Collision Statistics of Driven Polydisperse Granular Gases

Commun. Theor. Phys. (Beijing, China) 49 (2008) pp. 1333 1338 c Chinese Physical Society Vol. 49, No. 5, May 15, 2008 Collision Statistics of Driven Polydisperse Granular Gases CHEN Zhi-Yuan, 1,2, ZHANG Duan-Ming, 1 LI Zhong-Ming, 2 YANG Feng-Xia, 1 and GUO Xin-Ping 1 1 Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China 2 Department of Physics, Xianning College, Xianning 437100, China (Received June 4, 2007) Abstract We present a dynamical model of two-dimensional polydisperse granular gases with fractal size distribution, in which the disks are subject to inelastic mutual collisions and driven by standard white noise. The inhomogeneity of the disk size distribution can be measured by a fractal dimension d f. By Monte Carlo simulations, we have mainly investigated the effect of the inhomogeneity on the statistical properties of the system in the same inelasticity case. Some novel results are found that the average energy of the system decays exponentially with a tendency to achieve a stable asymptotic value, and the system finally reaches a nonequilibrium steady state after a long evolution time. Furthermore, the inhomogeneity has great influence on the steady-state statistical properties. With the increase of the fractal dimension d f, the distributions of path lengths and free times between collisions deviate more obviously from expected theoretical forms for elastic spheres and have an overpopulation of short distances and time bins. The collision rate increases with d f, but it is independent of time. Meanwhile, the velocity distribution deviates more strongly from the Gaussian one, but does not demonstrate any apparent universal behavior. PACS numbers: 61.43.Bn, 61.43.Hv Key words: fractal dimension d f of the disk size distribution, restitution coefficient e, standard white noise, nonequilibrium steady state, statistical properties 1 Introduction Granular materials represent a type of matter not well defined by conventional means. Although each granular particle is obviously solid, an assemblage of these particles shows distinctly nonsolid behavior when subjected to external forces. [1] Their collective behavior is peculiar and also different from other forms of matter, such as liquids, or gases. Although the ordinary statistical mechanical approach successfully deals with large assemblies of microscopic particles, it is not adequate for them. Generally speaking, granular materials cannot be described as equilibrium systems either from the configurational point of view or from the dynamical point of view. It is known that in fact these systems remain easily trapped in some metastable configurations which can last for long time intervals unless they are shaken or perturbed. [1] The understanding of the granular state still represents an open challenge. But, a granular system consisted of fluidizing grains behaves as a nonideal gas, which is relatively easier to be studied. The granular system can be regarded as granular gases. One crucial difference between ordinary gases and granular gases is represented by the intrinsic inelasticity of the interactions among the grains, which makes any theory based on energy conservation, e.g., for ideal gases, not suitable. However, several theoretical methods have been employed to deal with granular gases ranging from hydrodynamic equations, kinetic theories, to molecular dynamics. Usually researchers investigate the statistical properties of granular gases by experiments and computer simulations. They all show that the dynamics of granular gases stems from the inelastic nature of their collisions which leads to a variety of very peculiar phenomena, such as clustering, [2 5] non-maxwellian velocity distributions, [3,6 12] breakdown of energy equipartition in mixtures, [13] and so on, all of which have become one of the most active research topics in nonequilibrium statistical mechanics and fluid dynamics. It is well known that most models describing the statistical properties of granular gases are developed for monodisperse or bidisperse granular materials. In practice, however, actual granular materials are almost always polydisperse and thus the effect of it on the statistical properties should be of interest. Because the experimental study of the statistical properties in a real granular system presents a great challenge, the increasing power of computers awakens an interest in actual granular gases simulations. Recently, in the homogenous cooling state, Lambiotte and Brenig [13] have studied the nonequipartition of energy in granular fluids composed of an arbitrarily large number of components, and have discussed the asymptotic temperature ratio varying with the inelasticity of each species. Zhang et al. [14 17] presented a fractal model of polydisprse granular system. They studied that the fractal dimension d f of the particle size distribution influences the effective thermal conductivity due to the stochastic movement of the particles and the steady-state dynamic behaviors of the driven granular system, such as the granular temperature, kinetic pressure, velocity distribution and distribution of distances between nearest neighbors. While, in order to solve problems in an easier way, they only studied a one-dimensional case. Similar studies of polydisperse granular mixtures in higher dimensional systems are very scarce. The project supported by National Natural Science Foundation of China under Grant No. 10675048 and the Natural Science Foundation of Education Department of Hubei Province of China under Grant Nos. D200628002 and kz0627 Corresponding author, E-mail: czy2004hust@163.com

1334 CHEN Zhi-Yuan, ZHANG Duan-Ming, LI Zhong-Ming, YANG Feng-Xia, and GUO Xin-Ping Vol. 49 In this paper, we present a dynamical model of twodimensional polydisperse inelastic granular gases with fractal size distribution. As granular gases are dissipative, it is necessary to feed energy into the system to keep the disks agitated. To thermalize the system, we choose a random external force, which gives frequent kicks to each disk between collisions. Such a driving mechanism has been studied by many authors. [2,3,6,7,12,18] The advantage of this thermalization mechanism, besides its potential physical realizations, lies in the fact that the nonequilibrium stationary state is linearly stable against spatial inhomogeneities. [18] By Monte Carlo simulations, we mainly study the evolution of average energy in transient state and the effect of the inhomogeneity of the disk size distribution on the steady-state statistical properties, focusing on the distributions of path lengths and free times between collisions, collision rate and velocity distribution. 2 Model Let us consider a polydisperse granular mixture of N inelastic hard-disks with radius r i and mass m i (where 1 i N), in which the size distribution of the disks has the fractal characteristic. Fortunately, experimental results show that the particle size distributions of many granular materials exhibit the fractal characteristic, i.e. the size of particles following the fractal distribution. [19 21] Based on the fractal theory, [22] there is a fractal dimension d f as a measurement of the inhomogeneity in the particle size distribution, [20,21] the higher d f implies greater dispersion of the finer particles, which leads to more inhomogeneity in the particle size distribution, and the fractal dimension d f can be obtained, 1 < d f < 2, in two-dimensional case. So, it is reasonable that we consider the polydisperse granular mixture with the fractal characteristic of the disk size distribution, and use the fractal dimension d f to describe the inhomogeneity of the disk size distribution. Therefore, the disk size distribution in the granular system will be more and more inhomogeneous as the fractal dimension d f increases, and a more inhomogeneous disk size distribution indicates a more asymmetric mass distribution of disks. When n 0 /N 1, where n 0 is the number of disks with the maximum size r max, the disk size distribution of the granular system satisfies the size-frequency character by fractal theory [14] ( r ) Y nr (r) = 1 N 1 df n 0, (1) r max where Y nr is the ratio of n r to N, n r is the number of disks whose size is smaller than r, N is the total number of disks, and d f is the fractal dimension of disk size distribution, here 1 < d f < 2. For simplicity, we assume that the surface of disks is smooth and the mass density of disks is identical. Then, the mass of a disk with radius r is m = πr 2 ρ m, (2) where ρ m is the mass density of the disks. From Eqs. (1) and (2), the mass and the radius of any disk in the mixture can be respectively expressed as m = m max [ N n 0 (1 Y nr )] 2/df, (3) r = r max [ N n 0 (1 Y nr )] 1/df. (4) If the values of N, m max, r max, n 0 and d f are given, according to Eqs. (3) and (4) we can randomly evaluate the mass and the radius of every disk in the polydisperse granular mixture. In this work, an inelastic hard disk model is implemented. We simulate an ensemble of N polydisperse inelastic hard-disks in a two-dimensional horizontal box of side L with periodic boundary conditions. The area packing fraction φ of the system may be defined as φ = N / i=1 πr2 i L 2. The disk collisions are inelastic: the total momentum is conserved, while the component of the relative velocity parallel to the direction joining the center of the disks is reduced to a fraction e (0 < e < 1) of its initial value, lowering in this way the kinetic energy of the pair: m j v [ i = v i (1 + e) (vi v j ) ˆn ]ˆn, (5) m i + m j m i v [ j = v j + (1 + e) (vi v j ) ˆn ]ˆn, (6) m i + m j where v i and v j are the velocities of the colliding disks after the collision, ˆn = (x i (t) x j (t))/(r i + r j ) is the unit vector along the line of centers x i (t) and x j (t) of the colliding disks at contact. In the interval between two subsequent collisions, the motion of each disk i is governed by the following Langevin equation: [3,12] m i t v i (t) = m i ηv i (t) + 2m i ηt F ξ i (t), (7) t x i (t) = v i (t), (8) where the function ξ i (t) is a stochastic process with average ξ i (t) = 0 and correlation ξi α(t 1)ξ β j (t 2) = δ ij δ(t 1 t 2 )δ αβ (α and β being component indices), i.e., a standard white noise. This means that each disk feels a hot fluid with a temperature T F and a viscosity coefficient η. Therefore, the disks obtain the kinetic energy from the white noise forcing, execute Brownian motion between collisions, and dissipate the kinetic energy through inelastic collisions and dump. From Eqs. (5) and (6), the energy dissipation due to the inelastic collisions during one collision is E = (1 e2 )[(v i v j ) ˆn] 2 m i m j 2 m i + m j = (1 e2 )[(v i v j ) ˆn] 2 m i (m i + m ij ), (9) 2 2m i + m ij where m ij = m j m i. Obviously, both the mass difference m ij between the two colliding disks and the restitution coefficient e can influence energy dissipation. We know that, a larger value of the fractal dimension d f represents more inhomogeneity of size distribution, which makes the mass difference between any two colliding disks greater. Therefore, when the restitution coefficient e is

No. 5 Collision Statistics of Driven Polydisperse Granular Gases 1335 fixed (0 < e < 1), the greater the fractal dimension d f is, the more the energy dissipation E is. Likewise, if the fractal dimension d f remains the same, the smaller the restitution coefficient e is, the more the energy dissipation E is. 3 Simulations and Results The system considered here is a dilute inelastic gas with an area packing fraction φ < 0.4. By Monte Carlo method, we perform the simulations of the disk colliding and Brownian motion between collisions. In the actual simulations, we reduce any variable to dimensionless one. For definiteness of our discussion, we use m max = 20, r max = 1, n 0 = 1, T F = 1, and η = 0.1 in all simulations performed. The simulation scheme consists of a discrete time integration of the motion of the disks. At each time step of length t (where t τ c, and τ c is average collision time), the following operations are performed: [12] (i) Free streaming: equations (7) and (8) are integrated for a time step t disregarding possible interactions among the disks; (ii) Collisions: Every disk has a probability p = t/τ c of undergoing a collision. Its collision mate is chosen among the disks in a circle of fixed radius r B with a probability proportional to its relative velocity. Initially, the disks are randomly placed on the two-dimensional horizontal square box which is divided into L 2 identical square lattices. They are then randomly given an initial velocity which is consistent with their position by a Gaussian random generator. The disks subsequently move, if necessary, to eliminate disks overlap. The simulations proceed from collision to collision using an event-driven algorithm, [23] and the disk motion between two consecutive collisions is governed by Eqs. (7) and (8). After a long evolution time, the system reaches a nonequilibrium steady state which is a result of the balance between the dissipation of the kinetic energy due to inelastic collisions and dump, and the energy injection due to the driving mechanism. 3.1 Approach to Steady State Starting from any initial configuration, the system with dissipation (the restitution coefficient 0 < e < 1) reaches a steady state after a large enough number of collisions. To ensure that the system has reached a steady state, we relax the system until the average energy does not drift. Figure 1 shows how the average granular energy per disk relaxes with time toward a steady state for a system with L = 10, N = 50 000, e = 0.6, and d f = 1.5. It is found that the relaxation is exponential as E(t) = E( ) + [E(0) E( )] e t/τ B. (10) E(0) is the initial value of the average energy per disk, E( ) is the asymptotic value of the energy curve when the time approaches infinity, where E(0) = 1.65 10 3, E( ) = 1.21 10 3. We find that the relaxation time τ B (= 50.1) is independent of the initial state, and is controlled by the model parameters (i.e., the restitution coefficient e, the fractal dimension d f, the total number of disks N etc.). Fig. 1 Average energy relaxation starting from a random initial configuration for a system with L = 10, N = 50 000, e = 0.6, and d f = 1.5. The vertical axis is the average kinetic energy per particle (1/2) N m i=1 iυi 2 /N. The best fit using Eq. (11) is also shown with E( ) = 1.21 10 3, A = 4.56 10 4, and B = 0.02. In Fig. 1, the dotted line represents the first-order exponential fit. Therefore, the time-dependent relaxation is also given by E(t) = E( ) + A e Bt, (11) where A is a constant depending on the initial condition, B = 1/τ B = 0.02. Equation (11) further indicates that the average energy of the system evolves from the initial state into a stable state with an exponentially decaying form. When the system reaches a steady state, we start the simulations with the length of 100 000 collisions per disk. Thereafter, we can obtain the data with small statistical error. 3.2 Distributions of Path Lengths and Free Times Between Collisions By basic kinetic theory arguments, [24] the distribution of path lengths between collision events for an elastic hardsphere gas (and by a similar treatment the distribution of free times between collisions) is given by P(l) = (2 2φ) e 2 2φl. (12) Apparently, the distribution P(l) should follow a simple exponential form depending only on the packing fraction φ of the assembly. As in Ref. [23], the distribution of path lengths from the geometric distance between collision events is calculated and shown in Fig. 2(a) for all systems with different values of the fractal dimension d f at the same packing fraction φ. For comparison, we plot in Fig. 2(a) the corresponding distribution of an elastic gas which has the same fractal dimension d f and area packing fraction φ of our inelastic mixture. However, it is clear from the dashed line in Fig. 2(a) that the simple form given by Eq. (12) does not describe the behavior over all l in the inelastic case. Therefore, the presence of the inelasticity of collisions modifies

1336 CHEN Zhi-Yuan, ZHANG Duan-Ming, LI Zhong-Ming, YANG Feng-Xia, and GUO Xin-Ping Vol. 49 such a simple exponential law in the way from the symbol shown in Fig. 2(a). It is seen from Fig. 2(a) that, in the inelastic case, the distribution of path lengths deviates from the theoretical prediction for elastic hard spheres. In particular, P(l) shows a peak in the small l region, not present in elastic systems, demonstrating the inherent clustering in this system. From Fig. 2(a) it can also be found that, as the fractal dimension d f increases with respect to the same restitution coefficient e (0 < e < 1), the probability of the short-path increases. This indicates that when the value of the fractal dimension d f augments in the same inelastic case, the system becomes more and more clusterized. This phenomenon is consistent with our recent simulations in one-dimensional case. [17] the fractal dimension d f increases at the same restitution coefficient e (0 < e < 1), the probability of the short-time increases. Likewise, for comparison we plot in Fig. 2(a) also the corresponding distribution of an elastic gas. The effect of the shorter average collision time is also clearly visible. Such a behavior is similar to Refs. [8] and [9]. As we know, the higher d f implies greater dispersion of the finer particles, which leads to the particle number density increasing at the same packing fraction φ. Thus, more energy dissipation is caused by the inhomogeneity of the disk size distribution and the higher collision rates due to increased density in the same inelasticity. The effect of energy dissipation leads to strong spatial correlations of density and velocities, which reduce disk relative motion. This smaller value of disk relative velocity implies that the disks tend to cluster and stream when driven. Therefore, the physical reason for the distributions of paths and free times between collisions lies in the existence of strong spatial correlations of density and velocities, which lead to the presence of clusters where the path lengths and the free times are shorter than they are in a uniform system. 3.3 Collision Rate Fig. 2 (a) The probability distributions of path lengths P(l) vs. l, on a log-linear scale. The dashed line shows the theoretical form given by Eq. (12) derived for elastic particles, while the symbols represent the simulations. (b) The probability distributions of free times P(τ) vs. τ, on a log-linear scale. The symbols represent the simulations in the inelastic case, and the dashed line is the corresponding distributions for an elastic system. In both figures φ = 0.2, L = 10, η = 0.1, T F = 1, and e = 0.6. The distribution of free times between collisions P(τ) is also computed and shown in Fig. 2(b). It displays similar behavior to the path length distributions, namely an overpopulation of the short-time bins. One clearly sees from Fig. 2(b) that the probability density for which a disk suffers a collision in a short interval is enhanced with respect to the elastic case. It is also found that, when Fig. 3 (a) Real time vs. total number of collisions for a system with different values of the fractal dimension d f. (b) Collision rate ζ vs. d f. In both figures N = 50 000, L = 10, η = 0.1, T F = 1, and e = 0.6. In our simulations, we count the total number of collisions starting from a configuration in the steady state. Figure 3(a) shows that the total collision number (C) is linear with time for all systems with different values of the fractal dimension d f in the same inelasticity case

No. 5 Collision Statistics of Driven Polydisperse Granular Gases 1337 (0 < e < 1). The reciprocal slopes of the linear-fit lines are the collision rates for different fractal dimension d f. In the steady state the collision rate is constant, independent of time. This is in contrast to the time-dependent collision rate in the freely evolving monodisperse granular medium where the total number of collisions increases as ln(1 + t/t e ) for homogeneous cooling. [25,26] One would expect that the collision rate should be an increasing function with decreasing d f, as the velocity is larger and the collision is more likely to happen in the smaller d f system. However, we find from Fig. 3(a) that the collision rate increases as the energy dissipation increases. Figure 3(b) shows the dependence of the collision rate ζ as a function of the fractal dimension d f. The increase of collision rate with increasing d f may be linked with the clustering mechanism. [4,5] If the velocity distribution is a Gaussian then γ x = 3, and if the distribution is non-gaussian then γ x exceeds 3. In Fig. 4(b), γ x has been plotted as a function of d f for e = 1.0 and e = 0.6, respectively. In the elastic case, γ x would be 3 over all d f, this demonstrates the velocity distribution is a Gaussian. However, in the inelastic case, the increase in γ x shown in Fig. 4(b) implies that velocity distribution function deviates more strongly from a Gaussian as d f is increased. We find that the calculated values of the kurtosis changing with the fractal dimension d f have the same evolution as the above observation in Fig. 4(a). Thus, we can conclude that the dependence of γ x on d f may be due to the dissipation in the inelasticity. This analysis is similar to MD simulations of Brey and Ruiz Montero [11] and experimental investigations of D.L. Blair and A. Kudrolli. [9,10] 3.4 Velocity Distributions The distribution of velocities is normally displayed in a log-linear fashion to accentuate the tails of the velocity distribution function; however, this suppresses the deviations at low velocities. For comparison, all the disk velocities are rescaled by their mean square values. The distribution of the x-component of the rescaled velocities is plotted in Fig. 4(a) when the system with different values of the fractal dimension d f reaches a stationary state. In the elastic case, the velocity distribution is very well fitted by Gaussian. However, in the inelastic case, we observe that the velocity distribution displays strong deviations from Gaussian one both at low and high velocities. In fact, the kurtosis of the rescaled velocity distribution function rises higher distinctly with the fractal dimension d f increasing at the same restitution coefficient e (0 < e < 1), whereas, the fractal dimension d f is larger, the velocity distribution function has more extended tails. This phenomenon is consistent with our previous simulation results in one-dimensional case. [16,17] From our velocity distribution functions, whose corresponding inelasticity of collisions remains the same (0 < e < 1) and where the fractal dimension d f varies, we cannot find universality of the velocity distribution functions that describes the overall form. Such a lack of universality was also noticed in Refs. [3] and [8]: which is shown that the tails of the velocity distribution function become broader when the dissipation rate is increased. Here the mechanism is similar: the fractal dimension d f is larger, the dissipation rate is higher. The lack of universality was also noted experimentally by Blair and Kudrolli. [9] It is well known that the degree of the deviation from a Gaussian can be measured by the kurtosis of the velocity distribution function. To further show the deviation of the velocity distribution functions, we quantitatively calculate the kurtosis of the velocity distribution functions. The kurtosis is obtained by the following equation γ x = υ4 x υ 2 x 2. (13) Fig. 4 (a) The rescaled velocity distribution P(υ x /σ x ) vs. υ x /σ x (σ x = υ 2 x 1/2 ) in two different regimes (an elastic case with e = 1.0, and an inelastic case with e = 0.6) on a log-linear scale for a system with N = 50 000, L = 10, d f = 1.2, 1.5, and 1.8 respectively. In both cases, the symbols represent the simulations. The dotted line represents the Gaussian fit. (b) The kurtosis γ x calculated from P(υ x /σ x ) as a function of d f for e = 1.0 and 0.6, respectively. The dashed line is the values given by a Gaussian in the elastic case, and the solid line represents the values of non-gaussian in the inelastic case. The velocity distribution for the y-component P(υ y /σ y ) versus υ y /σ y for each d f is also calculated. We have found that the distribution of the y-component of the disk velocities resembles the velocity distribution function for the x-component. Therefore, our analysis shows that the present consensus emerging from various studies tends

1338 CHEN Zhi-Yuan, ZHANG Duan-Ming, LI Zhong-Ming, YANG Feng-Xia, and GUO Xin-Ping Vol. 49 to reach the conclusion of the absence of universality in the velocity distribution: [27] various experimental conditions and various energy injection modes lead to different distributions. 4 Summary and Conclusion In this paper, we present a dynamical model of polydisperse inelastic hard disks with the fractal size distribution, in which the disks are driven by standard white noise and constrained to move in a two-dimensional horizontal square box with periodic boundary conditions. The inhomogeneity of the disk size distribution is measured by a fractal dimension d f, and the higher d f implies greater dispersion of the finer disks, which leads to more inhomogeneity in the disk size distribution. By Monte Carlo simulations, we perform long-time simulations of the disk colliding and Brownian motion between collisions in the same inelasticity case (0 < e < 1). The simulation results indicate that the average energy of the system decays exponentially, with a tendency to achieve a stable asymptotic value, and the system finally reaches a nonequilibrium steady state after a long evolution time. Moreover, the inhomogeneity of the disk size distribution has great effect on the steady-state statistical properties of the system, focusing on the distributions of path lengths and free times between collisions, collision rate and velocity distribution. With the increase of the fractal dimension d f, the distributions of path lengths and free times between collisions are shown to deviate more obviously from expected theoretical forms for elastic spheres and have an overpopulation of short distances and time bins, and the probability of the short-path and short time bins increases. The collision rate increases with d f, but it is independent of time. As d f is increased, the velocity distribution deviates more strongly from the Gaussian one, but does not demonstrate any apparent universal behavior. Of course, these conclusions are based on the numerical studies of simplified models, so the universality is still to be considered as an open question from a point of experiment. Acknowledgments The authors would like to thank Prof. YU Bo-Ming for helpful discussions. References [1] H.M. Jaeger, S.R. Nagel, and R.P. Behringer, Rev. Mod. Phys. 68 (1996) 1259. [2] D.R.M. Williams and F.C. MacKintosh, Phys. Rev. E 54 (1996) R9. [3] A. Puglisi, V. Loreto, U.M.B. Marconi, A. Petri, and A. Vulpiani, Phys. Rev. Lett. 81 (1998) 3848; Phys. Rev. E 59 (1999) 5582. [4] M.A. Hopkins and M.Y. Louge, Phys. Fluids A 3 (1991) 47. [5] I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70 (1993) 1619. [6] T.P.C. Van Noije and M.H. Ernst, Granular Matter 1 (1998) 57. [7] S.J. Moon, M.D. Shattuck, and J.B. Swift, Phys. Rev. E 64 (2001) 031303. [8] D. Paolotti, C. Cattuto, U.M.B. Marconi, and A. Puglisi, Granular Matter 5 (2003) 75. [9] D.L. Blair and A. Kudrolli, Phys. Rev. E 67 (2003) 041301. [10] D.L. Blair and A. Kudrolli, Phys. Rev. E 64 (2001) 050301. [11] J.J. Brey and M.J. Ruiz-Montero, Phys. Rev. E 67 (2003) 021307. [12] A. Puglisi, A. Baldassarri, and V. Loreto, Phys. Rev. E 66 (2002) 061305. [13] R. Lambiotte and L. Brenig, Phys. Rev. E 72 (2005) 042301. [14] D.M. Zhang, Z. Zhang, and B.M. Yu, Commun. Theor. Phys. (Beijing, China) 31 (1999) 373; D.M. Zhang, Y.J. Lei, B.M. Yu, et al., Commun. Theor. Phys. (Beijing, China) 37 (2002) 231; D.M. Zhang, Y.J. Lei, B.M. Yu, et al., Commun. Theor. Phys. (Beijing, China) 40 (2003) 491. [15] D.M. Zhang, Y.J. Lei, G.J. Pan, and B.M. Yu, Chin. Phys. Lett. 20 (2003) 2221; D.M. Zhang, X.Y. Su, B.M. Yu, et al., Commun. Theor. Phys. (Beijing, China) 44 (2005) 551. [16] D.M. Zhang, R. Li, X.Y. Su, G.J. Pan, and B.M. Yu, J. Phys. A: Math. Gen. 38 (2005) 8861. [17] D.M. Zhang, Z.Y. Chen, Y.P. Yin, et al., Physica A 374 (2007) 187; Z.Y. Chen, D.M. Zhang, Z.C. Zhong, et al., Commun. Theor. Phys. (Beijing, China) 47 (2007) 1135. [18] I. Pagonabarraga, E. Trizac, T.P.C. Van Noije, and M.H. Ernst, Phys. Rev. E 65 (2001) 011303. [19] H. Xie, R. Bhasker, and J. Li, Minerals and Metallurgical Processing 2 (1993) 36; M.D. Normand and M. Peleg, Powder Technol. 45 (1986) 271. [20] L.M. Zhang, Coll. Surf. A: Physicochem. Eng. Aspects 202 (2002) 1. [21] J.P. Hyslip and L.E. Vallejo, Eng. Geol. 48 (1997) 231. [22] B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, New York (1982). [23] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, New York (1989). [24] C.E. Hecht, Statistical Thermodynamics and Kinetic Theory, Dover Publications, New York (1998). [25] P. Deltour and J.L. Barrat, J. Phys. I 7 (1997) 137. [26] J.A.G. Orza, R. Brito, T.P.C. Van Noije, and M.H. Ernst, Int. J. Mod. Phys. C 8 (1997) 953. [27] A. Barrat, E. Trizac, and M.H. Ernst, J. Phys.: Condens. Matter 17 (2005) S2429.