System Of Hard Disks Research Articles

Boundary effects in the stepwise structure of the Lyapunov spectra for quasi-one-dimensional systems.

Boundary effects in the stepwise structure of the Lyapunov spectra and corresponding wavelike structure of the Lyapunov vectors are discussed numerically in quasi-one-dimensional systems of many hard disks. Four different types of boundary conditions are constructed by combinations of periodic boundary conditions and hard-wall boundary conditions, and each leads to different stepwise structures of the Lyapunov spectra. We show that for some Lyapunov exponents in the step region, the spatial y component of the corresponding Lyapunov vector deltaq(yj), divided by the y component of momentum p(yj), exhibits a wavelike structure as a function of position q(xj) and time t. For the other Lyapunov exponents in the step region, the y component of the corresponding Lyapunov vector deltaq(yj) exhibits a time-independent wavelike structure as a function of q(xj). These two types of wavelike structure are used to categorize the type and sequence of steps in the Lyapunov spectra for each different type of boundary condition.

Proof of the Boltzmann-Sinai ergodic hypothesis for typical hard disk systems

We consider the system of $N$ ($\ge2$) hard disks of masses $m_1,...,m_N$ and radius $r$ in the flat unit torus $\Bbb T^2$. We prove the ergodicity (actually, the B-mixing property) of such systems for almost every selection $(m_1,...,m_N;r)$ of the outer geometric parameters.

Role of rigidity in the fluid-solid transition.

We examine the fluid-solid transition for a hard-disk system. By counting the near neighbors in the average configurations of a grand-canonical Monte Carlo simulation, this enables us to relate the thermodynamic transition with the rigidity theory, since we find that the coordination number in the fluid-solid transition is close to the coordination number predicted by a mean field rigidity theory, due to dynamical jamming of particles, where the contact region between disks is the radial ring outside a disk with a maximum allowed coordination number that is not bigger than six. Using these ideas, we were able to produce a continuous glass-like transition when nucleation of rigidity is suppressed.

Consistent cluster approach to the description of correlation functions and thermodynamic potentials

The new method for the construction of correlation functions, consecutively taking into account the cluster contributions of different orders, is developed. The concrete analysis of the binary correlation function for the system of hard disks in the approximation of two, three and four particle clusters is presented. The irreducible cluster contributions of different orders to the entropy of this system are studied.

Dynamic analysis of a disk-spindle system in a hard disk drive

Vibration analysis and experimental study of a disk-spindle system in hard disk drives are presented in this paper. The natural frequencies, mode shapes, and frequency response are calculated with the consideration of the flexibility of rotor and base structures of a disk-spindle system. The analytical model is verified by comparing the simulation results with the experimental data for a variety of motor configurations and sizes. It is shown that they agree very well with each other.

Realization of a hard disk drive head servo-positioning system with a voltage-driven voice-coil motor

In this paper we describe a head servo-positioning system for hard disk drives (HDDs), in which the usual current command for the voice coil motor has been replaced by a simpler voltage command. Current command of the voice coil motor (VCM) has been preferred so far, due to its good performance in terms of robustness. With this solution, in fact, the characteristics of the system to be controlled by the servo-controller are rather independent from all possible variations of electrical impedance of the VCM and power supply voltage. However, this solution has some drawbacks. In fact, in order to realize the current command, a current loop uses a shunt resistor as a current sensor and a linear high power operational amplifier with some phase-shaping network as a current driver. Both shunt resistor and linear amplifier are major sources of power dissipation. Moreover, linear amplifiers need a large area on the silicon chip, increasing the cost of the power device. On the other hand, a fully digital current loop would require a costly A/D converter, and this is one of the reasons why the current controller is still implemented in analog way, thus impeding the implementation of smart control strategies at this level. All the above considerations have led to the realization of a servo system with a voltage-driven VCM. We developed a solution in which current measurement is no longer needed in order to guarantee the correct behavior of the servo controller. The solution is based on a digital, multi-rate pre-filter, driving a voltage amplifier. The filter is designed in such a way that the transfer function between input signal and VCM current is close to that of a standard current loop, so providing a one-to-one replacement to standard current drivers. With the proposed solution, no measurement on the VCM current or voltage is required. It can be shown that with a proper tuning of the pre-filter, the proposed solution exhibits a good robustness against variations of VCM parameters. Since the proposed solution is fully digital, digitally driven switching power stages can be used. Experimental results, including those related to robustness issues, show that the HDD servo-positioning performance obtained with the new voltage-driven system matches that obtained with a standard current loop, in both seek and track following operations. Finally, in order to adapt the pre-filter to operating conditions, an on-line parameter estimation procedure, capable of determining VCM windings resistance, is presented.

An Evolutionary Algorithm with Advanced Goal and Priority Specification for Multi-objective Optimization

This paper presents an evolutionary algorithm with a new goal-sequence domination scheme for better decision support in multi-objective optimization. The approach allows the inclusion of advanced hard/soft priority and constraint information on each objective component, and is capable of incorporating multiple specifications with overlapping or non-overlapping objective functions via logical 'OR' and 'AND' connectives to drive the search towards multiple regions of trade-off. In addition, we propose a dynamic sharing scheme that is simple and adaptively estimated according to the on-line population distribution without needing any a priori parameter setting. Each feature in the proposed algorithm is examined to show its respective contribution, and the performance of the algorithm is compared with other evolutionary optimization methods. It is shown that the proposed algorithm has performed well in the diversity of evolutionary search and uniform distribution of non-dominated individuals along the final trade-offs, without significant computational effort. The algorithm is also applied to the design optimization of a practical servo control system for hard disk drives with a single voice-coil-motor actuator. Results of the evolutionary designed servo control system show a superior closed-loop performance compared to classical PID or RPT approaches.

Velocity distribution of inelastic granular gas in a homogeneous cooling state.

The velocity distribution of inelastic granular gas is examined numerically on a two-dimensional hard disk system in nearly elastic regime using molecular dynamical simulations. The system is prepared initially in the equilibrium state with the Maxwell-Boltzmann distribution, then after several inelastic collisions per particle, the system falls in the state that the Boltzmann's equation predicts with the stationary form of velocity distribution. It turns out, however, that due to the velocity correlation the form of the distribution function does not stay time independent, but gradually returns to the Maxwellian immediately after the initial transient till the clustering instability sets in. It shows that, even in the homogeneous cooling state (Haff state), where the energy decays exponentially as a function of collision number, the velocity correlation in the inelastic system invalidates the assumption of molecular chaos and the prediction of the Boltzmann's equation fails.

Phase transitions in model colloids in reduced geometry

The nature of the melting transition for a system of hard disks with translational degrees of freedom in two spatial dimensions has been analyzed by a novel finite size scaling technique. The behavior of the system is consistent with the predictions of the KTHNY theory. Hard and soft disks in external periodic potentials show rich phase diagrams including freezing and melting transitions when the density of the system is varied. Here we report on extensive Monte Carlo simulations for detailed finite size scaling analyses of various thermodynamic quantities like the order parameter, its cumulants, etc. in order to map the phase diagram of the system for various values of the density and the amplitude of the external potential. For hard disks we find clear indication of a re-entrant liquid phase over a significant region of the parameter space. Our simulations therefore show that the system of hard disks behaves in a fashion similar to charge stabilized colloids which are known to undergo an initial freezing, followed by a re-melting transition as the amplitude of the imposed, modulating field produced by crossed laser beams is steadily increased. Detailed analysis of our data shows several features consistent with a recent dislocation unbinding theory of laser induced melting. The differences and similarities of systems with soft potentials (DLVO, 1/ r 12) is discussed.

Settling Control with Reference Redesign for Dual Actuator Hard Disk Drive Systems

This paper considers settling control for a dual actuator system for hard disk drives consisting of a coarse actuator(voice coil motor) and a fine actuator(peizoelectric transducer, PZT). The design method proposed in this paper is called the dual actuator reference trajectory re-design(RTRD). In the RTRD, the design is divided into three steps. In the first step, the coarse actuator loop is designed to be stable via active damping and achieve the basic performance. In the second step, the fine actuator path is designed by loop shaping for superior performance of the overall system. Finally the reference signals are generated for both VCM and PZT actuators. In the dual rate RTRD : the coarse actuator works at slow sampling frequency and the fine actuator works at fast sampling frequency. The “soft switching” technique is applied for gradually introducing the fine actuator(PZT actuator), which may shorten the settling time and achieve smoother settling response. The effectiveness of the proposed approach is evaluated by simulations.

Local structure, fluctuations, and freezing in two dimensions

Starting from the concept of local solidlike order in two-dimensional (2D) liquids we introduce and quantify the corresponding ensembles of fluctuations, using a probabilistic-based method of local-structure analysis (LSA). A systematic LSA (including size dependence) was performed for a hard disk and 2D Lennard-Jones systems, simulated using Monte Carlo and molecular-dynamics methods. We find that the onset of freezing is accompanied by a dramatic crossover between ensembles of fluctuations. Some universal features related to the onset of freezing in two dimensions are found and corresponding freezing criteria are formulated: (i) the liquid starts to freeze when the concentration of solidlike atoms constitutes 0.50\char21{}0.56, and (ii) a Lindemann-like freezing criterion: the rms fluctuation constitutes, at the onset, 0.12\char21{}0.13. Those criteria offer an effective method for a localization of the onset of freezing in computer simulations. We point out, in this context, that in computer simulations there is a possibility that all quantitative characterizations of the onset of freezing are related to a metastable range. This important methodological topic is discussed briefly in the light of recent results both for 2D and 3D systems.

Nonequilibrium relaxation analysis of two-dimensional melting.

The phase diagram of a hard-disk system is studied by observing nonequilibrium relaxation functions of a bond-orientational order parameter using particle dynamics simulations. From a finite-time scaling analysis, two Kosterlitz-Thouless transitions can be observed when the density is increased from the isotropic fluid phase to closest packing. The transition densities are estimated to be 0.901(2) and 0.910(2), where the density denotes the fraction of area occupied by the particles, the density is normalized to one for the quadratic packing configuration. These observations are consistent with the predictions of the Kosterlitz-Thouless-Halperin-Nelson-Young theory.

Melting in two dimensions: first-order versus continuous transition

In the framework of the theory, based on the density functional theory of freezing, the melting of the system with Yukawa potential and the hard-disk system with square-well attraction is considered. It is shown that melting scenario crucially depends on the shape of the potential. For the soft-core Yukawa potential melting mainly occurs by means of two continuous transitions, however, there is the critical value of the radius of the potential where the crossover from two continuous to one first-order transition occurs. It should be noted that the first-order transition exists only for very short-range interaction. In the case of hard-core square-well potential situation is more complex, and there are several crossover points, where the melting scenario changes from continuous to first-order transition, and vice versa.

How do dynamic heterogeneities evolve in time?

We present simulations of a hard disk system and analyze the time evolution of the dynamic heterogeneities. We characterize the time evolution of slow regions and slow particles individually. The motion of slow clusters turns out to be very restricted, i.e. a cluster is generated and annihilated in a spatial region four times the size of its maximum instantaneous extent. The residual motion of the cluster can be traced back to subdiffusive motion of the constituent particles and the process of absorption and loss of adjacent particles. The subdiffusive dynamics are independent of how long the particles remain slow. Clusters of fast particles show an even smaller reach, which seems to be due to their short life time.

Stepwise structure of Lyapunov spectra for many-particle systems using a random matrix dynamics.

The structure of the Lyapunov spectra for the many-particle systems with a random interaction between the particles is discussed. The dynamics of the tangent space is expressed as a master equation, which leads to a formula that connects the positive Lyapunov exponents and the time correlations of the particle interaction matrix. Applying this formula to one- and two-dimensional models we investigate the stepwise structure of the Lyapunov spectra that appear in the region of small positive Lyapunov exponents. Long range interactions lead to a clear separation of the Lyapunov spectra into a part exhibiting stepwise structure and a part changing smoothly. The part of the Lyapunov spectrum containing the stepwise structure is clearly distinguished by a wave-like structure in the eigenstates of the particle interaction matrix. The two-dimensional model has the same step widths as found numerically in a deterministic chaotic system of many hard disks.

Phase transitions of hard and soft disks in external periodic potentials: A Monte Carlo study

The nature of the melting transition for a system of hard disks with translational degrees of freedom in two spatial dimensions has been analyzed by a novel finite size scaling technique. The behavior of the system is consistent with the predictions of the KTHNY theory. Hard and soft disks in external periodic potentials show rich phase diagrams including freezing and melting transitions when the density of the system is varied. Here we report on extensive Monte Carlo simulations for detailed finite size scaling analyses of various thermodynamic quantities like the order parameter, its cumulants, etc. in order to map the phase diagram of the system for various values of the density and the amplitude of the external potential. For hard disks we find clear indication of a reentrant liquid phase over a significant region of the parameter space. Our simulations therefore show that the system of hard disks behaves in a fashion similar to charge-stabilized colloids which are known to undergo an initial freezing, followed by a re-melting transition as the amplitude of the imposed, modulating field produced by crossed laser beams is steadily increased. Detailed analysis of our data shows several features consistent with a recent dislocation unbinding theory of laser-induced melting. The differences and similarities of systems with soft potentials (DLVO, 1/r 12 ) are discussed.

Phase equilibria in systems of hard disks with thickness polydispersity.

We study isotropic-nematic (I-N) phase equilibria in the Onsager (-Parsons) model for systems of hard colloidal disks allowing for arbitrary polydispersity in thickness. The phase behavior is investigated by analyzing the exact phase equilibrium equations for Gaussian orientational distribution functions. We observe a strong fractionation effect, with the thicker disks found preferentially in the isotropic phase. Due to this effect, the system may undergo an I-N density inversion indicating that the mass density of the isotropic phase becomes higher than that of the coexisting nematic phase. This phenomenon has been observed explicitly in experiment. We also encounter a divergence of the I-N coexistence region for Schulz-distributed parents with polydispersities larger than 46%. An implication of this phenomenon is that the system cannot become fully nematic at high densities but will continue to split off a small fraction of a dilute isotropic phase predominantly containing very thick species.

FREE VIBRATION ANALYSIS OF A SPINNING FLEXIBLE DISK–SPINDLE SYSTEM SUPPORTED BY BALL BEARING AND FLEXIBLE SHAFT USING THE FINITE ELEMENT METHOD AND SUBSTRUCTURE SYNTHESIS

Free vibration of a spinning flexible disk–spindle system supported by ball bearing and flexible shaft is analyzed by using Hamilton's principle, FEM and substructure synthesis. The spinning disk is described by using the Kirchhoff plate theory and von Karman non-linear strain. The rotating spindle and stationary shaft are modelled by Rayleigh beam and Euler beam respectively. Using Hamilton's principle and including the rigid body translation and tilting motion, partial differential equations of motion of the spinning flexible disk and spindle are derived consistently to satisfy the geometric compatibility in the internal boundary between substructures. FEM is used to discretize the derived governing equations, and substructure synthesis is introduced to assemble each component of the disk–spindle–bearing–shaft system. The developed method is applied to the spindle system of a computer hard disk drive with three disks, and modal testing is performed to verify the simulation results. The simulation result agrees very well with the experimental one. This research investigates critical design parameters in an HDD spindle system, i.e., the non-linearity of a spinning disk and the flexibility and boundary condition of a stationary shaft, to predict the free vibration characteristics accurately. The proposed method may be effectively applied to predict the vibration characteristics of a spinning flexible disk–spindle system supported by ball bearing and flexible shaft in the various forms of computer storage device, i.e., FDD, CD, HDD and DVD.

The Properties of Voronoi Tessellations for 2D Ising Fluid Near the Second-Order Magnetic Phase Transition

The topological and geometrical properties of Voronoi cells generated for 2D fluid of hard disks with Ising-like spins near the second-order phase transition from the paramagnetic to ferromagnetic phase are described for different disk densities. The comparison with Voronoi cells generated for the random hard disk system is given.

Melting of two-dimensional systems: Dependence of the type of transition on the radius of the potential

In the framework of the two-dimensional melting theory based on the density functional approach, we use a Monte Carlo computer simulation to study melting of a two-dimensional hard disk system with a rectangular-well attraction potential. We show that depending on the attraction radius, the melting can occur via a single first-order transition as well as continuously in accordance with the Kosterlitz–Thouless–Halperin–Nelson–Young theory.