction of the clusters. To fully understand the nonequilibrium properties in such a sistem, we should pay attention to the time-dependent structures of fluctuations, which are absent in most of the previous studies. In this paper, we numerically study the one-dimensional driven granular system in detail from the dynamical viewpoint. Unexpectedly, we find that the collective diffusion is anomalous address: yasuaki@daisy.phys.s.u-tokyo.ac.jp 1/11

2 where the usual Fickian diffusion mechanism breaks down. However, as we will demonstrate below, the unusual finding will be reasonably understood in terms of a relation between the static spatial correlation function and the one-point time correlation function, both of which are also calculated in the present simulations. 2. Model Oursystem iscomposed ofn identical particles of diameter d distributedon alineoflength L. Mutual passages of the particles are prohibited so that their order remains unchanged. Each particle is driven by Gaussian random forces which are uncorrelated spatially and temporally. Energy input by random forces can be balanced with the dissipation due to inelastic mutual collisions and frictional forces. The equation of motions for the particles are given by ẍ i = γẋ i + Θ ξ i (t) [ ( θ(d r ij ) κ r ij r ) ] ij r ij d ηṙ ij, (1) j=i±1 where x i is the position of i-th particle, θ(x) is the Heaviside step function, r ij = x i x j,θis the strength of the noise and γ is the drag force coefficient. The random driving force ξ i (t) is assumed to be the Gaussian white noise with zero mean and satisfy ξ i (t)ξ j (t ) = δ ij δ(t t ). It should be emphasized that ξ i (t) is not a source of fluctuation which ensures the fluctuationdissipation relation around the thermal equilibrium state but just an external force in a nonequilibrium steady state. This way of energy input was originally introduced by Williams and MacKintosh. 10) The last term in eq. (1) represents inelastic collisions between particles. 14) κ is the elastic constant and η is the friction coefficient. κ and η are determined from the coefficient of restitution r and the contact time t col during a collision by κ =(π 2 +(lnr) 2 )/2t 2 col and η = ln r/t col. A prohibition rule of mutual passages of the particles is not explicitly included in our model. By choosing proper values of κ and η, however, we can avoid the mutual passages. We chose d as the unit length and t col as the unit time. As for the other parameters, we set N = 100, L = 10000, Θ = and γ =10 4, whereby we limited our concern to rarefied gas with low drag force. We have verified that the mutual passage do not occur in the range of r used in our simulations. To integrate eq. (1) we use the explicit Euler scheme with the time step t =0.01. This value of the time step provides the good accuracy and efficiency of our calculation, and we have also confirmed that using smaller values of t do not change our results. Throughout our simulations, periodic boundary conditions were imposed. In order to decide that the system is in the steady state, we calculated the total kinetic energy E = N j=1 ẋ2 j /2 and verified that it conveged to the constant value. We also confirmed that the two-point static spatial correlation function (see below) converged at the same time. 2/11

3 All of our simulations use the data after this transient period. 3. Simulation Results Fig. 1. Time evolution of a distribution of particles in the steady state. (a)r =0.9; (b)r =0.01.The horizontal axis and the vertical one denote the space and the time respectively. The direction of thetimeaxesisfromtop(t = t 0 ) to bottom (t = t ). Figure 1 shows the typical space-time plots in the steady state for r = 0.9(a) and r =0.01(b). As is well known, and also supported by our simulations, for small r the density profile shows strong inhomogeneity and the particles form clusters due to the inelastic collisions. In addition, in Fig. 1(b) the system exhibits a much more dynamic behavior; each cluster moves around and repeats coalition and breakdown. In order to study such highly inhomogeneous spatial structures more quantitatively, we consider the correlation properties of this system. The spatial distribution can be characterized by the two-point static spatial 3/11

4 0.1 r=0.001 r=0.5 r=0.9 g(x,0) 0.01 x x Fig. 2. Plots of the two-point correlation function g(x, 0) versus distance for several values of r. The fitting lines represent g(x, 0) x α with α =0.49 for r =0.001, α =0.32 for r =0.5, and α =0.009 for r =0.9. correlation function g(x, 0), which is defined by g(x, 0) = n(x 0,t 0 )n(x 0 + x, t 0 ), (2) where n(x, t) = 1 N N j=1 δ(x x j(t)) is the number density of the particle at position x at time t, andthebrackets denote the averages with respect to the random force ξ i and the reference position x 0. Figure 2 shows that g(x, 0) follows the power law with its exponent close to 0.5 as r approaches 0. A similar power law behavior has also been obtained for an inviscid case. 10) This finding suggest that the density fluctuation becomes long-ranged and the system is strongly correlated. Nevertheless, the equal-time spatial correlation is not sufficient to describe the dynamical aspect of the system, which is our main concern. We thus calculated the one-point time correlation function g(0,t) defined by g(0,t)= n(x 0,t 0 )n(x 0,t 0 + t). (3) Figure 3 shows that g(0,t) also clearly exhibits the power law behavior for small r. Asr 0, the exponent approaches 0.5, which is the same value as obtained for g(x, 0). Letting g(x, 0) x α and g(0,t) t β, we show the dependence of the exponents on r showninfig.4. To further characterize the dynamics of the density fluctuation we calculated the dynam- 4/11

5 1 r=0.001 r=0.1 r=0.9 g(0,t) t Fig. 3. Plots of the one-point time correlation function g(0,t) for several values of r. The fitting line represents g(0,t) t β with β =0.45 for r = ical structure factor S(q,t), which determines the relaxation of density fluctuations. S(q,t) is defined as S(q,t) = n(q,t)n( q,0) 1 = N j 0 e iq(x j(t) x 0 (0)), (4) where n(q,t) denotes the spatial Fourier transform of n(x, t), and the brackets are the same as that of eq.(2). In the following we set r =0.01. Figure 5 shows the time dependence of S(q,t) for several values of q; q = 10q min, q = 15q min,an = 20q min,whereq min = 2π/L. The plotted data are well fitted by exp( D(q)t 2 )atleastuntils(q,t) diminishes by half. Therefore the main process of the relaxation of fluctuations can be described by S(q,t) exp( D(q)t 2 ). D(q) are plotted in Fig. 6 as a function of q 2. The plotted data were well fitted by the line D(q) =Dq 2.From these results we can determine the form of S(q,t) ass(q,t) =S(q,0) exp( Dq 2 t 2 ). This result is in marked contrast to the usual Fickian behavior S(q,t) exp( D c q 2 t), where D q is the collective diffusion constant. This implies anomalous collective diffusion. The characteristic relaxation time τ of a cluster with size l is defined as τ = l/ D, so the relaxation of 5/11

6 Fig. 4. Dependence of α and β on r, whereα and β are the exponent of the correlation functions g(x, 0) and g(0,t) respectively. As r 0, the two exponents approach the same value, 1/2. fluctuations is not diffusive but ballistic. 4. Discussion It is surprising that the collective motions of the particles are not diffusive but rather ballistic in the steady state. We also studied an individual motion of a tagged particle in the steady state with r =0.01. The mean square distance defined as x(t) 2 = 1 N N i=1 (x i(t + t 0 ) x(t 0 )) 2,where is the same as before, is shown in Fig. 7. This shows that particles lose their ballistic property at τ c 600, because of the prohibition of mutual passages in one dimensional system. 16, 17) But the anomaly of the collective diffusion continues beyond τ c (see Fig. 5). Therefore, the ballistic collective behavior cannot be attributed to the individual particle motions. The origin of this anomaly is still under investigation. The finding successfully explains why g(x, 0) and g(0,t) exhibits the same scaling behavior. By definition, g(x, t) = Using the form of S(q,t) obtained above, we obtain g(0,t)= 2π S(q,t) = 2π S(q,t)e iqx. (5) t 2 2π S(q,0)e Dq2 6/11

7 S(q,t) t Fig. 5. Plots of dynamical structure factor S(q, t) averaged over configurations in the steady state. Each curve corresponds to q =10q min, q =15q min,an =20q min (q min =2π/L) from top to bottom. Fitting curves have the function form exp( D(q)t 2 ). = 2π dx g(x, 0)e iqx e Dq2 t 2. (6) Substituting the asymptotic form g(x, 0) x α into the above equation, we obtain g(0,t) 2π dx x α e iqx e Dq2 t 2 2π qα 1 e Dq2 t 2 const. t α const. (7) Comparing the asymptotic form of the time correlation function, g(0,t) t β,weobtain α = β. This indicates that the spatial correlations has a close relation to the time correlations via the anomalous collective diffusion. We believe that such a consideration is quite reasonable, because nonequilibrium steady states generally involve various relaxational processes of their internal degrees of freedom, and cannot be characterized only by time-independent properties. The ballistic collective diffusion has been further confirmed by another simulation result. Figure 8 shows the spatial distribution of particles initially close-packed in a small region with their center of mass at the origin. All the parameters are the same as before, except that the system size is large enough for the particles not to reach an end of the system. 7/11

8 -6 8x 6x D(q) 4x 2x (q/qmin) 2 Fig. 6. Plots of D(q) onq. The fitting line represents D(q) =Dq 2 with D = Rescaled as x/t x and tn n, all the three plots at the different moments sit on the same curve n(x, t) = 1 2D t exp( x /D t). This indicates again that the lifetime of the cluster is proportional to its size, or the relaxation process is ballistic. In conclusion, we numerically investigated the nonequilibrium steady state of a onedimensional driven granular system and found that the dynamic structure factor S(q,t) decayed as exp( Dq 2 t 2 ), as opposed to the usual Fickian behavior exp( Dq 2 t). This causes the coincidence of the scaling exponent between the spatial density correlation and the temporal density correlation. Acknowledgement We would like to thank Yoshihiro Murayama and Hirofumi Wada for fruitful discussions. 8/11

9 Fig. 7. The mean square distance of a particle. The two straight lines represent t and t 2, respectively. 9/11

10 Fig. 8. Density distribution of particles initially close-packed in a region with their center at the origin, rescaled as x/t x and tp (x/t) P (x). +:t = 1000; :t = 2000; + :t = The fitting curve is A exp( x /b) witha = and b = /11

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