2 Abstract In this thesis, the numerical study on the dynamics of granular fluidization in a simple model is presented. The model consists of two-dime

Transcription

1 Phase Changes in an Inelastic Hard Disk System with a Heat Bath under Weak Gravity for Granular Fluidization A DISSERTATION SUBMITTED TO KYUSHU UNIVERSITY FOR THE DEGREE OF DOCTOR OF SCIENCE BY MASAHARU ISOBE July 2, 2000

2 2 Abstract In this thesis, the numerical study on the dynamics of granular fluidization in a simple model is presented. The model consists of two-dimensional hard disks, which undergo inelastic collisions; the system is under the uniform external gravity and is driven by the heat bath. The competition between the two effects, namely, the gravitational force and the heat bath, is carefully studied. In the simulation, the event driven method is employed. By improving the various techniques, we succeed in developing the fast algorithm; its complexity is O(log N) with N being the number of particles. This is the same with the existing algorithm but its coefficient is somewhat smaller than the previously published one. This algorithm allows us to perform the simulation over awiderange of parameter region. From the numerical simulations, we found the system shows three phases upon increasing the external driving, which is described by the ratio of the heat bath to the gravitational force. These phases are named the condensed phase, the locally fluidized phase, and the granular turbulent phase. In the condensed phase, most of the particles are aggregated around the bottom of the system with an almost closed packed density, and the state of a dense packing layer is relatively stable, which means the potential energy is dominant and the system is in weakly excited states. In the locally fluidized phase, the dense packing layer is locally broken by excitation of the heat bath. The high-speed particles are blown upward from the holes in the layer. The location of holes is fairly stable. In the granular turbulent phase, the positions of the hole become unstable in time. The average density is quite low, but it is different from the ordinary molecular gas phase. The density fluctuation is large and this fluctuation causes turbulent motion due to the gravity. The holes become dynamically unstable, and the condensed layer, appears only temporally. Over the whole region, the flatness f(y) hv 4 xi=hv 2 xi 2 at the height of the maximum packing fraction layer is different from 3, which means the velocity distribution deviates from the Gaussian. It is remarkable that f becomes very large, as large as 20, in the locally fluidized phase. The transition from the condensed phase to the locally fluidized phase is distinguished by the existence of fluidized holes. On the other hand the transition from the locally fluidized phase to the granular turbulent phase is understood by the destabilization transition of the fluidized holes due to the mutual interference.

3 CONTENTS 3 Contents 1 Introduction Background Statistical Characters of Granular Materials Density Profile Probability Distribution FunctionofVelocity Spatial Velocity Correlation Function Motivation of Our Study and Organization of the Thesis Modeling of Granular Systems Elements Characters of Granular Particles Distinct Element Method Inelastic Hard Sphere Model Non-Equilibrium Steady States D Model with a Heat Bath without Gravity D Model with a Uniform Heat Bath without Gravity D Model with a Heat Bath under Gravity D Model with a Heat Bath without Gravity D Model with a Uniform Heat Bath without Gravity D Model with a Heat Bath under Gravity IHSHG Model Advantages of IHSHG Model Some Problems of the Inelastic Hard-Core Model Inelastic Collapse Multiparticle Interactions Absence of Static Limit Algorithms for Event-Driven Method Introduction Event-Driven Molecular Dynamics with Hard Disks Interaction of Hard Disk Basic Equation of Hard Disk Molecular Dynamics Four Categories of the Efficient Techniques

4 CONTENTS Subcell Method, Linked Cell Method, Neighbor List Event List Event List Scheduling Coordinate Updating Extended Exclusive Particle Grid Method NeighborListandDynamical Upper Time Cut-off Analysis of Complexity Empirical Evaluation Some Examples Freely Cooling Process in the Inelastic Hard Disk System Hard Disk Liquid in the Elastic Binary Particles System Extension to Infinite System Summary Numerical Simulations of IHSHG model Numerical Setting System Behavior upon Changing External Driving Packing Fraction and Excitation Ratio Snapshots Collision Rate Probability Distribution FunctionofVelocity Flatness Parameter Fitting Functions of Velocity PDF Height Dependence of Flatness and Packing Fraction Spatial Velocity Correlation Function Displacement ofparticles Horizontal Averaged Particle Current in CP Surface Wave in CP Dynamics of Fluidized Holes Discussion Size Dependence Packing Fraction & Flatness Fitting Functions of Velocity PDF Dynamics of Fluidized Holes

5 CONTENTS CP-LFP transition LFP-GTP transition Various Limit of IHSHG model Concluding Remarks 93 A Error Bar Estimation of Flatness 96

6 CONTENTS 6 Acronyms BST CBT CP DEM DUTC DSA EDMD (E)EPGM ICW IHS IHSHG GTP LCM LFP LMA LRV NESS NL PDF RB SVCF TSDMD Binary Search Tree Complete Binary Tree Condensed Phase Distinct Element Method Dynamical Upper Time Cut-off Delayed State Algorithm Event-Driven Molecular Dynamics (Extended) Exclusive Particle Grid Method Independent Collision Wave Inelastic Hard Sphere Inelastic Hard-disk System with a Heat bath under weak Gravity Granular Turbulent Phase Linked Cell Method Locally Fluidized Phase Local Minima Algorithm Largest Relative Velocity Non-Equilibrium Steady State Neighbour List Probability Distribution Function Rayleigh-Bénard Spatial Velocity Correlation Function Time-Step-Driven Molecular Dynamics

7 1 INTRODUCTION 7 1 Introduction 1.1 Background The dynamics of fluidized granular systems has attracted much attention in physics communities as a non-equilibrium statistical system [1 3]. Due to the fact that an element particle in granular systems is already macroscopic, there are two major differences in their dynamics from that of an ordinary molecular system. First, thermal fluctuation does not play any role because relevant energy scales for the kinetic and potential energy are much larger than the thermal energy. Second, the dynamics is dissipative because the degrees of freedom we are dealing with are coupled with microscopic processes that are not treated explicitly. Because of these properties, granular systems require an energy source in order to be in a steady state and the external gravitational force plays an important role in their dynamics. In most experimental situations, the systems are excited by a vibrating plate. This is called Granular Vibrated Bed", in which the effect of gravity is very large [4 7]. Granular vibrated bed has been studied in many simulation works [8 16]. The convection in the vibrated bed has been theoretically studied [17]. It has been found that various patterns and localized oscillations appear as the intensity and frequency of the vibration are changed [18 22]. The freely cooling process", that is, the time evolution of the system without the gravity has been studied by many numerical simulations [23 30] and theoretical analyses [31 40]. In this situation, system has been found to show the peculiar pattern of density fluctuation, which reminds one of the galaxy distributions in the universe. Other topics for granular flow such as stratification [41], rotational drum [42, 43] and granular friction [44] have been studied. The study of granular materials requires new theoretical ideas beyond those in the standard statistical mechanics, hydrodynamics, or traditional solid mechanics. For the progress of understanding the dynamics of granules, a simple model system to realize the non-equilibrium steady state is required. A one-dimensional system with a heat bath at one end has been investigated and it has been found that the spatial distribution of particles develops a singularity in a certain situation. Therefore, the hydrodynamic description of the behavior does not appear to be possible [45]. It has been also demonstrated that the velocity distribution of the particles does not have the Maxwell-Boltzmann distribution, reflecting the fact that the system is not in equilibrium [46 48].

8 1 INTRODUCTION 8 Systems with uniform excitation, in which all the particles are agitated by the Langevin force, are also studied [49, 50] because these systems are statistically homogeneous and have a thermodynamic limit, even though such situations may not be easily realized experimentally. Large density fluctuation and non-gaussian velocity distribution have been observed in such systems [49] in the region where the effect of particle collision is larger than that of the Langevin force. Even small gravitation should have a large effect on the system with such density fluctuation, but there have been very few studies that focus on the external field effects on the system dynamics. Then since the gravitational force should have a large effect on clustering, it would be of interest to observe how the system behavior changes as the gravity sets in. This is one of the motivating factors behind this work. 1.2 Statistical Characters of Granular Materials The ordinary Navier-Stokes eq. for viscous fluid is not valid for the granular system, and phenomenological hydrodynamic theory for it is still yet to be developed. In this section, we review some of the statistical properties that have been examined Density Profile Warr et al. [51] calculated the static structure factor and the dynamical density autocorrelation function over a range of wave vectors and found that the local behavior of the vibrated granular medium exhibits strong similarities with the density fluctuations in an atomistic fluid in the thermal equilibrium at the equivalent particle packing fraction. The density profiles in the vertical axis of a vibrated bed were observed in the experiment and has been shown to obey a universal function that is independent of the phase of oscillations [52]. A simulation based on the Distinct Element Method (DEM) [53] has also shown this result. Hayakawa and Hong argued [54] that the density profile should follow the Fermi distribution with an appropriate effective temperature. The theory is examined by two-dimensional MD simulation [55] and one-dimensional experiment [56] and it has been found that the agreement with the theory is excellent.

9 1 INTRODUCTION Probability Distribution Function of Velocity In the equilibrium, the Probability Distribution Function (PDF) of the particle velocities should be the Maxwell-Boltzmann, or the Gaussian distribution, but in the granular systems, there is no reason for velocity PDF to be Gaussian. Actually, Taguchi and Takayasu [46] showed that this is the case for the first time using DEM simulation. By examining the flatness parameter (f hv 4 xi=hv 2 xi 2 ), Murayama and Sano [48] have found that velocity PDF changes its shape from Gaussian to the exponential as the packing fraction is increased. In the study of the fluidized bed, Ichiki and Hayakawa [47] have found that the velocity PDF of particles becomes power distribution. It has been established that the deviation from Gaussian is rather general feature in the granular system, but there has not been any theoretical understanding for the actual forms of velocity PDF Spatial Velocity Correlation Function The Fourier transform of the spatial velocity correlation is the kinetic energy spectrum and it is particularly interesting in the connection with the famous Kormogolov scaling law in the fluid turbulence. Taguchi found the k 5=3 spectrum in the simulation of twodimensional vibrated beds [57]. The spatial correlation functions have been studied in the two-dimensional uniformly driven system by Peng and Ohta [58], and Bizon et al. [59]. Bizon et al. found that the spatial velocity correlation function strongly depends on the types of driving force. 1.3 Motivation of Our Study and Organization of the Thesis In the conventional setting of vibrated bed, the frequency of external driving plays an important role and this gives variety of patterns phases. Experiments on pattern formation in granular vibrated bed [18, 19], have demonstrated the existence of subharmonic stripe, square, and hexagonal patterns [18], as well as localized structures called oscillons [19], as the frequency and amplitude of the vibration are varied. The patterns are similar in spatial structure to those observed in fluid dynamical systems, most notably parallel convection rolls in a thin layer of fluid heated from below (Rayleigh- Bénard convection) and standing surface waves in a vertically vibrated liquid layer (the Faraday instability).

10 1 INTRODUCTION 10 However, the vibration complicates the situations. Since energy input from the vibration depends on both the amplitude a and the angular frequency!, the accurate quantities of input energy cannot be estimated from the external control parameter = a! 2 =g, which is usually used as the control parameter in the granular vibrated bed. Therefore, it is difficult to construct the theory for the dissipative particles system in the non-equilibrium steady state. From the numerical point of view, the DEM, which is the soft-core particles model (Sec. 2.1), has been often used for the studies of granular vibrated bed. However, if we want to obtain the data for the statistical properties during the simulation, the sampling data must be taken at the same phase of external vibration cycle. Therefore, DEM simulation for the granular vibrated bed is not appropriate as the simplest model for the study of the non-equilibrium steady state. The granular vibrated bed has these disadvantages to study the statistical characters of the granular system from the aspects of both theory and simulation. Although there already have been many studies on the granular system for fluidization, most of these studies were a collection of the various phenomena from the experimental and numerical results, and these were discussed individually for each study. Therefore, it becomes difficult to capture what is the intrinsic characters of the granular system. Since the study of granular system requires new theoretical ideas beyond ordinary standard statistical mechanics or hydrodynamics, a simple model system for the non-equilibrium steady state is required for the progress. Recently, a series of works to find some universal framework have been appeared (Sec. 2.2). The onedimensional model system with a heat bath at one end for the non-equilibrium steady state has been firstly investigated by Kadanoff's group [45]. In their study, the granular particles are regarded as a collection of inelastic hard spheres that interact with each other via hard sphere potential. The assumption of an ideally hard sphere potential is also used in kinetic theory and the Boltzmann or Enskog approaches [60 63], which facilitates comparisons between theory and simulation. As denoted in Sec. 1.2, there have been a few studies about the statistical characters of particles in the granular system. Since the gravitational force should have a large effect on clustering, it would be of interest to observe how the system behavior changes as the gravity setsin.the main purpose of our study is to examine the statistical and dynamical characters of the inelastic hard disk system in the non-equilibrium steady state under gravity, and to provide extensive numerical results for theoretical studies to construct a base of the universal framework. In this thesis, we particularly focus

11 1 INTRODUCTION 11 on the competition between the effect of external driving and that of the gravitational field using the event-driven molecular dynamics simulation, systematically. The distinct element method and the inelastic hard sphere model are described in more detail in Sec In Sec. 2.2, we will review a series of previous works using the inelastic hard sphere model with the heat bath in the non-equilibrium steady states. Our model is described in Sec. 2.3, which is based on these series of works (i.e. using inelastic hard sphere model with the heat bath under gravity). Then, we clarify what are the advantages and disadvantages of our model in Sec. 2.4 and Sec. 2.5, respectively. In Sec. 3, we develop a new algorithm for the event-driven method, with which we achieve a very efficient computation and are able to simulate the model for a wide range of parameter regions. In Sec. 4, results of numerical simulations to our model are presented. Discussion on the characteristics of the phase transition are described in Sec. 5. Finally, we summarize our work in Sec. 6.

12 2 MODELING OF GRANULAR SYSTEMS 12 2 Modeling of Granular Systems 2.1 Elements Characters of Granular Particles There are two particularly different aspects from the molecular liquid, whichcontribute to the unique properties of granular systems. One is the ordinary thermal fluctuation plays no role because of the macroscopic size of particles, and the other is the interaction between particles are dissipative. Two types of simple models have been studied in the granular dynamics, which are Distinct Element Method (DEM) and Inelastic Hard Sphere (IHS) model. The DEM is established as one of tool to simulate granular materials [8]. Most experimental situations related in the granular systems can be simulated, such as convection [9, 10], heap formation, size segregation, pattern formation [13], granular friction [44]. A problem of DEM, however, is that it contains too many adjustable parameters to define interaction. The other is the IHS model. This model uses a simple discrete potential and particles collide inelastically; the collision rule is characterized only by the restitution coefficient. Because of this simplicity, many physicists use this model. Goldhirsh and Zanetti [23] firstly simulated freely cooling process using this model and they found that particles were unstable to the formation of high-density clusters. The clustering mechanism has been studied by others in detail [24, 26]. Since the IHS model has an advantage about the simplicity, the many studies from the theoretical point of view have been done. Actually, many theoretical predictions both freely cooling and nonequilibrium steady states have been studied. These studies in the non-equilibrium steady states are briefly reviewed in Sec In this subsection, we will review these two models and point out the advantages and disadvantages of these models in actual simulation Distinct Element Method Here, we describe the DEM in detail [8]. The equations of translation and rotational motion under gravity g are described by

13 2 MODELING OF GRANULAR SYSTEMS 13 m i ẍ i = X <i;j> I _! i = d 2 (d jx i x j j)(f ij n + F ij t ) m i g; (2. 1) X <i;j> (d jx i x j j)(n F ij t ); (2. 2) where (x) is a step function, x i and! i are the position and the angular momentum vector of the i-th particle with diameter d, andi is moment of inertia. F ij n and F ij t are normal and tangential contact forces, which are defined by F ij n = nff repl n n (v i v j )g; (2. 3) 8 < F ij t = non slip <μfn ij : F non slip F ij μtf ij otherwise n (2. 4) F ij non slip = tff fric t [t (v i v j )+ d 2 (! i +! j )]g; (2. 5) where v is velocity, and k and are the elastic constant and the friction coefficient of the velocity dependent damping force (i.e. viscosity coefficient), respectively. Usually, F repl = kx a is used for repulsive force between particles as the function of the overlap length x, especially the case with a = 3=2 is called Hertz force. For the tangential force Mindlin force is usually used, F fric (x; x 0 )= k 0 x a x 0b, where x 0 is the total shear displacement during a contact, and k 0 is the virtual tangential spring constant. In eq. (2.4), Coulomb friction is also considered. For simplicity, MD simulations in granular system have been often performed using the no-rotational linear spring-dashpot model(i.e. a = 1), which was proposed by Taguchi [9]. In this model, the overlap length x along the line connecting the centers of the particles obeys the linear differential equation, ẍ +2 _x +2kx =0; (2. 6) where and k are adjustable parameters. When we integrate these equations of motion eq. (2.1) (2.5) with time step, we obtain the dynamics of granules. Since DEM is solved by the time-step-driven scheme, we must introduce a finite time step. In the dynamics of dissipative particles, the effect of different time step is not negligible. This is a crucial effect to the particle behavior especially when the system is in the high density. Therefore, DEM seems to have

14 2 MODELING OF GRANULAR SYSTEMS 14 the practical disadvantage to study the correct statistical properties and dynamical behavior in the dissipative system Inelastic Hard Sphere Model The IHS model is simple one, which is often used as the reference system of molecular liquid/gas in equilibrium. An Event-Driven (ED) method is usually used to simulate the dynamics of rigid hard spheres with infinite intrinsic material elasticity, because the duration of contact is zero. The IHS model is used for two reasons: Firstly, it is the simplest system to study the instability of the dissipative particle system. The statistical properties are easy to compare with the theoretical analysis. To establish a physical picture, it is desirable to reduce the number of parameters, which are essential to the granular phenomena. Secondly, even if the typical distance between the particles is short, the ED simulation scheme is quite efficient (Sec. 3) and has smaller numerical error compared with time-step-driven method. Since the collision times are calculated directly, we can obtain the collision rate rigorously. 2.2 Non-Equilibrium Steady States To achieve the Non-Equilibrium Steady State (NESS) in the granular system, energy must be injected into the system from outside. Previous studies for NESS in the granular system may be categorized by the following six groups. Firstly, we categorize the models by their dimensionality (1D or 2D). Secondly, we consider the way in which the system is linked to the heat bath (i.e. wall boundary or uniform excited). Finally, we classify the existence or non-existence of gravity field. In this subsection, we will briefly review these models for each category D Model with a Heat Bath without Gravity Du et al. [45] studied the dynamics of quasi-elastic particles constrained to move on a line with energy input from boundaries. They showed that the system violates equi-partition of energy and the simple hydrodynamic approach fails to give a correct description of the system. Similar models were also studied from the theoretical point of view [60 63].

15 2 MODELING OF GRANULAR SYSTEMS D Model with a Uniform Heat Bath without Gravity Some papers investigated the properties of the IHS fluid that is heated uniformly by random force so that it reaches a spatially homogeneous steady state. This way of forcing was proposed by Williams and MacKintosh [64] for inelastic particles moving on a line. Since the dissipation due to inelastic collisions is not effective enough in balancing the increase of energy coming from the random kicks, the authors had to subtract the average velocity of the center of mass of the system from the velocity of each particle at every time step in order to avoid energy divergence and to conserve of the total momentum. Giese and Zippelius [65] regarded the internal degree of vibrating rods as a thermalized bath. They calculated the restitution coefficient r as a function of the relative length of the colliding rods, the center of mass velocity, and the degree of excitation of the internal vibrations. Puglisi et al. [49, 66] examined a model consisting of N particles on a ring of length L with the Langevin force. They found two regimes: when the typical relaxation time fi of the driving Brownian process is small compared with the mean collision time fi c the spatial density is nearly homogeneous and the velocity PDF is Gaussian, but in the opposite limit fi fl fi c one has strong spatial clustering, with a fractal distribution of particles, and the velocity PDF strongly deviates from the Gaussian D Model with a Heat Bath under Gravity Luding et al. [67] showed that 1D granular system also shares some of the phenomena of higher-dimensional systems such as transition from the condensed to the fluid state. Kurtze and Hong [68] studied the effect of dissipation on the density profile of 1D granular gas under the gravity. Perturbative analysis of the Boltzmann equation reveals that the correction of the density profile along the height z due to dissipation resulting from inelastic collisions is positive for 0» z <z c and negative for z>z c with z the vertical coordinate. They also obtained the expression for z c, whose value is given by mgzc k B T ο 1:1613. Using Boltzmann equation and molecular-dynamics simulations, Ram rez and Cordero [69] studied the 1D column of N inelastic point particles, in the quasielastic limit, under the influence of gravity. They showed that the physical properties (e.g. density and temperature profile) in the hydrodynamic limit (N!1) depend solely on the product

16 2 MODELING OF GRANULAR SYSTEMS 16 (1 r)n, where r is the restitution coefficient, which was confirmed by both the theory and simulation D Model with a Heat Bath without Gravity Esipov and Pöschel [70] studied the kinetic energy distribution function satisfying the Boltzmann equation and construct the phase diagram of granular systems on the 2D particle system in a circle wall at constant temperature. Their study has suggested that the granular hydrodynamics is valid for simple flows or for any flows in dilute and sufficiently elastic systems. Grossman et al. [71] studied the steady state theoretically. Their predictions compared well with numerical simulations in the nearly elastic limit. Although the system is no longer close to equilibrium even in the low-density limit, the scaling behavior of the velocity distributions showed that a hydrodynamic treatment could still be useful in describing the system in the elastic limit. The steady state of a low-density IHS gas confined between two parallel walls with same temperature is studied by Brey and Cubero [72]. For larger values of the degree of inelasticity, their theoretical predictions are in conflict with the simulation results. Their analysis suggests that the discrepancy is due to the asymptotic but divergent character of the expansion, similar to what happens when the usual Chapman-Enskog expansion is applied to molecular fluid. Soto et al. [73] also has studied the heat conduction between two parallel plates by molecular dynamics simulation of the IHS model. Their results show that Fourier's law is not valid and a new term proportional to the density gradient must be added D Model with a Uniform Heat Bath without Gravity Peng and Ohta [50, 58] studied the 2D granular system where an external driving force is applied to each particle in the system in such away that the system is driven into a steady state by balancing the energy input and the dissipation due to inelastic collisions between particles. Experimentally, steady state of clustering, long-range order, and inelastic collapse are observed in vertically shaken granular monolayers, which is correspond to the system driven by the uniform heat bath in the limit of the frequency!!1[74, 75]. In 2D the model may be considered to describe the dynamics of light disks moving

17 2 MODELING OF GRANULAR SYSTEMS 17 on an air table. A similar IHS model with random external accelerations has been studied by Bizon et al. [76] to test continuum theories for vertically vibrated layers of granular material. To investigate the long-scale correlation and structure factor, van Noije et al. [40] present a theory of the randomly driven IHS fluid in 2D simulation, and characterized its nonequilibrium steady state. As another way to realize the steady state, Komatsu [77] introduced a new observation frame with a rescaled time. This operation is the same as the velocity scaling method, which is often used as the temperature control method of the molecular dynamics simulation in equilibrium. He found that a phase transition with symmetry breaking, which is the circulation of the particles in the box with elastic walls, occurs when the magnitude of dissipation is greater than a critical value D Model with a Heat Bath under Gravity In 2D system under the gravity, Luding et al. [78] compared the results of DEM and IHS model. The system of their study is driven by vibrations like a granular vibrated bed. The results of the DEM simulation were good agreement with that of ED simulation in the fluidized regime. They also found that the results of the DEM simulation depend strongly on the time t c during which the particles are in contact. Although there exist only a few recent laboratory experiments for 2D system under the gravity, an experiment on the stainless steel particle system in two dimensions was studied by Kudrolli et al. [79]. The system was driven by the periodically moving wall and slightly inclined to apply the weak gravity. This experiment showed that the clustering occurs around the opposite side of driving wall when there is no gravity, but the cluster migrates downward when the system is inclined even by a very small angle. Hong [80] derived the theoretical equation of motion for the equilibrium density profile at a temperature T. Recently, aninteresting three-dimensional experiment under weak gravity was done in outer space [81]. They reported that in the case of a dilute granular medium, there was a spatially homogeneous gaslike regime, and observed that the pressure scales like the 3=2 power of the vibration velocity. However, when the density is increased, the spatially homogeneous fluidized state is no longer stable and displays the formation of a motionless dense cluster surrounded by low particle density regions.

18 2 MODELING OF GRANULAR SYSTEMS IHSHG Model To study the statistical properties of the granular system under an external field with energy input in non-equilibrium steady states, we propose a simple model system on the 2D inelastic hard disk system under the gravity with a heat bath. The system considered here consists of inelastic hard disks of mass m and diameter d in 2D space under uniform gravity. We employ the periodic boundary condition in the horizontal direction. The particles collide with each other with a restitution coefficient r. All the disks are identical, namely the system is monodispersed. For simplicity, we neglect the rotational degree of freedom. Because of the hard-core interaction, collision is instantaneous and only binary collisions occur. When two disks, i and j, with respective velocities v i and v j collide, the velocities after the collision, v 0 i and v 0 j, are given by v 0 i = v i 1 2 (1 + r)[n (v i v j )]n (2. 7) v 0 j = v j (1 + r)[n (v i v j )]n; (2. 8) where n is the unit vector parallel to the relative position of the two colliding particles in contact. Between the colliding events, the particles undergo free fall motion with the gravitational acceleration g (0; g) following parabolic trajectories. The system is driven by a heat bath with temperature T w at the bottom of the system; a disk hitting the bottom bounces back with velocity v =(v x ;v y ) chosen randomly from the probability distributions ffi x (v x )andffi y (v y ) [82]; ffi x (v x ) = ffi y (v y ) = s m 2ßk B T w exp ( mv2 x 2k B T w )( 1 <v x < 1) (2. 9) m k B T w v y exp ( mv2 y 2k B T w ) (0 <v y < 1); (2. 10) where k B is the Boltzmann constant. Note that the wall temperature T w here is just a parameter to characterize the external driving and is not related to the thermal fluctuation. We call this system Inelastic Hard disk System with a Heat bath under weak Gravity" (IHSHG) [83, 84] to emphasize the importance of the competition between excitation by the heat bath and the gravity.

19 2 MODELING OF GRANULAR SYSTEMS Advantages of IHSHG Model The IHSHG model is so simple that the system is completely characterized with only four following dimensionless parameters; the restitution coefficient r, the driving intensity Λ k B T w =mgd, the system width in the unit of disk diameter N w L x =d, and the particle number of the initial layer thickness N h N=N w, where L x is the system width and N is the total number of disks. A typical snapshot of the IHSHG model with the parameters (r; Λ;N w ;N h )=(0:9; 182; 100; 20) are shown in Fig. 1. Now we compare our IHSHG model with other typical dissipative system in the NESS. Rayleigh-Bénard(RB) convection is the typical continuous fluid system, where the energy source is the heat bath at the bottom, the dynamics is described by Navier- Stokes equation, and the system is driven by the difference of temperature between the top and the bottom. The dissipation comes from the viscosity of fluid. In this system, the control parameters are well-known as Rayleigh Number and Prandtl Number. There are some previous MD studies of RB convection using elastic hard disk [85 91]. y A max. HEAT BATH A Figure 1: A typical snapshot of the simulation and the average packing fraction profile as a function of height y. The parameters are (r; Λ;N w ;N h )=(0:9; 182; 100; 20). The black disks are the disks in the layer where the packing fraction reaches a maximum value. The ordinary granular vibrated beds using DEM [9 13,46,48,57], are the discrete

20 2 MODELING OF GRANULAR SYSTEMS 20 soft particles systems; the energy source is the vibration of the bottom, and the dynamics obeys Newton's equation of motion under the gravitational field. The dissipation appears between inelastic particles, and the control parameter is =a! 2 =g, where a and! are the amplitude and the angular frequency respectively. Recently Aoki and Akiyama [92] proposed that the state of granular convection involving multipairs of convection rolls is governed by a dimensionless parameter A = = a!2 ff, where D D=ff gd and ff are the self-diffusion constant and the energy diffusivity ff, respectively. Our IHSHG model is the discrete hard disks system, the energy source is the heat bath at the bottom and the dynamics obeys Newton's equation of motion under the gravitational field. The dissipation comes from inelastic collision of particles, and the control parameter is Λ = k B T w =mgd. The present system is analogous to the ordinary granular vibrated bed but simpler because it does not have an external time scale. Since the IHSHG model has no external periodical cycles and there are only four dimensionless parameters in the system, the IHSHG model is easy to deal with numerical analysis and theoretical approach. As described in Sec , there is a recent experiment [79] on the stainless steel particle system in 2D similar to the IHSHG model. Based on these experimental results, we firstly reproduced the clustering phenomenon and the behaviors of non-equilibrium steady states to simulate the IHSHG model. Consequently, we found that the IHSHG model could reproduce Kudorolli's experiment quantitatively. Note that the IHSHG model is categorized in 2D model with a heat bath under gravity denoted in Sec Some Problems of the Inelastic Hard-Core Model Although it is simple and easy to deal with, there are some problems in the IHSHG model. Since the IHSHG model based on IHS model, all problems come from essentially one reason: the potential between two colliding particles is unusually stiff. The instantaneous collisions imply that an interaction potential is zero when the particles are not in contact and suddenly becomes infinite when the distance between the colliding particles is zero. Therefore, momentum exchange occurs instantaneously. However, in a real system, the situation is different, each contact takes a finite length of time and the force is large but finite. The infinitely stiff hard-core interaction is an idealization or simplification of smooth repulsive pair-potential.

21 2 MODELING OF GRANULAR SYSTEMS Inelastic Collapse When the restitution coefficient is low (i.e., a dissipation rate is high) or kinetic energy is relatively smaller then potential energy under external gravity force, the number of collisions during a finite time often diverges. This is called Inelastic Collapse. It was first discovered while studying the 1D model system of a column of dissipative particles hitting a wall. The result of inelastic collapse is that the particles come into contact without interparticle forces or cohesion. For instance, if the restitution coefficient 0 < r < 7 4 p 3 then the collapse takes place for the system with only three particles; if 7 4 p 3 < r < 3 2 p 2 then it requires at least four particles for the collapse to take place in 1D. The occurrence of the inelastic collapse can be estimated using the product of the number of particles N and the dissipation (1 r). The effective dissipation ο = N(1 r) has a critical value of ο c ο ß above which collapse occurs. The value of ο c was calculated with the Independent Collision Wave (ICW) model [93]. With slightly different arguments using the cushion model" [94], the value was evaluated as ο c ο ln [4=(1 r)]. The ICW model seems to work better in the inelastic limit, whereas the cushion model is superior in the elastic limit. One might suspect that inelastic collapse is pathology of one dimension. McNamara and Young [95] showed, however, inelastic collapse is also present even in twodimensional system (2D). In freely cooling systems only three particles are enough to lead to the collapse, if dissipation is large enough [96]. In larger assemblies, the inelastic collapse occurs, but it involves just a few particles arranged almost along a line. This leads to the conclusion that the inelastic collapse is mainly a 1D effect [70] and that the one-dimensional predictions for the critical ο should work also in 2D. In fact, inelastic collapse in 2D unforced simulations can be predicted reasonably well by using the 1D criteria with ο =(νl=d)(1 r) (ν is the packing fraction, l is the length of the system, and d is the diameter of the disk) [24]. In a container under the presence of gravity, this expression for ο is equivalent to the number of layers of particles when the granular material rests motionless on the ground. With these boundary conditions, the inelastic collapse likely occurs for small energy input and large ο. Vibrated beds with large filling heights cannot be simulated with the IHS model [97]. The inelastic collapse is thought as artificial pathology and several proposals exist to avoid it. Particles with relative energy below a critical threshold can be merged into a cluster" by setting their relative velocity and separation to zero. If another particle hits such a cluster, the momentum transport inside the cluster takes place

22 2 MODELING OF GRANULAR SYSTEMS 22 instantaneously in the sequence of the largest relative velocity (LRV) [67]. A stochastic addition of rotational energy, as soon as the relative velocity after a collision, drops below a critical value [26]. The internal modes of every colliding particle may be agitated, and their energy is dissipated during a time-scale longer than the duration of a contact. If a particle suffers an additional collision within this time, then energy can be transferred from the internal modes back into transnational motion. At least in a freely cooling system in 1D the inelastic collapse is prevented by this method [28, 65]. A frequently used way to reduce dissipation in the low velocity regime (which has also been observed experimentally) is a velocity dependent restitution coefficient r(v) based on the assumption of viscoelastic contacts [98, 99]. The velocity dependent restitution coefficient was studied to avoid the inelastic collapse in the absence of walls and external forces [96], however, it had not been proved for other boundary conditions. Instead of feeding energy into the system, another approach is proposed to switch off dissipation if the next collision partner of a particle is detected within a critical distance c [24]. A similar argument for switching off dissipation is that two collisions should not be treated as separate events if they take place within a short time-interval t c that corresponds to the duration of a contact [67, 100]. These models mentioned above have no theoretical background, except for the approach in involving internal modes. The LRV method has no reasonable static limit where stresses can be defined. The stochastic approaches require the choice of an a priori unknown source of fluctuation energy. The velocity dependence of r was experimentally measured for binary collisions, and is not necessarily important for multiparitcle contacts. Though the quantitative study of the internal modes of a granular particle is a hot topic [101], the inelastic collapse has never appeared when the dissipation is low and the system is highly excited. Therefore, the IHSHG model in this paper does not need to take care of it Multiparticle Interactions Since the dynamics proceeds by only the event of binary collision, multiparticle interactions are impossible to realize. In a real system, each contact takes a finite time so that multiparticle contacts are possible. The difference between binary and multiparticle contacts was examined for one- and two-dimensional model systems [67, 98], and it was found that small energy is dissipated in multiparticle events compared with binary collisions.

23 2 MODELING OF GRANULAR SYSTEMS 23 Nevertheless, in the fluidization study, multiparticle collision can be thought as a rare event. Hence, we can ignore this effect Absence of Static Limit Another problem with the inelastic hard-core model is that the static limit does not exist, i.e. there is no way to represent enduring contacts between particles like sandpile. For example, in the framework of the inelastic hard-core model, a particle cannot rest motionless on the ground. More specifically, consider a realistic granular material inside a box, under the influence of gravity and in the absence of energy sources. Due to dissipation, the particles will loose energy. The potential and elastic energies will approach constantvalues while the kinetic energy tends to zero. Since a certain number of contacts are necessary to allow for a stable static configuration, the elastic energy is larger than zero. If the particles would be dissipative hard particles, the inelastic collapse will occur long before the kinetic energy vanishes so that the system will never reach a static configuration. Since in this artificial limit of the infinite elastic modulus is defined, contact forces and stresses are also not properly defined. This is an argument against the IHS model itself, which has by construction no static or zero-temperature" limit. However, since the IHSHG model has a heat bath at the bottom of the system, the kinetic energy always stay finite. In the simulation of NESS, the dynamics has little effect from this problem.

24 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 24 3 Algorithms for Event-Driven Method 3.1 Introduction The molecular dynamics method (MD) was performed for the first time by Alder and Wainwright in 1957 [102], and they subsequently published a series of works [ ]. Their simulations were done for the system with many hard disks (or hard spheres in 3D) near the fluid-solid phase-transition point, and they found that the system was crystallized despite the fact that the particle had only repulsive force. These discoveries overturned the common view of those days, and greatly influenced the development of the study in computer simulations. In the hard disk system, the dynamics consists of only collisions and straight-line movement. Since distinct events occur one after another in time, we do not need to integrate the differential equations with a constant time step based on Newton's equation of motion. The method that is based on the finite constant time step and integration with the equations of particles step by step in time is sometimes called Time-Step-Driven Molecular Dynamics"(TSDMD). On the other hand, in the hard disk system, the simulation that proceeds based on events is called Event-Driven Molecular Dynamics" (EDMD). The algorithm of EDMD simulation is completely different from that of TSDMD simulation. We need the knowledge of an advanced algorithm and a data structure to perform the efficient simulation in EDMD. The strategy of direct computation of particle-pairs results in the complexity O(N 2 ) per event for particle number N. To improve the efficiency in the simulation of the hard disk system, we have to deal with the complicated data structures for the list of future collision events. The substantial reduction of complexity in the algorithm of large-scale hard disk system was achieved by Rapaport [110, 111]. His algorithm is based on the concept of sub-cell method [112], and both Collision Event and Cell-Crossing Event are stored into Multiple-Event Time List. Then the minimum time is searched by Binary Search Tree(BST) [113]. When an event occurs, the particle-pair or the particle sub-cell respectively relevant to Collision Event or Cell-Crossing Event is deleted, and collision time for the particle relevant to the event is re-computed and new nodes are created. The BST is reconstructed by inserting these new nodes, and the minimum time is searched. With this algorithm, the averaged complexity per event becomes O(log N), and the reduction of a large amount of computation is realized.

25 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 25 However, since the algorithm of Rapaport is very complicated and difficult to understand, several algorithms to simplify a data structure and improve the efficiency in the large-scale molecular dynamics simulation were proposed in the 90s [ ]. Mar n et al. [117] developed the technique of Local Minima Algorithm (LMA) to avoid additional re-computation for Event List.When we actually schedule future event list, LMA put only the minimum event time relevant to each particle into Complete Binary Tree (CBT). In 1995, Mar n and Cordero [118] compared systematically various O(log N) searching algorithms in actual EDMD simulations. They concluded that the efficiency of Complete Binary Tree (CBT) was the most suitable for hard disks system in all density regions. If the number of particles increases, for CBT it was clearly shown that efficiency increased significantly when compared with other searching algorithms. When one compares CBT with BST, their complexity are the same, i.e., O(log N); but the algorithm of CBT is much simpler than that of BST, consequently CBT requires less memory space and realizes better actual performance. In this section, we developed an algorithm based on a strategy different from that of Cell-Crossing type [119, 120]. The algorithm is extended to Exclusive Particle Grid Method (EPGM) developed by Form et al. [121], (Sec. 3.4). Then, a bookkeeping method [122] is applied (Sec. 3.5). Compared with the Cell-Crossing type, our algorithm extended the concept of Linked-Cell Method [123, 124] and Neighbor List, which are often used in TSDMD to carry out an efficient simulation [125]. From the analysis of complexity, we show our algorithm is smaller than the complexity of Cell-Crossing type. By empirical evaluation of the simulation in hard disk systems, our code will be shown to perform better than that of any previously published works. The extension of the sub-cell method to an infinite volume system is usually considered impossible because the required memory is proportional to the volume [115]. We developed the method of compressing information on the infinite sub-cells into finite arrays. In addition, the hashing method, which is considered the most efficient searching algorithm, is applied to our method in order to pull out the information on a neighbor cell in high speed. It is found that the hashing method can be applied especially easily on our method. Various applications in a very wide-ranging field will be possible by changing the external field and the collision rule in the large-scale EDMD. Typical examples performed by EDMD so far are: phase transition in the equilibrium system (solid-liquid transition and two-dimensional melting) [ ], the nonequilibrium fluid system (e.g., Rayleigh-Bénard convection) [85 91], the nonequilibrium

26 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 26 chemical-reaction system [ ], the nonequilibrium dissipative system (granular system) [21 26,70,95, ], random disk packing [138, 139], glassy system [ ], polymer system [140, 141]. In many cases, however, their physical interpretations are seriously limited because of the inability to perform the large-scale simulations, therefore, any improvement of efficient algorithm for EDMD becomes very important. The outline of this section is as follows: in Sec. 3.2, basic equations of molecular dynamics for the hard disk system are described. The various efficient techniques for EDMD are reviewed in Sec In Sec. 3.4 and Sec. 3.5, algorithms developed are explained. The comparisons with the algorithm of Cell-Crossing type by analyzing the complexities are shown in Sec The empirical evaluation is given in Sec Some examples are shown in Sec The extension to the infinite system based on developed algorithms is explained in Sec In Sec. 3.10, a short summary of this section is presented. 3.2 Event-Driven Molecular Dynamics with Hard Disks The motion of hard disk is unusual insofar as velocities change only at collision times; between collision times, the particles move along straight lines at constant speeds. The velocities after a collision are obtained analytically from the positions and velocities of particles before the collision Interaction of Hard Disk The potential function for Hard Disks is discontinuous and is given by ffi(r) =f 1 0 <r ij» ff ij (3. 1) 0 r ij >ff ij (ff ij = ff i + ff j ); where ff i and ff j represent the radii of disks i and j. When the distance of two hard disk r ij between i and j equals ff ij, a collision occurs. Although hard-core potential is artificial one, this has been often used by physicists because of its simplicity to confirm the result from the theory.

27 3 ALGORITHMS FOR EVENT-DRIVEN METHOD Basic Equation of Hard Disk Molecular Dynamics Instead of integrating the ordinary differential equation of motion with a finite time step, the dynamics of hard disks are obtained simply by solving algebra equation. Now we denote the velocities of disks i and j are v i and v j, respectively. The time between collisions is described by t c, and we define the velocities after the collision as v 0 i; v 0 j. Since a particle moves at a line except for the collision, the displacement of particle r i (t) is given by r i (t) =r i (0) + v i t (t <t c ); (3. 2) where r i (0)isthereference position of particle at a last time of the collision. We determine the collision time. Now the relative position and the relative velocity are defined as r ij = r i (0) r j (0) and v ij = v i v j, respectively; b ij = r ij v ij is also defined. At a moment of the collision, the following equation is satisfied; From eq. (3.2) and eq. (3.3), the collision time is given by jr i (t c ) r j (t c )j = ff i + ff j : (3. 3) q t c = b ij b 2 ij vij(r 2 2 ij ffij) 2 : (3. 4) vij 2 The velocities after the collision can be obtained from both the momentum and the energy conservation law; v 0 i = v i 2b ij ff 2 ij v 0 j = v j + 2b ij ff 2 ij m j ( )r ij ; (3. 5) m i + m j m i ( )r ij : (3. 6) m i + m j This is called Collision Rule". In dissipative systems, these collision rules have to be rewritten with the restitution coefficient r. The dynamics of hard disk system is simple. If we want to simulate the hard disk system, we only compute the collision times between all pair of disks in the system and then search the minimum time among them. After that, all disks are proceeded by its minimum time. This iteration process can by principle simulate the hard disk system. However, this strategy is inefficient and needs a lot of computation because

28 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 28 the complexity of this process is O(N 2 ). If the system has a large number of particle, the computation will not finish in a finite CPU time. 3.3 Four Categories of the Efficient Techniques In this subsection, various efficient algorithms in the hard disk system are reviewed in detail. The efficient techniques for the EDMD simulation are categorized into the following four subsection Subcell Method, Linked Cell Method, Neighbor List First, we review the various methods to reduce the candidates of colliding particle-pair. We know the following tendency that the collision time is relatively small when the distance between particles to collide is short. Therefore, it is appropriate that the only nearest particles are adopted as the candidate. If the computation of the collision time between particle-pair is reduced, the efficiency will increase. Erpenbeck et al. [112] proposed the method that the system divided into small cells in the EDMD simulation(subcell Method). In their algorithm, the length of small cells much larger than the diameter of particle, and only particles along to the current cell and eight adjacent cells are regarded as neighbor particles in 2D. This is called Linked Cell Method (LCM)" [123, 124]. In this method, the complexity becomes O(N) instead of O(N 2 ). This method is often used in the large-scale molecular dynamics simulation. Erpenbeck et al. [112], Rapaport [110] developed an efficient algorithm in EDMD simulation based on this Subcell Method. It is an idea that the system divided into sub-cell and Link List is constructed by LCM, and they regarded the event as not only the collision time between the particles registered by Link List but also the time which the particle moves from the current sub-cell to the neighbor sub-cell. Therefore, there are two type of events, which are called Collision Event" and Cell-Crossing Event", respectively. When the Cell-Crossing Event occur, except for the sub-cell boundary which the particle have crossed, the times until the particle will cross another three boundaries of sub-cell are computed. These three times are regarded as the Cell-Crossing Event times. These techniques have been established as the standard technique to compute the efficient EDMD simulation [ ]. On the other hand, another conventional efficient algorithm development by Verlet [122], called Bookkeeping Method" with Neighbor List(NL) in TSDMD were ap-

29 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 29 plied by Smith et al. [141] to EDMD in polymer system, and they also realized high efficiency computation Event List In this section, we do not use Subcell Method for simplicity to explain the concept of Event List". As mentioned by the end of Sec. 3.2, we must iterate N(N 1)=2 times computation to obtain the collision times for each event because all particle-pairs in the system must be listed to collide for searching the minimum collision time. This strategy is not suitable to simulate the large-particle number system. To overcome this difficulty, the technique to store the collision time and its collision pairs into lists was proposed by Alder et al. [103]. This lists are called Future Event List", which are updated by every event. Only the collision times relevant to the particles to collide are calculated again, and these are re-stored to the Future Event List. Since the complexity using Future Event List becomes O(N), the efficiency increases drastically. There are two types of Future Event List. One is that all particle-pairs are stored into the list. If there are N j candidates for particle j to collide, all particle-pairs are stored into the list, whose size becomes N N j. This type of Event List is called Multiple-Event Time List". If we use Multiple-Event Time List, since the size of list is unknown, we must evaluate the size of Future Event List by trial and error. N j Therefore, if possible, we want to cut down the Future Event List to the fixed size. Additionally, the reduction of the size for Event List becomes the large amount of the reduction both the computation and the memory use. The other Future Event Lists, which are a fixed size, are proposed by several workers. Shida et al. [115, 142] proposed the method of cutting down the size of Future Event List to 2N. Then, Lubachevsky [114] proposed the method of decreasing the size to N, which was stored by only the minimum collision time for each particle i. Allen et al. [125] also proposed the same method in their source code of their book. This list is called Single-Event Time List". The memory use of Single-Event Time List is much smaller than that of Multiple-Event Time List, and a data structure is very simple. This simple data structure has the great merit, because the Event List is easy to be dealt with, especially when we want to use additional efficient techniques based on Future Event List such as the Event List Scheduling reviewed by Sec However, the method of storing only the minimum collision time for each particle into the list reduces too much information. Therefore, the event time and its partner of

30 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 30 another particles, which is not relating to the event, must also be re-computed. Mar n et al. [117] proposed the technique of Local Minima Algorithm (LMA) to avoid this additional re-computation of event time. Their method is, firstly, Multiple-Event Time List is constructed. Next, at the stage of scheduling Event List, only the minimum event times for each particle i are listed into the queue. They compared LMA with two types of Event List (i.e., Multiple-Event Time List and Single-Event Time List), and insisted that LMA was more efficient than others. Note that when we use Subcell Method, Cell-Crossing Event is also stored into Event List as the same thing of Collision Event Event List Scheduling After the Event List is constructed, we must search the minimum event time from the list. The simplest method is the linear searching method, which we compare the time from the beginning of the list one after another. The complexity of this method is O(N). In 1980, Rapaport [110] proposed the revolutionary strategy of searching the minimum time in EDMD adopted by the Binary Search Tree (BST), which is known as the most efficient method to obtain the minimum or maximum value from the list. Rapaport succeeded in the improvement of an algorithm in the hard disk system, and he achieved more efficient computation than previous methods. This algorithm was developed on the base of Subcell Method, and both Collision Event and Cell-Crossing Event were stored into Multiple-Event Time List. Then BST was constructed. Note that each element of Event Time List was regarded as nodes. When the event occurs, the nodes about the particle-pair or the particle sub-cell respectively relevant to Collision Event or Cell-Crossing Event was deleted, and the event times relating to the event particle was re-computed, and this new nodes were inserted into BST. Then the next minimum time were obtained. (i.e., the most left node of BST is the minimum event time.) On this algorithm, the averaged complexity for 1 Event becomes O(log N) instead of O(N), and realized the reduction of the large amount of computation. Note that the bottom of log is always 2 in the field of algorithm. However, the algorithm of Rapaport is very complicated, and one node needs too many arrays (i.e., two integer arrays as the Future Event List, which store the particlepair numbers or the particle the sub-cell boundary, and three integer arrays for pointer which connect the nodes of BST, and four integer arrays for pointer(circular

31 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 31 List) which delete the invalid nodes, and one real number array in order to put the collision time). After the event occurs, the branch of the node must be changed to delete invalid nodes of BST. This is very complicated, and it is difficult to code this algorithm. This had been the big barrier to study the hard disk system using EDMD simulation. In 1991, Lubachevsky [114] proposed the strategy using Implicit Heap instead of BST. In 1995, Mar n and Cordero [118] compared various O(log N) searching algorithms actually in EDMD simulation, systematically, which were Complete Binary Tree, Implicit Heap, Binary Search Tree, Pairing Heap, Skew Heap, Splay Tree, Binary Priority Queue, Leftist Tree, Binomial Queue, Priority Tree. From the empirical computation, they found that the Complete Binary Tree (CBT) was the most efficient for hard disks system. Moreover, CBT was showed that the efficiency increase significantly from other searching algorithms when the particle number is increased. CBT is the strategy of arranging Event List in a line, and performing the tournament well known as a sport event. For simplicity, we use Single-Event Time List. When Event List is constructed, the event times in the list are compared with the neighbor one, the smaller one becomes a winner and goes up to the upper level. By repeating this process, the minimum event time will remain in the root after N 1 times comparison, and a winner becomes the root. (The complexity of CBT construction is O(N)) When an event occurs, since Event List is changed only lists corresponding to the event particle, we can use the previously constructed CBT structure. After the recalculations of the event time, they are reinserted into the bottom level of CBT. Then, we go up only the changed event list to the upper level toward a root, and we obtain the new CBT winner. Above this, the complexity of CBT is always O(log N). On the one hand, when the node structure of BST has a regular arrangement, the complexity becomes the worst case O(N). In this point, CBT is always much efficient than BST. In addition, CBT has an additional advantage to BST. We do not need the pointers, which connect the nodes such as BST. Here, the nodes corresponding to the local minimum event time for each particle are stored in the size of 2N 1 array from the index number n to 2n 1. Except for the bottom level, the node n in CBT has the children both index number 2n and 2n +1. Moreover, the upper level node can be expressed by [ n ], which is the winner between n and n +1. Since any pointer arrays 2

32 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 32 for connecting a searching tree is not needed, the reduction of a memory is realized. When Single-Event Time List is used, not only the insertion and deletion of new event nodes but also a re-connecting of the branch do not need in CBT. A data structure of CBT becomes quite simple and the difficulty of coding is reduced very much. When we compare the BST of Rapaport and CBT of Mar n et al., although the complexity is the same order O(log N), a simplicity, efficiency, and memory reduction are much different in the EDMD simulation Coordinate Updating After searching the minimum time t min in the Event List, we need to proceed the dynamics by t min. The particles related to the event are treated according to its kind of the event (i.e., Collision or Cell-Crossing) and the rest of particles are only moved in a straight-line by eq. (3. 2). However, since the complexity of this coordinate updating process is O(N), a large amount of CPU time is needed when we actually want to simulate in the large particle system. Since the method of changing all particle positions for each event is inefficient, Erpenbeck et al. [112] proposed the efficient algorithm that the particle related to the event is treated at a moment and the rest of particles are left. In the paper of Smith et al. [141], this algorithm is denoted as the efficient algorithm called False Positioning". The detail of this method is that the dynamics of particles, which collide each other, are reversed from the current positions by the collision time t c after the collision. Therefore, the times of all particles in the system are always fixed at the particular system time (for example, t = 0). An alternative approach proposed by Rapaport [110], which is the method to assign the local time to each sub-cell. Mar n et al. [117] also proposed the similar method to assign the local time to each particle, which is called Delayed State Algorithm(DSA)". In these methods, since the computation of next collision time needs a particle position at the system time, not the local time, we must carefully proceed the neighbor particles to the system time. At the end of simulation, we obtain the collect positions of all particles to proceed positions by the time between the difference of local time of each particle and the system time. The False Positioning has a little excessive part of computation than the Delayed State Algorithm. However, since the whole amount of computation is hardly influenced, it can be regarded as the same efficiency. On the other hand, False Positioning has

33 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 33 the advantage with memory use because it is not necessary to prepare the additional arrays for storing the local time of each particle or sub-cell. These techniques realize the complexity of Coordinate Updating per 1 event becomes O(1) instead of O(N). 3.4 Extended Exclusive Particle Grid Method The sub-cell method is often used to achieve the significant reduction of computation. In the EDMD, the complexity of producing and updating future event list for restoring event time of each particle-pair are reduced to O(N) and O(1), respectively, by the sub-cell method. LCM is the method of dividing the system into small sub-cells. When the size of the sub-cell is bigger than a particle diameter, we will have a difficulty in coding a program, because we do not know how many particles enter in each sub-cell. Therefore, link lists must be prepared for the particles in each sub-cell. On the other hand, another efficient sub-cell method, called Exclusive Particle Grid Method (EPGM), was independently developed by Buchholtz and Pöschel (1993) [143] and Form, Ito, and Kohring (1993) [121] to simulate the soft-core granular particle system in TSDMD. In this method, there is only one particle in each sub-cell. Though the EPGM is essentially the method of putting a particle to one sub-cell in LCM, it does not need to use pointers for neighbor sub-cells or link list. Here sub-cells in EPGM are called grid". In the EPGM, the length of grid l gx is determined by the following inequality: ff<l gx < p 2ff; (3. 7) where ff is the radius of particle. One example of the number of grids n gx ;n gy in the system of length l x ;l y and the length of grid l gx ;l gy can be calculated by the following equations. l gx = INT(l x =( p 2ff)) + 1 (3. 8) l gy = INT(l y =( p 2ff)) + 1 (3. 9) n gx = l x =l gx (3. 10) n gy = l y =l gy : (3. 11)

34 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 34 Figure 2: A typical example of the mapping pattern using EPGM when packing fraction ν =0:70. Both the position of hard disks and the occupied lattices are shown. Note that the total number of grids is n g = n gx n gy. As an analogy to the lattice spin system, EPGM is regarded as N + 1-states Potts model of the square lattice system. This is because the particles are completely mapped into each lattice (i.e., one grid corresponds to one particle respectively), and we put 0 into the rest of grids in which there is no particle (Fig. 2). Since continuous and random positions of the particles are mapped into the lattice, the specification of neighbor particles becomes very easy. Form et al. [121] applied this algorithm in the high-density soft-core granular system with short-distance interaction in TSDMD. They achieved high efficiency on a vector computer. Based on this algorithm, the extension of EPGM to EDMD in the hard disk system is developed. When the system is in high-density, EPGM can be simply applied to EDMD. For a candidate of the next colliding particle-pairs, we need to search only 24 neighbor grids, which form the square mask. If neighbor grid is not 0, the collision times of candidates of colliding particle-pairs are computed only by the registered particle number in the square mask. We call these 24 neighbor grids MIN, because this is the minimum mask

35 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 35 Figure 3: Minimum mask (MIN) and larger masks, which are generated by the algorithm of making circle on the computer display, are shown. The solid lines in each mask denote the minimum distance from the central lattice of mask to the frame of mask. in the simulation of EDMD. Note that if the smaller mask is used, the computation will break down during the simulation, since a possibility of overlap between a central particle in the mask and particles outside the mask occurs. When EPGM is applied in the high-density system, the computation is optimized because a sufficient number of particles contained in the mask MIN. These are enough to be candidates of particlepairs of collision, and only the required particles are registered in the mask MIN; the efficiency increases as a result of the computational reduction of collision time for neighbor particle-pairs. On the other hand, when the system is of low density, no sufficient number of particles as candidates of collision is registered in the mask MIN. Under such a situation, the computation will break down, since collision time between the central particle in the mask and particles out of the mask will become the next minimum time. In order to prevent this breakdown, the extension of EPGM is developed. The region must be

36 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 36 extended to look for candidates of collision particle-pairs bigger than MIN. Since the rigorous isotropy of neighbors in EDMD is not necessarily demanded, the shape of the mask might be approximated by a rough circle. It is found that the mask approximated by the lattice-like grid with the circle can use the algorithm describing the circle on the discrete space of computer display. Figure 3 displayed the circles from R= 3 to R= 10 on the discrete space of computer display, which MIN is also showed over Fig. 3. The total number of neighbor grids (mask) is 24 (MIN), 36 (R=3), 68 (R=4), 96 (R=5), 136 (R=6), 176 (R=7), 224 (R=8), 292 (R=9), 348 (R=10), respectively. This extension of EPGM is called Extended Exclusive Particle Grid Method (EEPGM). Compared with LCM, EEPGM is simple, because rough neighbor particles can be simply regulated by grids so that only the necessary minimum may be taken. In EEPGM, since link list and pointers for neighbor sub-cell is not necessary, the required memory is also sharply reduced, the number of operations for setting EEPGM is small, and the program becomes very simple. Moreover, the extension to the infinite volume system is easy when using a hashing method to EEPGM. This is explained in detail in Sec Neighbor List and Dynamical Upper Time Cut-off Though EEPGM produces a significant reduction of computation compared with considering all particle-pairs, it is inadequate when large number of particle simulation are performed. In this subsection, the next step in the strategy of the increase in efficiency based on EEPGM is developed. Here, the concept of Neighbor List (NL) [122] is adopted. Since the grids correspond to each particle, we can regard the registration of Neighbor List as already being completed. Therefore, we need only to search neighbor particles along in the form of a mask. In the usual way of LCM+NL, after the system is divided into sub-cells, particles are listed into the link list. Then neighbor particles within radius r NL are entered into Neighbor List from the link list. However, since the length of both lists is unknown, they must be estimated by trial and error. Although registration of NL is completed by one operation in EEPGM, LCM+NL must use two different unknown size lists, which is accompanied by a complicated procedure and requires difficult programming. Therefore, EEPGM (+NL) is simple, which means that both debugging and extension do not require excessive effort; moreover, only the minimum nearest particles can be seen, because the system is divided into the minimum sub-cells, i.e., the grid. Since

37 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 37 MIN. R=3 R=4 R=5 R=6 R=7 R=8 R=9 R=10 p p p p p p p p r min =l gx Table 1: The minimum distance for each mask. registration of NL is completed by one operation in EEPGM, efficiency is better than LCM+NL at a result. The next improvement in speed is that the computation of collision time only from particle-pairs which are registered in the mask of EEPGM during the time t NL. Then the complexity of EEPGM for every event is reducible to O(1) instead of O(N). After time t NL, the grids are again re-mapped in order to update neighbor particles. The time of Neighbor List t NL must be determined such that the central particle does not collide with a particle completely out of the mask. If t NL is too long and such a event happens, the program picks up a wrong pairs of particles to collide, which could result in negative collision time. This conventional determination of t NL needs a lot of trial and error. In order to overcome these difficulties, t NL is determined by the following procedure. First, after completing EEPGM, the maximum velocity v max in the system is searched, and the value of its velocity is restored (the complexity of this searching is the same order of EEPGM O(N)). Next, time t NLmin of the system is calculated. In this calculation we suppose both the central particle and the particle out of the mask have the maximum velocity and those particles undergo head-on collision. If t NL = t NLmin, a counting mistake of collision pairs in the system never occurred during the time t NLmin. The minimum NL distance r NLmin is required when t NLmin is calculated, which becomes clear from the geometry of the adopted mask shown in Fig. 3. Therefore, t NLmin is given by t NLmin = r NLmin 2ff max 2v max ; (3. 12) where ff max is the maximum radius of particle in the system. The minimum distances r min for each mask are shown in Table 1. When we simulate in the equilibrium system, this strategy will work well because t NLmin hardly changes. However, in the nonequilibrium system (e.g., the relaxing process or the system with heat bath) it breaks down because the maximum velocity changes drastically at every step. To overcome this difficulty, we must check the max-

38 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 38 imum velocity for each event with energy increase. Fortunately the complexity of this checking process is O(1). Therefore, in the simulation of the nonequilibrium system t NL will change one after another. We call this change in t NL techniques Dynamical Upper Time Cut-off (DUTC). The development of DUTC, EEPGM became applicable in the nonequilibrium system. Although we do not need to update the grid pattern for a long time in the highdensity system, we should often do so when we use the mask MIN in low-density systems because the grid pattern changes drastically. In order to reduce the frequency of updating grid pattern, we can only use a bigger mask. 3.6 Analysis of Complexity Analysis of complexity is one of important factor in estimating the efficiency of algorithms. In this subsection, a comparison of analysis of complexity between the algorithm of the Cell-Crossing type and the EEPGM + DUTC is shown. The difference between the algorithm of Cell-Crossing type and the strategy of EEPGM + DUTC is Cell-Crossing Event itself. Therefore, especially with regard to this point, both complexities with a constant coefficient k are estimated in the case of A N collisions being actually simulated. Note that the particle number N is supposed to be quite a large number and the techniques of improvement in the speed are also used in both algorithms described by Mar n et al. [117], ffl Cell-Crossing type (LCM + Cell-Crossing Event) The initial and last step ( 1) Linked-Cell Method k LCM N Computation of Event Time k PP 9 N c N + k PC 4 N Construction of Complete Binary Tree k CBT N Update the final position of particles k UPDATE N Iteration Step (loop) (A N) (k PP 9 N c +k CBT log N)+(B N) (k PC 3+k CBT log N) where N c is the number of particles per sub-cell. The most important point is that the Cell-Crossing Event occurs at a certain ratio to Collision Events. Therefore, the additional computation of B N times Cell-Crossing Events are needed when

39 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 39 we want to simulate A N times Collision Events. Since the complexity of the terms related to Cell-Crossing Events is always O(N), this is not negligible in the actual simulation. ffl EEPGM + DUTC Update of EEPGM ( C) EEPGM k EEPGM N Computation of Event Time k PP N g N Search the Maximum Velocity k MV S N Construction of Complete Binary Tree k CBT N Update the final position of particles k UPDATE N, Iteration Step (loop) (A N) (k PP N g + k CBT log N) where N g is the averaged particle number of the mask. Actually the value of C is an order C ο A, which is the same order of the frequency of updating EEPGM. Now, the comparison with order N in both algorithm with constant coefficient is as follows: ffl Cell-Crossing type (k LCM + k PP 9 N c (A +1)+k PC (3 B +1)+k CBT + k UPDATE ) N + (A k CBT + B k CBT ) N log N ffl EEPGM + DUTC (k EEPGM +k PP 2 N g +k MVS +k CBT +k UPDATE ) C N +(A k CBT ) N log N The most striking point is that the complexity of Cell-Crossing Events is of order of O(N log N) (i.e., B k CBT N log N). This result of analysis suggests that the efficiency of EEPGM + DUTC is better than for Cell-Crossing when the simulation with an enormous number of particle is performed. On the other hand, in the comparatively small particle system, the coefficient of C N terms in EEPGM + DUTC may be larger than Cell-Crossing type. However, the difference might be quite small, and it is impossible to estimate an exact coefficient of algorithms analytically. To the author's knowledge, the coefficient of N terms is strongly dependent on the ratio of

40 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 40 N c to N g, and the rough estimation shows both algorithms are same when N g =N c ο 4. Actually, the increase in computing the Cell-Crossing Events has almost no effect to the CPU time as the simulation is actually performed in a system with very large number particles. 3.7 Empirical Evaluation When actually performing the simulation, the order of the complexity is not so reliable, because it depends strongly on a constant coefficient when the number of the particle is relatively small. Moreover, the efficiency of the code changes significantly by the ability of a computer, the complexity of algorithm, the performance of compiler, and the ability of programmers. Though a perfect comparison of efficiency of codes developed by the past workers is impossible, some publications showed how many particle collisions per CPU-hour can be computed with their codes on their computers. Mar n et al. [117] simulated with hard disk system, and they also compared their code with the codes based on two main high-speed algorithms proposed by Rapaport [110] and Lubachevsky [114], respectively. As a result, it was shown that the efficiency of the code of Mar n et al. was higher for the entire density region. Therefore, the code proposed here should be compared with the code of Mar n et al., and is computed with equal number of particles (N = 2; 500). Mar n et al. achieved the maximum performance of millions of collisions per one CPU hour on a SUN690 workstation. Note that only the highest performance is shown because it is different for different densities of the systems. On the other hand, the code proposed here achieved a maximum performance of 460 million collisions per one CPU hour (Alpha600 compatible, DEC-Fortran). It was found that high efficiency was realized even if the performance of the machine was reduced. Note that the workstation of our laboratory could actually simulate a 2,500,000- particle system. In this simulation, the amount of installed memory was 250 megabytes and the computation performance was 210 million collisions per one CPU hour. Figure 4 shows that size dependence of the number of collision, calculated during a CPU hour by efficiency of our code on Alpha500 compatible using g77 compiler. Since the horizontal axis was scaled by logarithmic, we found that the complexity of our code was actually O(log N), which was consistent with the analysis discussed by Sec. 3.6.

41 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 41 Million Collisions per CPU Hour VT Alpha 500 g77 O Particle Number Figure 4: The number of collisions calculated during acpuhour as a function of the particle number. 3.8 Some Examples In this subsection, some examples using our code will be shown on the large-scale hard disk molecular dynamics simulation Freely Cooling Process in the Inelastic Hard Disk System Here, the freely cooling process of the inelastic hard disk system is actually simulated. The inelastic hard disk system is drastically different from the elastic equilibrium system because the energy dissipation occurs by the collision. We fixed the restitution coefficient r(= 0:9), bywhich the normal direction of the relativevelocitywas decreased at a moment of collision between the particles. The two simulations were done with the number of the particles N = 1; 000; 000 and packing fractions were ν = 0:20 and 0:75, respectively. The snapshots after the collision number per particle C=N = 100 are shown in Fig. 5. These large-scale EDMD simulation indicates that vortex or string like structure appears, which can never be seen in the small system. In these simulation, the amount of a memory use was about 100 megabyte and the total CPU time

42 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 42 was about 30 minutes (in Alpha600 compatible, Linux, Dec-Fortran). Figure 5: The snapshots of the density fluctuation in the freely cooling process of inelastic hard disk system. The particle number is N = 1; 000; 000, the restitution coefficient is r = 0:9, the packing fraction are ν = 0:75(left) and ν = 0:20(right), respectively Hard Disk Liquid in the Elastic Binary Particles System The dense liquid composed of binary particles was much studied by soft-core MD simulation in the supercooled state. Although there are many theoretical studies in hard-core binary liquid state at higher density, MD simulation in hard-core system is very few. Here, we consider the similar model liquid introduced by Froböse et al. [136], which contains an binary mixture of 4; 096 hard disks with a size ratio ff 1 : ff 2 =1:1:3. The packing fraction of the system is ν = 0:77 with periodic boundary conditions. Note that time t is scaled by the unit time fi, whichischosen to make the mean-square velocity equal to unity. The displacement pattern obtained from our simulation is shown in Fig. 6. This is quite similar to the Fig.1 of Froböse et al. [136]. We found the clear collective flow exists.

43 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 43 We performed the same simulations for larger system; the system size N was changed from 4; 096 to 10; 000. In N = 10; 000 particles system, the supercooled binary liquid of soft" particle were performed by Muranaka and Hiwatari [137]. They introduced the correlated motion coefficients" to calculate the collective flow quantitatively within the cut-off of neighbor particles. The correlated motion coefficients is defined by CM i (r c ;t)= 1 N i XN i j=1 r i (t) r j (t) ; (3. 13) r 2 (t) where r i (t) is the displacement vector of the i-th atom for an elapsed time t, and r 2 (t) is the mean square displacement of an elapsed time t. N i is the number of atoms within the distance r c centered at the i-th atom for t =0. In Fig. 7, the initial pattern of binary particles are shown. In this simulation the segregation cannot be found. The spatial pattern of CM i (2:0ff 1 ; 30fi)isshown in Fig. 8. We also show the same flow pattern using HLS color model to compare the pattern of CM i in the right of Fig. 9. It is obviously highly correlated cluster exists, which is shown in soft particle system. These simulations take only 30 seconds in Alpha600 compatible, Linux, Dec-Fortran. This is very efficient compared to the soft-core time-dependent MD. In this subsection, we showed that a large-scale computation became really possible based on our algorithm proposed here in a small workstation of our laboratory. 3.9 Extension to Infinite System In this subsection, a simple example in the open boundary system that does not have a ceiling is considered. This is the case when there is an energy source at the bottom of the system under uniform gravity. The system is also divided into grids by EEPGM. However, because the top of the system is open, the grid goes to the top of the system infinitely. This means that the number of arrays for the grid becomes infinity, and the simulation is impossible from finite memory. To overcome this difficulty, the hashing method, which is well known as the fastest searching algorithm, is applied to keep the number of arrays in finite size and to simulate the dynamics of the system with high-efficiency. ffl Construction of Data Structure

44 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 44 Figure 6: The typical example of displacement pattern after the collision number C=N = 670 in N = 4; 096, which is reported by K. Froböse et al. Figure 7: The snapshots in the initial condition of the particles. The dark circles and white circles are small and large particles, respectively.

45 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 45 Figure 8: The correlated motion coefficient CM i (2:0ff; 30:0fi) is shown in N =10; 000.

46 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 46 Figure 9: The same displacement pattern is displayed using HLS color model.

47 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 47 Firstly, the serial number of grid N G (i g ;j g ) is defined by N G = N gx (j g 1) + i g ; (3. 14) where N gx and i g (1» i g» N gx ) are the total number of grids and the index of grid in the horizontal direction, respectively; j g (1» j g»1) refers to the index of grid in the vertical direction. In addition, the maximum number of the serial grid N Gmax is calculated by N Gmax = N gx (j gmax 1)+i gmax ; the pair (i gmax ;j gmax ) is at the maximum grid pair of containing the particle. The serial grid N G (1» N G» N Gmax ) is a one-dimensional array,inwhich particle number or 0 is listed. This is called the one-dimensional Virtual Array. When j max is large value, there are many 0's in the one-dimensional Virtual Array. Though there is only information that a particle is not just in grid, this 0 relates to the memory capacity directly. Now Virtual Array is compressed. After all, the only information necessary is the particle number and its index of grid. Therefore, one-dimensional integer arrays are prepared for A(N);BX(N);BY(N), and grids of 0 in Virtual Array are ignored, and then packed in order from the end; A(N) stores the particle numbers only,andbx(n) andby (N) stores indexes of i g and j g for each particle, respectively. If you want to know whether there is a particle in grid (i 0 g;j 0 g), you need only search the index-pairs correspond to (i 0 g;j 0 g) in the lists of BX(N);BY(N) linearly. However, this process is inefficient because the complexity of O(N) is necessary in searching. Therefore, the hashing method, otherwise known as an algorithm which can realize the searching with O(1) is applied. The following simplest hashing function is explained here as an example though various hashing functions can be considered and there is still room for development. First, a hashing function is defined by k = INT( N G 1 )+1; (3. 15) L which means that the Virtual Array is equally divided by the length L (e.g., 5 ο 10) and key k is calculated corresponding to serial number N G. Then, k max = INT(N Gmax =L) is calculated using N Gmax. Then additional arrays

48 3 ALGORITHMS FOR EVENT-DRIVEN METHOD 48 C(k max );D(k max ) are prepared, these arrays are restored in C(k) where k begins in A(N) and in D(k) how big the arrays for each k. In this case, necessary arrays are A(N);BX(N);BY(N);C(k max );D(k max ), and these are confined to a finite value. Since additional arrays needed to use the hashing method are only C(k max );D(k max ) (k max <N), the amount of memory used is only slightly increased in comparison with the linear searching. ffl Searching Process In order to know what the particle number is in the grid (i 0 g;j 0 g), the following process is carried out. First, a serial number is calculated by N 0 G = N gx (j 0 g 1)+i 0 g. Next, k 0 is calculated by the equation of hashing function (3. 15). Then, the searching ranges of BX;BY are limited only to index of [C(k 0 ) ο C(k 0 )+D(k 0 ) 1] (» L). If there are equal pairs of grids (i 0 g;j 0 g) as a result of the searching of BX;BY, the index s of BX;BY reveals A(s), which is the particle number in the grid (i 0 g;j 0 g). When equal pair is not found in BX;BY, there is no particle in the grid. This procedure becomes possible in high-speed simulation. The complexity becomes O(1) instead of the linear search O(N), because a search is only carried out on the length of L with hashing method. A computation is possible for other boundary conditions with the same procedure if one-dimensional Virtual Array can be created. Therefore, high-speed simulation is possible in principle for any kind of boundary to be applied. There is room for improvement in hashing functions, which is the easiest when dividing equally, because it is obviously inappropriate when particles are distributed heterogeneously. The strategy of EEPGM has an advantage that it is easily extended. The ease of a development is an important factor of the evaluation of an algorithm. One problem in low-density systems is the overwhelming increase of the arrays assigned to grid in comparison to the number of the particles. This is not desirable as the memory capacity is the same of as that of infinite system. However, the arrays for the grid are compressed to the size of the particle number in the same way as described above. Since supplementary arrays are made by the hashing method and information on neighboring grids is efficiently obtained, there is no problem for both efficiency and memory capacity.

49 3 ALGORITHMS FOR EVENT-DRIVEN METHOD Summary In this section, we developed an algorithm for a hard disk system without using Cell- Crossing Event; the algorithm is simple, efficient and easy to extend [119, 120]. EEPGM is easy to extend to the various system because of its simplicity, which can never be realized in LCM. One example is the system that hard disks with various size of diameters coexist. Though there was a limitation in the degree of the poly-dispersion with EPGM described to Buchholtz and Pöschel [143], EEPGM can be applied easily to those systems. First, the grid is made based on the smallest particle radius in the system. Next, we have only to check the nearest grids by using a suitable mask of the bigger level when the poly-dispersity increases. This way, EEPGM has a wider application than EPGM. This code achieved a very efficient computation; about 460 million of collisions per CPU hour for the 2500 disk system on the VT-Alpha-600. Since the order of complexity per event is O(log N), the increase of complexity isslow when the particle number increases. Now, we can carry out large-scale molecular dynamics simulation (N ο 10 6 ) on the usual Workstation in our laboratory. Finally, the algorithm in this section is suitable for the scalar machine, and the development of an algorithm for the parallel machine is the subject of a future study. Note the extension to the 3D system is easy.

50 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 50 4 Numerical Simulations of IHSHG model In this section, we report the results of numerical simulation on IHSHG model described in Sec. 2.3 using the simple and efficient algorithm developed in Sec Numerical Setting Most of the simulations presented in this section were performed on the system with r =0:9, N w = 200, and N h = 20 (i.e. N =4; 000). The driving intensityλ k B T=mgd was varied from 25.0 to to study the change of the system behavior. To ensure that the system had reached a steady state, we relaxed the system until the total energy did not drift. Therefore, the system reaches the non-equilibrium steady state. Thereafter, we started the simulations with the length of 100,000 collisions per particle; We could obtain the data with small statistical error. In addition, we performed the simulations with N w = 400 (i.e. N = 8; 000) and N w = 1; 000 (i.e. N = 20; 000) to study the size-dependence of the system behavior. These run lengths were 30,000 and 10,000 collisions per particle, respectively. The size-depencence of the system is discussed in Sec. 5. Note that an inelastic collapse (Sec ) did not occur in the present system during the simulation time for the parameter region studied here. 4.2 System Behavior upon Changing External Driving In Fig. 1 of Sec. 2.3, a typical snapshot of the simulation is shown for Λ = 181:8 with (r;n w ;N h )=(0:9; 100; 20). The average packing fraction profile as a function of height y is also shown. It can be seen that the density is small near the bottom because of the excitation by the heat bath, and the packing fraction reaches the maximum value A max at a certain height y = H Amax. In the actual simulation, we reduced any variables to dimensionless ones by the following three basic units (k B T; m; d), which are all fixed at unity. We, then, performed the simulations for various values of the dimensionless external driving Λ k B T=mgd by changing reduced gravity g; the range of Λ is 25:0 ο 909:1 (Table 2).

51 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 51 g(= Λ 1 ) Λ g Λ g Λ g Λ g Λ Table 2: The reduced gravity and its corresponding Λs.

52 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Packing Fraction and Excitation Ratio In order to characterize the steady states of the system, firstly, we measure the maximum packing fraction A max at the height y = H Amax as a function of the driving intensity Λ. In the system with (r;n w ;N h )=(0:9; 200; 20), there are two cusps around Λ = 200 and 380(ffi in Fig. 10), suggesting phase transitions with changing Λ. These transitions should be related to the excitation structure of the state, therefore, we define the excitation ratio μ by μ K=U, namely, the ratio of the total kinetic energy K = m=2 P i vi 2 to the potential energy U = mg P i y i. The cusps appear at the same points for μ, which confirms that some transitions occur at these points ( in Fig. 10). 1 A max. A max., 0.5 Locally Condensed Fluidized Granular Turbulent Figure 10: The maximum packing fraction A max and the excitation ratio μ vs. the driving intensity Λ.The system parameters are (r;n w ;N h )=(0:9; 200; 20).

53 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Snapshots In order to conceptualize the underlying physical mechanism of the system behavior in each phase separated by the transitions, three snapshots for typical values of Λ for each phase are shown in Fig. 11. For Λ = 181:8 (a1 a3 in Fig. 11), most of the particles are aggregated around the bottom of the system with an almost closed packing density, and the state of a dense packing layer is relatively stable, which means the potential energy is dominant and the system is in weakly excited states. We call the phase in Λ» 188:7, the condensed phase (CP). It can be seen from the snapshots, however, that collective motion appears on the surface of the layer. In the second phase, Λ = 250:0 (b1 b3 in Fig. 11), the dense packing layer is locally broken by excitation of the heat bath. The high-speed particles are blown upward from the holes in the layer. In Fig. 11, we see only one hole in the system, but for a larger system, there are certain cases where we can observe more than one hole. The holes migrate and occasionally become more active; sometimes they almost close temporarily, but the structure is fairly stable. We call this phase the locally fluidized phase (LFP). For the case of Λ = 666:7 in the third phase (c1 c3 in Fig. 11), the layer is completely destroyed. The average density is quite low, but it is very different from the ordinary molecular gas phase. The density fluctuation is large and this fluctuation causes turbulent motion driven by the gravity. At some time, the whole system gets excited with some relatively smaller density fluctuations, but the very next moment, a large proportion of the particles travels downward and forms a layer like structure. This structure, however, is destroyed immediately. We call this phase the granular turbulent phase (GTP).

54 4 NUMERICAL SIMULATIONS OF IHSHG MODEL a1 a2 a3 b1 b2 b3 c1 c2 c3 54 Figure 11: The series of snapshots for the three typical values of, a1 a3 ( = 181:8), b1 b3 ( = 250:0), c1 c3 ( = 666:7). The system parameters are (r; Nw ; Nh) = (0:9; 200; 20).

55 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Collision Rate Since the simulation was performed in the non-equilibrium steady state, the collision rate takes constant value during the whole run. Therefore, when the total number of q md collision C total (= 100; 000 N) was fixed, the total times T total ([fi = 2 k B ]=1:0) for T each run were different for each Λ. We found that the system in the nonequilibrium steady state can be also characterized by the collision rate because it seems to correspond to the pressure of molecular system in the equilibrium. Though in elastic hard disk system the collision rate is only the function of packing fraction of the system, in an inelastic hard disk system it depend on the competition between the inelasticity r and the external driving Λ. In the Fig. 12, the collision rate (C total =T total ) vs. the external driving Λ in the log-linear scale is shown. We found that the collision rate decrease exponentially in the LFP and GTP (Λ > 188:7). However the collision rate deviates from exponential fitting in the CP, which is in the weakly excited state. It diverges when Λ! 0.

56 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Collision Rate 10 3 Locally Condensed Fluidized Granular Turbulent Figure 12: The collision rate C total =T total vs. the driving intensity Λ.

57 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Probability Distribution Function of Velocity These inhomogeneous behaviors should result in a non-maxwell-boltzmann distribution of velocity (Sec ). Figure 13 shows the horizontal velocity PDFs at the horizontal layer around the height of the maximum packing fraction (y = H Amax ) for each phase in the log-linear scale. The data are plotted in the unit of q k B T=m. The distributions for Λ = 250:0 and666:7 deviate from the Gaussian and are more or less in exponential form in the tail region. It can be seen that the distribution for Λ = 250:0 in the locally fluidized phase consists of two parts; the central part that originates from the dense packing layer, and the wide tail that comes from the fluidized holes. The central parts for all cases are very close to that for Λ = 181:8. Note that these velocity PDFs are evaluated over 6 orders of magnitude. These high-resolution results can be obtained only by the long time computation, which was never shown by the previous workers.

58 4 NUMERICAL SIMULATIONS OF IHSHG MODEL =181.8 =250.0 =666.7 Velocity PDF Vx Figure 13: The horizontal velocity probability distribution functions for the three typical values of Λ. The other parameters are (r;n w ;N h )=(0:9; 200; 20).

59 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Flatness Parameter In order to quantify the deviation from Gaussian distribution, we next calculated the flatness parameter as a function of the height f(y) defined as f(y) hv 4 xi=hv 2 xi 2 ; (4. 1) which is often used in the study of the intermittency of the fluid turbulence. dependence of the external driving of the maximum flatness f(h fmax ) at the horizontal layer is shown in Fig. 14. Over the whole region, the value of f(h fmax ) is different from 3, which is the value for the Gaussian distribution, but it is remarkable that f becomes very large, as large as 20,inLFP. In the Fig. 14, the statistical error bars are also plotted for each data. Since the flatness were taken over each runs of finite length, there are the statistical errors in the mean values. Especially, the flatness involves in the calculation about the higher moment. The statistical errors of the time averages for molecular dymanics simulation were estimated from the block average method described in Computer Simulation of Liquids" by Allen and Tildesley[125]. The detail explanations are denoted in Appendix A. The

60 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Locally Condensed Fluidized Granular Turbulent Maximum Flatness Value Exponential Gaussian Figure 14: The flatness f(h fmax ) vs. the driving intensity Λ. The other parameters are (r;n w ;N h ) =(0:9; 200; 20). The values for the Gaussian distribution (f =3)and the exponential distribution (f = 6)are indicated in the figure.

61 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Fitting Functions of Velocity PDF The unexpected large value of f, however, can be understood as follows. Assume the velocity PDF ffi(v) has two components, the narrow Gaussian distribution ffi G (v) with the weight 1 p and the broader stretched exponential distribution ffi S (v) with the weight p; where ffi S (x) andffi G (x) are defined by ffi S (x) ffi(v) =pffi S (v)+(1 p)ffi G (v); (0» p» 1); (4. 2) fi 2x 0 (1=fi) exp ( j x x 0 j fi ); ffi G (x) Then, the flatness of this distribution is given by 1 p exp ( x2 ): (4. 3) 2ßffG 2ffG 2 f = [ (5=fi)= (1=fi)]~x4 0p + 3(1 p) ([ (3=fi)= (1=fi)]~x 2 0p +(1 p)) 2 ; ~x 0 x 0 ff G (4. 4) because the second and the fourth moment of ffi S (x) are given by (3=fi)= (1=fi) x 2 0 and (5=fi)= (1=fi) x 4 0, respectively. If we fit the parameters in eq. (4.2) to the data for Λ=250:0 for example, we obtain ff G =0:048; ~x 0 =1:7, fi =0:85, and p =0:25, which yields f ' 19:0 (Fig. 17(b)). The enhancement of the flatness by the superposition of the broader distribution can even be drastic for a small value of p as can be seen if eq. (4.4) is plotted as a function of p. This observation indicates that f can be used as a sensitive index to detect the appearance of a small weight of the broad component in the distribution. The sharp rise of f in Fig. 14 around Λ = 200 is clear evidence of the appearance of the fluidized holes. The other PDFs with fitting function for Λs in LFP and GTP are shown in Fig. 15(a),(b) and Fig. 16(c),(d), respectively. From these p, we obtain the p for each Λ in Fig. 17(a) and its flatness values obtained from both fitting function and real data in Fig. 17(b). Near the CP-LFP critical point Λ CL, the weight ofp takes a certain finite value, not 0 (Fig. 17(a)) In LFP, theweight p increases linearly when Λ increases, which means the fluidized holes region increases linearly. However, there is a plateau p ο 0:5 when the transition from LFP to GTP.

62 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 62 (a) (b) 10 0 = =250.0 fitting curve fitting curve Velocity PDF (x ) Velocity PDF (x ) Vx/ Vx/ Figure 15: Scaled velocity PDF in LFP. (a)λ = 222:2 with the parameters, x 0 =ff G = 1:7, fi =0:85, and p =0:19. (b)λ = 250:0 with the parameters, x 0 =ff G =1:7, fi =0:85, and p =0:25.

63 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 63 (c) (d) 10 0 = =476.2 fitting curve fitting curve Velocity PDF (x ) Velocity PDF (x ) Vx/ Vx/ Figure 16: Scaled velocity PDF in GTP. (c)λ = 400:0 with the parameters, x 0 =ff G = 1:7, fi =0:85, and p =0:49. (d)λ = 476:2 with the parameters, x 0 =ff G =1:7, fi =0:85, and p =0:49.

64 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 64 (a) (b) 1 LFP GTP 30 LFP GTP simulation fitting function 20 p 0.5 flatness Figure 17: (a) The values of weight p for each Λs. (b) The flatness values obtained from both the real simulation and the fitting function for PDFs.

65 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Height Dependence of Flatness and Packing Fraction The height dependences of the packing fraction A and the flatness f(y)( hv x4 i=hv x2 i 2 ) are calculated in the horizontal layers with the thickness of the disk diameter d; we refer H fmax and H Amax to the positions where the flatness and the packing fraction take their maximum values (Fig. 18(a)). In LFP and GTP, the both of the parameters reach their maximum values at the almost same height (H fmax ' H Amax )(Fig. 20), but in CP the flatness shows its maximum value at the lower position than the packing fraction does (H fmax <H Amax )(Fig. 19); in CP the value of the packing fraction where the flatness takes its maximum value is very close to the value of the Alder transition point A c of the elastic hard disk system (Fig. 18(b)). The fact is somewhat intriguing, but we could not find obvious reason for it. (a) (b) Condensed Phase CL ( 189.0) Locally Fluidized Phase LG ( 357.0) Granular Turbulent Phase 1 Closed Packing CL ( 189.0) 0.2 packing fraction peak height y 0.1 flatness Packing Fraction 0.5 Alder Transition maximum value of packing fraction packing fraction at flatness peak Condensed Phase Figure 18: (a) The heights in the maximum values both packing fraction and flatness are shown for each Λ. (b) The maximum values of packing fraction and the packing fraction at flatness peak in CP are shown for each Λ. In the granular vibrated beds with DEM, Taguchi and Takayasu[46] showed that velocity PDF becomes power distribution in the fluidized region, but Gaussian in the

66 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 66 Packing Fraction (layer number) 6 Condensed Phase (Weakly Excited Region) =25.0 Packing Fraction flatness 4 2 Flatness Packing Fraction (layer number) Condensed Phase (Highly Excited Region) =175.4 Packing Fraction flatness Flatness y y 0 Figure 19: The height dependence of the packing fraction and flatness in CP (layer number) Locally Fluidized Phase 1 24 (layer number) Granular Turbulent Phase Packing Fraction 0.5 =250.0 Packing Fraction flatness Flatness Packing Fraction 0.5 =400.0 Packing Fraction flatness Flatness y y 0 Figure 20: The height dependence of the packing fraction and flatness in LFP and GTP.

67 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 67 solid (closed packing) region. While, Murayama and Sano [48] found flatness varied Gaussian to Exponential when the packing fraction became higher. These results seem to disagree with each other. In CP of IHSHG model, we obtained two different characters about velocity PDF. In CP, we can distinguish the following two regions; (i) The region 0 <y<h fmax, where the packing fraction increases from 0 to A c and the flatness increases, (ii) the region H fmax <y<h Amax, where the packing fraction (> A c ) reaches nearly the closed packing and the flatness decreases.

68 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Spatial Velocity Correlation Function In the region (ii) of CP (i.e. H fmax < y < H Amax ), we found the Spatial Velocity Correlation Function (SVCF) C vxv x (R) = hv x (x + R)v x (x)i takes positive value for all R(0 < R < L x =2) (Fig. 21), which means particle flows collectively in the same horizontal direction. Spatial Velocity Correlation Function =25.0 =175.4 =222.2 =243.9 =270.3 =303.0 =344.8 =370.4 =400.0 =434.8 = R (=Lx/2) Figure 21: Spatial velocity correlation function at the height ofh Amax in the horizontal direction vs. the relative displacement R. In Fig. 21, the SVCFs at the height of H Amax in LFP and GTP for each Λ are also shown. The negative minimum of the SVCF becomes larger when Λ is increased, which

69 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 69 means that the size of fluidized hole grows large. When Λ > 357:1, there is no negative minimum in C vxv x (R) for all R's less than half of the system size(l x =2). This means the size of fluidized hole grows larger than the half of system size.

70 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Displacement of Particles The time-dependence of displacement vectors for each particle from the initial position are shown in Fig. 22 at t=fi = 100; 200; 300; 400 when Λ = 25:0. The red or blue vectors mean whether the particle moves left or right from the initial position, respectively. This is a clear evidence of the existence of the domain of collective particle motion. And the feedback of collective motion can be seen from the time t=fi =200to t=fi = 400. Figure 22: The time-dependence of displacement vectors for each particle from the initial position are shown at t=fi = 100; 200; 300; 400 in Λ = 25:0. Red and blue vectors mean whether the particle moves left and right from the initial position, respectively.

71 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 71 The time-dependence of displacement vectors for each particle from the initial position are shown in Fig. 23 at t=fi = 150 when Λ = 175:4. The domain size of collective motion becomes smaller than that of Fig. 22. Since the particle has a room to move freely compared with weakly exited state (Λ = 25:0), the convective flow can be seen. This convectional mode is similar with the granular vibrated bed. Figure 23: The displacement vectors for each particle from the initial position are shown at t=fi = 150 in Λ = 175:4. In LFP, particle moves vortex-like convection at the region of broken layers. The most of excited particles drops on the packed layer near the broken point. Figure 24: The displacement vectors for each particle from the initial position are shown at t=fi =50inΛ=250:0.

72 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 72 In GTP, the highly excited particles at the broken layer can reach another side of condensed layer through the periodic boundary condition. This motion must have effects on the velocity PDF and spatial velocity correlation function (Sec. 5.2, Sec. 5.3). Figure 25: The displacement vectors for each particle from the initial position are shown at t=fi =50inΛ=400:0.

73 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Horizontal Averaged Particle Current in CP Since the exclusive volume effect is dominant in CP, the collective motion of particle appears. As mentioned in Sec. 4.5, the displacement of particles in CP move to the same direction in the short time scale (Fig. 22). This is also confirmed by the fact that the averaged particle current field summed up in the packed region (ii) described in Sec. 4.3, j(x g ; t) = 1 X N sample packedregion X i v i ( t)ffi(x i ( t) x g ); (4. 5) where N sample is the sample number of particles, x g is grid and t = N=(CollisionRate), fluctuate around zero in the short time scale (Fig. 26). Figure 26 shows that the timedependent of the horizontal averaged particle current for one particle when Λ = 25:0 and Λ = 175:4. In the relatively short time, the current takes positive values. However, we found the current fluctuate around zero in the long run. The time scale that the current reversed become longer when Λ becomes near the critical point. This means that the collective motion appears strongly when the system is in the weakly excited state.

74 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 74 Horizontal Avaraged Particle Current Nw=200, =175.4 Nw=200, = t/ Figure 26: The time-dependent of the horizontal averaged particle current for one particle in CP is shown.

75 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Surface Wave in CP For Λ» 189:0 incp, the dense packing layer is formed and it is stable in time because excitation is weak and the potential energy is dominant. However, it is apparent that collective motion appears on the surface of the condensed layer (Sec ). What is the mechanism of the surface wave-like motion in CP? There must be no surface tension in the macroscopic dissipative system, therefore the restoring force should originate from the balance between the gravity and the excitation, but it is not clear if it could be defined as an ordinary gravitational wave in fluid. To study the dynamical behavior of surface wave in detail, simulations were done on the system with r = 0:9, N w = 200, and N h = 20 (i.e. N = 4; 000). The driving intensity Λwas varied from 25.0 to 188.6, in q which the system is in CP. We performed md simulations until the time T = 10240fi (fi = 2 k B T w =1:0) for each run. Figure 27 shows the spatio-temporal pattern of surface wave in CP far from the CP-LFP critical point ((a) Λ = 25:0 and (b) 100:0). And Fig. 28 shows the pattern near the CP-LFP critical point ((c) Λ = 180:0 and (d) 188:6). The horizontal and vertical axis corresponds to the space expansion (x =0ο L x ) and the time evolution (T = 0 ο 2000fi), respectively. There are many domains, which represent the collective motion of surface appears. The typical domain sizes, which are estimated by Fig. 28(d), are about x ο L x =2 and t ο 850. These are correspond to (k(= 2ß=(L x =2));!(= 2ß=850)) = (0:0628; 0:00739), respectively. To study this collective motion quantitatively, we calculate the dynamical structure factor of the surface wave in CP. The dynamical structure factor S(k;!) is described as S(k;!) = 1 Z T X hh y (x i ;t)h y (x j ; 0)i T e ik(x i x j ) e i!t dt; (4. 6) 2ßN gx T i;j where h y (x; t) is the surface height of condensed layer; N gx (= 200) is the total number of grids with the width of disk diameter d in the horizontal direction and the discrete positions are taken by x i =(L x =N gx )=2+(i 1)(L x =N gx ); (i =1:::N gx ). The dynamical structure factor of surface wave at k = 2ß=(L x =2) is shown in Fig. 29(b). When Λ = 188:6, the strongest peak in the k! space was found at (k;!) ο (2ß=(L x =2); 0:0065). We foundthese values were consistent with the estimation of the domain sizes in the spatio-temporal pattern (Fig. 28(d)). When Λ becomes low, we found the domain size in the vertical direction became small. While, the domain size in the horizontal direction was not changed very much.

76 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 76 (a) (b) Figure 27: The spatio-temporal pattern of surface wave far from the critical point ((a) Λ = 25:0 and (b) 100:0). The horizontal and vertical axes correspond to the space expansion (x = 0 ο L x ) and the time evolution (T = 0 ο 2000fi), respectively. The colors of spatio-temporal pattern correspond to the height of surface wave. The right color is higher than the left one.

77 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 77 (c) (d) Figure 28: The spatio-temporal pattern of surface wave near the critical point ((c) Λ = 180:0 and (d) 188:6). The horizontal and vertical axes correspond to the space expansion (x = 0 ο L x ) and the time evolution (T = 0 ο 2000fi), respectively. The colors of spatio-temporal pattern correspond to the height of surface wave. The right color is higher than the left one.

78 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 78 This means that periodical cycle of surface becomes shorter in low Λ. These results are also confirmed by the computation of the dynamical structure factor. When the strongest peak (k max ;! max ) was plotted in k! space for each Λ, we found that the k max stays around k = 2ß=(L x =2) and increases only slightly upon decreasing Λ (Fig. 30(a)), but! max becomes large as Λ to 0 (Fig. 30(b)). (a) (b) 2 Amplitude fluctuation of surface wave =188.6 k=2 /(Lx/2) the most strong peak 2 1 Intensity Figure 29: (a) The amplitude fluctuation of surface wave for each Λ is plotted. (b) The dynamical structure factor S(k;!) of the surface vs. the frequency! at k =2ß=(L x =2) near the critical point Λ = 188:6 is shown. We conclude the periodical behavior of the surface wave in CP shows the critical slowing down when Λ! Λ CL. Note that the value of! at the critical point takes a certain finite value, not 0 (Fig. 30(b)). It can be understood the transition occurs before the break down of the condensed layers, which means the transition seems to be subcritical. We also found the amplitude of the surface fluctuation becomes large near the transition point Λ CL (Fig. 29(a)). This is an interesting point that the system shows large fluctuation around the transition like the critical fluctuation in equilibrium.

79 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 79 (a) (b) k (wave number) /(Lx/2) (angular frequency) Figure 30: The Λ dependence of the most strong peak (k max ;! max ) is plotted. (a) The wave number k vs. external driving Λ. (b) The angular frequency! vs. external driving Λ.

80 4 NUMERICAL SIMULATIONS OF IHSHG MODEL Dynamics of Fluidized Holes The localized excitation in LFP should resemble a circle in 3D system and reminds us of an oscillon[19], but they are different; the external vibration frequency is essential for the oscillon dynamics, but the localized excitation here does not have such a characteristic frequency. In the large N w simulation, we can observe the merging process of two or more excitations, but their mode of interaction is not clear. An interesting question here is if the transition from LFP to CP and/or the transition to GTP can be understood in terms of the local excitation; if LFP transform to CP when the distance between the excitations diverges? Does LFP transform to GTP at the point where they condense? Toinvestigate the excitation dynamics, we calculate the density fluctuation for each position with the width of particle diameter d in the system. The driving intensity Λ was varied from to 476.2, in which the systems are in LFP and GTP. We performed simulations until the time T = 10240fi for each run. The density is defined by the particle number in each column, which is summed up from the bottom to the top. If the particle number is less than the averaged layer number N h, we regard it excitation occurs (black in Fig. 31). If the column has the particles more than the averaged layer number N h, these columns are regarded as condensed layers; they are colored by its corresponding density (in Fig. 31). Though it is not likely rigorous definition of fluidized holes and condensed layers, in the really simulation we confirm that the fluidized holes are distinguished very well. In Fig. 31, the spatio-temporal patterns of the density fluctuation are shown in LFP and GTP (N w = 200; (a) Λ = 200:0, (b) 250:0, (c) 333:3 and (d) 476:2). Near the critical point between CP and LFP (Λ = Λ CL ), the life time of a fluidized hole does not continue through the simulation (Fig. 31(a)). When Λ is increased, the region of fluidized holes also increases (Fig. 31(b),(c)). These results are consistent with the increase of the weight of the broader distribution p for the fitting functions of velocity PDF (Sec ). Above LFP-GTP transition point (Fig. 31(d)), the local excitation domain fluctuates largely and sometimes expands to the length of the system. We found that the spatio-temporal patterns of fluidized holes changes significantly through the transition from LFP to GTP.

81 4 NUMERICAL SIMULATIONS OF IHSHG MODEL 81 (a) (b) (c) (d) Figure 31: The spatio-temporal patterns of the density fluctuations are shown in LFP and GTP.(N w = 200; (a) Λ = 200:0, (b) 250:0, (c) 333:3 and (d) 454:5). The horizontal and vertical axes correspond to the space expansion (x = 0 ο L x ) and the time evolution (T = 0 ο 10240fi), respectively. The black region correspond to the fluidized holes. The colors of spatio-temporal pattern correspond to the density of the condensed layer.

82 5 DISCUSSION 82 5 Discussion 5.1 Size Dependence In this subsection, we discuss the changes of the statistical properties in IHSHG model upon increasing the system size N w Packing Fraction & Flatness We performed the almost same simulations described by Sec. 4 for the systems with larger N w,which means the aspect ratio in the system is changed. In Fig. 32, the height dependences of the packing fraction (black) and the flatness (blue) for Λ = 250:0 are shown for the systems with N w = 200 (filled circles) and N w = 400 (squares), respectively. The difference of the packing fraction between N w =200and N w = 400 cannot be seen. However, we found that the flatness around the height of the maximum packing fraction (y = H Amax )inn w = 400 takes larger values than that of the flatness in N w = = Packing Fraction Flatness y Figure 32: The height dependence of the packing fraction (black) and flatness (blue) in Λ=250. Filled circles and squares denote that the system size are N w = 200 and N w = 400, respectively.

83 5 DISCUSSION 83 To examine the size dependence further, the numerical simulations for various Λ with N w = 400 (i.e. N = 8; 000) and N w = 1; 000 (i.e. N = 20; 000) are performed. The simulation lengths for these data are 30,000 (N w = 400) and 10,000 (N w =1; 000) collisions per particle, respectively. The N w dependences of the maximum packing fraction and the excitation ratio are shown in Fig. 33(a). The N w dependences of the flatness at the height H Amax are shown in Fig. 33(b). The packing fraction and excitation ratio do not change when N w is increased. However, although the statistical errors are large in the flatness data for N w = 1; 000, the flatness clearly becomes larger when N w is increased. In general, the flatness takes large value when the spatial heterogeneity become large. This might be the reason why flatness takes larger value when the system size become larger. (a) (b) 1 A max. 30 Nw=1000 Nw=400 Nw=200 A max., 0.5 Nw=200 Nw=400 Nw=1000 f Figure 33: The N w dependence of (a) the maximum packing fraction, the excitation ratio, and (b) the flatness at the height H Amax are shown when the driving intensity Λ is increased.

84 5 DISCUSSION Fitting Functions of Velocity PDF In Fig. 34(a), the velocity PDFs ffi(v) with fitting function in N w = 200 and N w = 400 are shown when Λ = 250:0. The difference of velocity PDF between N w = 200 and N w = 400 is very small. However, the flatness becomes clearly larger when N w is increased (Fig. 33(b)). We change fi for N w = 400(fi = 0:82) to fit slightly broader distribution than that of N w = 200(fi = 0:85). The other fitting parameters for N w = 400 remain the same as in N w = 200, i.e. ff G = 0:048; ~x 0 = 1:7, and p = 0:25 (Sec ). The fitting function seems good agreement with the simulation data. Using these parameters, we calculated the flatness value described in eq. (4.4) of Sec In Fig. 34(b), the flatness values both fi =0:85 and fi =0:82 are plotted as functions of p. We found that the flatness values with fi =0:82 are larger than that with fi =0:85. This is consistent to the fact that the flatness in N w = 400 takes larger value than that in N w = 200 (Fig. 33(b)).

85 5 DISCUSSION 85 (a) (b) 30 Velocity PDF (x ) 10 0 =250.0 Nw=200 ( =0.85) Nw=400 ( =0.82) flatness 20 =0.82 (Nw=400) =0.85 (Nw=200) Vx/ 0 1 p Figure 34: (a) Scaled velocity PDFs with fitting function both N w = 200 and N w = 400 are shown when Λ = 250:0. (b) The flatness values for each fitting parameters (fi = 0:85 and fi =0:80) by changing weight p are shown Dynamics of Fluidized Holes Although there is one hole in the system when N w = 200 (Sec. 4.8), there should be more holes in the limit N w! 1 to reach finite density of holes. What an average density of the holes? It might have some relationship to the width of the surface wave in CP (Sec. 4.7). In Fig. 35 and Fig. 36, the spatio-temporal pattern of the density fluctuations are shown in LFP (N w = 400; (a) Λ = 200:0, (b) 250:0, and (c) 333:3) and GTP (N w = 400; (d) Λ = 454:5, (e) 588:2, and (f) 833:3), respectively. Near the critical pointλ CL (Λ = 200:0), the lifetime of a fluidized hole is shorter than the simulation length (Fig. 35(a)), which is similar situation in the case of N w = 200 (Sec. 4.8). When Λ is increased, the width of the holes becomes wider. However, the system seems to contain two fluidized holes (Fig. 35(b),(c)), which is the different situation if it is compared with the case of N w =200. Above the LFP-GTP transition

86 5 DISCUSSION 86 point (Λ > 357:1), the local excitation domain largely fluctuates (Fig. 36(d),(e)). We can see the high-speed motion of the high density cluster when Λ = 833:3 (Fig. 36(f)). When N w is increased, since there are many fluidized holes in the system, the fluctuation of hole width might become larger. In this situation, velocity PDFs for each hole have different variances for the width of holes, which meansvarious broader distributions of horizontal velocity are coexisted with in the system. This spatial heterogeneity of the velocity PDFs at fluidized holes might be the reason why the flatness takes larger value when N w is increased (Fig. 33(b)).

87 5 DISCUSSION 87 (a) (b) (c) Figure 35: The spatio-temporal patterns of the density fluctuations are shown in LFP. (N w = 400; (a) Λ = 200:0, (b) 250:0, and (c) 333:3 ).The horizontal and vertical axes correspond to the space expansion (x = 0 ο L x ) and the time evolution (T = 0 ο 10240fi), respectively. The black region correspond to the fluidized holes. The colors of spatio-temporal pattern correspond to the density of the condensed layer.

88 5 DISCUSSION 88 (d) (e) (f) Figure 36: The spatio-temporal patterns of the density fluctuations are shown in GTP. (N w = 400; (d) Λ=454:5, (e) 588:2, and (f) 833:3 ). The horizontal and vertical axes correspond to the space expansion (x = 0 ο L x ) and the time evolution (T = 0 ο 10240fi), respectively. The black region correspond to the fluidized holes. The colors of spatio-temporal pattern correspond to the density of the condensed layer.