Hard disks correlations

Figure 60. Pair correlation function for dense random packings of hard disks. The inset shows an expanded view.

The earliest work was done for hard disks and spheres, with no longer range forces. In that case there is no direct effect due to interactions with distant (but correlated) parts of the periodic system. There is nevertheless some dependence of the results on the size and shape of the periodic sample. This arises in various ways, which of course also persist for systems with other forces. For example, the number of particles within any of the periodic boxes is fixed at N. This is a serious constraint on the density fluctuations for small N, and will lead to errors in the resulting thermodynamic averages, in particular to a diminished entropy. (For mixtures, the concentration fluctuations are similarly constrained, and this could be a still more severe problem.) With small samples the range of structural fluctuations may be similarly constrained. This will clearly be the case if significant interparticle configurational correlations have a... 

On the microscopic level, both spatial and temporal fluctuations in number density are to be expected. The decay of these should be rapid compared to the time and distance scales appropriate to the macroscopic self-diflusion phenomenon. Thus to study the presence of a steady state, the M observations of JVi and /i are pooled AM at a time so that the resulting observations appear to be uncorrelated. Typically (AM)fi 200to seems to be satisfactory for this purpose at a reduced volume t = 3 for hard disks (but pooling over even longer intervals is frequently used to assure the independence of the observations). While there are rather long-time correlation effects, there seems little doubt that on any macroscopic time scale the flow is steady. 

Fig. 26. The velocity correlation between a central particle and its neighborhood at a density corresponding to of close packing for 224 hard disks. Because of symmetry, only the upper half of the neighborhood is shown. The size of the central particle is shown by the smallest half-circle, and the sizes of the other concentric circles are chosen so as to include about six neighboring particles. These circles have been partitioned into four parts. The arrows give the direction and magnitude of the average velocity in each section after about 10 collision times. The scale of the velocity is indicated as 0.01 of the initial velocity. A number of additional arrows are included in the figure to show the extension of the vortex pattern outside the areas covered by the molecular dynamics calculations. These arrows are obtained by a hydrod)mamic simulation of the molecular flow. [After B. J. Alder and T. E. Wainwright, Phys. Rev. AI, 18 (1970).]...

Finally, we mention the fact that the molecular dynamics calculation of Po t) for a gas of hard disks exhibited a vortex type of velocity correlation between the tagged molecule and the surrounding molecules that is very similar to the hydrodynamic flow field surrounding a moving volume element in a fluid initially at rest. This vortex pattern, illustrated in Fig. 26, suggests that a fraction of the momentum transferred by the tagged particle to the particles in front of it is eventually returned to it from behind. This process causes the velocity autocorrelation function to be larger than it would be if these vortices did not occur, and it is connected with the slow decay of the velocity autocorrelation function. 

For a gas of hard disks or hard spheres, if one neglects all the effects of correlated successive binary collisions and includes only the excluded volume... 

Usually, in computer simulations the time correlation functions are calculated after the run is finished, using the information saved during the run (i.e., in a postprocessing stage). One typically saves the instantaneous values of the quantities of interest during the run in one or several files on the hard disk, which are processed after the simulation run. 

Understanding molecular-scale adhesion, friction, lubrication, and wear is crucial to modern technologies, such as microelectromechanical systems (MEMS) and hard disk drives.With atomic force/friction force microscopy (AFM/FFM),several studies have shown the correlation of frictional properties of self-assembled monolayers (SAMs) of alkanethiols with their chain lengths and terminal groups. The long chain monolayers (more than 12 carbon units) have lower friction coefficients compared with their short chain counterparts. Recently, Kim et al investigated the effect of the size of chain termination on frictional properties and found that the difference in friction arises predominantly from the difference in the size of the terminal groups. The AFM/FFM has also been applied to study frictional... 

The computational cost for CIS is similar to that of the ground-state HF calculation and can also be performed directly in the AO basis thus minimizing hard-disk I/O. The accuracy of CIS for excited states is roughly that of HF for the ground state, that is, dynamic electron correlation effects are neglected (for ground and excited states). 

Donsmark J, Rischel C (2007) Fluorescence correlation spectroscopy at the oil — water interface hard disk diffusion behavior in dilute p-lactoglobulin layers precedes monolayer formation. Langmuir 23(12) 6614—6623... 

NATIONAL INSTRUMENTS, Texas, USA), and were stored on the hard disk of a personal computer. With recorded EGG signal, R-R intervals (RRI) were calculated using a cross-correlation fimetion to detect R waves. And analysis of time series of RRI to estimate heart rate variability (HRV) was performed with STFT as well as analysis of EGG. Indexes of HRV were defined as below. The component of low frequency (LF) and high frequency (HE) were given as area of frequency spectrum from 0.05 to 0.15 Hz and 0.15 to 0.4 Hz respectively, and then ratio of LF to HE (LF/HF) reflects the activity of sympathetic nervous system and HE reflects the activity of parasympathetic nervous system. Obtained time series of HE and LF/HF were divided for sub-periods as well as DP and DF. 

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