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Journal articles on the topic 'Velocity autocorrelation function'

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Published: 10 March 2023

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1

Leegwater, Jan A. "Velocity autocorrelation function of Lennard‐Jones fluids." Journal of Chemical Physics 94, no. 11 (June 1991): 7402–10. http://dx.doi.org/10.1063/1.460171.

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2

Chakraborty, D. "Velocity autocorrelation function of a Brownian particle." European Physical Journal B 83, no. 3 (October 2011): 375–80. http://dx.doi.org/10.1140/epjb/e2011-20395-3.

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3

Cichocki, B., and B. U. Felderhof. "Velocity autocorrelation function of interacting Brownian particles." Physical Review E 51, no. 6 (June 1, 1995): 5549–55. http://dx.doi.org/10.1103/physreve.51.5549.

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4

CHANG, KEH-CHIN, CHIUAN-TING LI, and HSUAN-JUNG CHEN. "EXPERIMENTAL INVESTIGATION OF VELOCITY AUTOCORRELATION FUNCTIONS IN TURBULENT PLANAR MIXING LAYER." Modern Physics Letters B 24, no. 13 (May 30, 2010): 1361–64. http://dx.doi.org/10.1142/s0217984910023621.

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The velocity autocorrelation coefficient correlates the velocity in the time domain but at the same spatial position. Turbulent planar mixing layer consists of two types of turbulence, that is, shear turbulence in the central shear layer and nearly homogeneous turbulence in both the high- and low-speed free stream sides. It is interesting to know what kind of function forms can be used to represent faithfully the experimental observations of the velocity autocorrelation coefficients in the mixing layer. Various velocity autocorrelation functions are tested with the measured data. It is found that the Frenkiel function family is the most proper form to represent the measured velocity autocorrelation coefficients in both the shear layer and free stream regimes.
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5

Mroczek, Stefan, and Frederik Tilmann. "Joint ambient noise autocorrelation and receiver function analysis of the Moho." Geophysical Journal International 225, no. 3 (February 19, 2021): 1920–34. http://dx.doi.org/10.1093/gji/ggab065.

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SUMMARY In the field of seismic interferometry, cross-correlations are used to extract Green’s function from ambient noise data. By applying a single station variation of the method, using autocorrelations, we are in principle able to retrieve zero-offset reflections in a stratified Earth. These reflections are valuable as they do not require an active seismic source and, being zero-offset, are better constrained in space than passive earthquake based measurements. However, studies that target Moho signals with ambient noise autocorrelations often give ambiguous results with unclear Moho reflections. Using a modified processing scheme and phase-weighted stacking, we determine the Moho P-wave reflection time from vertical autocorrelation traces for a test station with a known simple crustal structure (HYB in Hyderabad, India). However, in spite of the simplicity of the structure, the autocorrelation traces show several phases not related to direct reflections. Although we are able to match some of these additional phases in a qualitative way with synthetic modelling, their presence makes it hard to identify the reflection phases without prior knowledge. This prior knowledge can be provided by receiver functions. Receiver functions (arising from mode conversions) are sensitive to the same boundaries as autocorrelations, so should have a high degree of comparability and opportunity for combined analysis but in themselves are not able to independently resolve VP, VS and Moho depth. Using the timing suggested by the receiver functions as a guide, we observe the Moho S-wave reflection on the horizontal autocorrelation of the north component but not on the east component. The timing of the S reflection is consistent with the timing of the PpSs–PsPs receiver function multiple, which also depends only on the S velocity and Moho depth. Finally, we combine P receiver functions and autocorrelations from HYB in a depth–velocity stacking scheme that gives us independent estimates for VP, VS and Moho depth. These are found to be in good agreement with several studies that also supplement receiver functions to obtain unique crustal parameters. By applying the autocorrelation method to a portion of the EASI transect crossing the Bohemian Massif in central Europe, we find approximate consistency with Moho depths determined from receiver functions and spatial coherence between stations, thereby demonstrating that the method is also applicable for temporary deployments. Although application of the autocorrelation method requires great care in phase identification, it has the potential to resolve both average crustal P and S velocities alongside Moho depth in conjunction with receiver functions.
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6

Kumari, Shikha, and Syed Rashid Ahmad. "Velocity autocorrelation function in uniformly heated granular gas." EPJ Web of Conferences 140 (2017): 04007. http://dx.doi.org/10.1051/epjconf/201714004007.

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7

Balucani, U., J. P. Brodholt, and R. Vallauri. "Analysis of the velocity autocorrelation function of water." Journal of Physics: Condensed Matter 8, no. 34 (August 19, 1996): 6139–44. http://dx.doi.org/10.1088/0953-8984/8/34/004.

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8

Cichocki, B., and B. U. Felderhof. "Rotational velocity autocorrelation function of interacting Brownian particles." Physica A: Statistical Mechanics and its Applications 289, no. 3-4 (January 2001): 409–18. http://dx.doi.org/10.1016/s0378-4371(00)00532-x.

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9

Chtchelkatchev, N. M., and R. E. Ryltsev. "Complex singularities of the fluid velocity autocorrelation function." JETP Letters 102, no. 10 (November 2015): 643–49. http://dx.doi.org/10.1134/s0021364015220038.

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10

Lee, M. H. "Comment on 'Velocity autocorrelation function in fluctuating hydrodynamics'." Journal of Physics: Condensed Matter 4, no. 50 (December 14, 1992): 10487–92. http://dx.doi.org/10.1088/0953-8984/4/50/037.

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11

Krishnan, S. H., and K. G. Ayappa. "Modeling velocity autocorrelation functions of confined fluids: A memory function approach." Journal of Chemical Physics 118, no. 2 (January 8, 2003): 690–705. http://dx.doi.org/10.1063/1.1524191.

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12

Iwashita, Takuya, Yasuya Nakayama, and Ryoichi Yamamoto. "Velocity Autocorrelation Function of Fluctuating Particles in Incompressible Fluids." Progress of Theoretical Physics Supplement 178 (2009): 86–91. http://dx.doi.org/10.1143/ptps.178.86.

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13

Masters, A. J. "Long‐time behavior of the angular velocity autocorrelation function." Journal of Chemical Physics 105, no. 21 (December 1996): 9695–97. http://dx.doi.org/10.1063/1.472810.

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14

Imam, Mohamed A., and Mohamed A. Osman. "Autocorrelation function of velocity fluctuations and noise in diamond." Diamond and Related Materials 2, no. 1 (February 1993): 15–18. http://dx.doi.org/10.1016/0925-9635(93)90136-p.

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15

Hoef, M. A. Van Der, M. Dijkstra, and D. Frenkel. "Velocity Autocorrelation Function in a Four-Dimensional Lattice Gas." Europhysics Letters (EPL) 17, no. 1 (January 1, 1992): 39–43. http://dx.doi.org/10.1209/0295-5075/17/1/008.

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16

Lowe, C. P., M. H. J. Hagen, and D. Frenkel. "Response to “Rotational velocity autocorrelation function of interacting Brownian particles”." Physica A: Statistical Mechanics and its Applications 289, no. 3-4 (January 2001): 419–21. http://dx.doi.org/10.1016/s0378-4371(00)00533-1.

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17

Singla, B., K. Tankeshwar, and K. N. Pathak. "Velocity autocorrelation function of a two-dimensional classical electron fluid." Physical Review A 41, no. 8 (April 1, 1990): 4306–11. http://dx.doi.org/10.1103/physreva.41.4306.

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18

Garland, Gregory E., and James W. Dufty. "Computer simulation of the velocity autocorrelation function at low density." International Journal of Quantum Chemistry 22, S16 (June 19, 2009): 91–100. http://dx.doi.org/10.1002/qua.560220811.

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19

Kawski, A., Z. Kojro, P. Bojarski, and J. Lichacz. "Rotational Depolarization of Fluorescence of Prolate Molecules." Zeitschrift für Naturforschung A 45, no. 11-12 (December 1, 1990): 1357–60. http://dx.doi.org/10.1515/zna-1990-11-1222.

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AbstractThe theory of rotational depolarization of fluorescence, taking into account the inertial effect, has been verified experimentally for two angular velocity autocorrelation functions obtained for both the approximate and the exact Langevin equation. To this end, the emission anisotropy was investigated as a function of viscosity for PPO, POPOP, and α-NOPON in liquid n-paraffins (from « = 5 to n = 11) at 304.5 K. For these three luminescent molecules with mean lifetimes of about 1 ns, and the viscosities ranging from 0.22 to 0.9 cP, no essential difference was found between the experiment and the theory for the angular velocity autocorrelation functions employed.
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20

Wang, H., R. J. Barthelmie, P. Doubrawa, and S. C. Pryor. "Errors in radial velocity variance from Doppler wind lidar." Atmospheric Measurement Techniques 9, no. 8 (August 29, 2016): 4123–39. http://dx.doi.org/10.5194/amt-9-4123-2016.

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Abstract. A high-fidelity lidar turbulence measurement technique relies on accurate estimates of radial velocity variance that are subject to both systematic and random errors determined by the autocorrelation function of radial velocity, the sampling rate, and the sampling duration. Using both statistically simulated and observed data, this paper quantifies the effect of the volumetric averaging in lidar radial velocity measurements on the autocorrelation function and the dependence of the systematic and random errors on the sampling duration. For current-generation scanning lidars and sampling durations of about 30 min and longer, during which the stationarity assumption is valid for atmospheric flows, the systematic error is negligible but the random error exceeds about 10 %.
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21

Wallace, Duane C. "Liquid dynamics theory of the velocity autocorrelation function and self-diffusion." Physical Review E 58, no. 1 (July 1, 1998): 538–45. http://dx.doi.org/10.1103/physreve.58.538.

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22

Erpenbeck, Jerome J., and Wiilliam W. Wood. "Molecular-dynamics calculations of the velocity autocorrelation function: Hard-sphere results." Physical Review A 32, no. 1 (July 1, 1985): 412–22. http://dx.doi.org/10.1103/physreva.32.412.

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23

Montfrooij, Wouter, and Ignatz de Schepper. "Velocity autocorrelation function of simple dense fluids from neutron scattering experiments." Physical Review A 39, no. 5 (March 1, 1989): 2731–33. http://dx.doi.org/10.1103/physreva.39.2731.

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24

Verkerk, P., J. Westerweel, U. Bafile, L. A. de Graaf, W. Montfrooij, and I. M. de Schepper. "Velocity autocorrelation function of dense hydrogen gas determined by neutron scattering." Physical Review A 40, no. 5 (September 1, 1989): 2860–63. http://dx.doi.org/10.1103/physreva.40.2860.

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25

Huang, Y. X., F. G. Schmitt, Z. M. Lu, and Y. L. Liu. "Autocorrelation function of velocity increments time series in fully developed turbulence." EPL (Europhysics Letters) 86, no. 4 (May 1, 2009): 40010. http://dx.doi.org/10.1209/0295-5075/86/40010.

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26

Friedman, B., and R. F. Martin. "Behavior of the velocity autocorrelation function for the periodic lorentz gas." Physica D: Nonlinear Phenomena 30, no. 1-2 (February 1988): 219–27. http://dx.doi.org/10.1016/0167-2789(88)90108-x.

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27

Beijeren, H. van, and K. W. Kehr. "Correlation factor, velocity autocorrelation function and frequency-dependent tracer diffusion coefficient." Journal of Physics C: Solid State Physics 19, no. 9 (March 30, 1986): 1319–28. http://dx.doi.org/10.1088/0022-3719/19/9/005.

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28

Singla, B., K. Tankeshwar, and K. N. Pathak. "Erratum: Velocity autocorrelation function of a two-dimensional classical electron fluid." Physical Review A 42, no. 6 (September 1, 1990): 3642. http://dx.doi.org/10.1103/physreva.42.3642.

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29

Kirkpatrick, T. R. "Does the velocity autocorrelation function oscillate in a hard-sphere crystal?" Journal of Statistical Physics 57, no. 3-4 (November 1989): 483–96. http://dx.doi.org/10.1007/bf01022818.

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30

Oleksy, Czesław. "The velocity autocorrelation function for a random walk on percolation clusters." Physica A: Statistical Mechanics and its Applications 205, no. 4 (April 1994): 487–96. http://dx.doi.org/10.1016/0378-4371(94)90215-1.

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31

Parker, Kevin, Jonathan Carroll-Nellenback, and Ronald Wood. "The 3D Spatial Autocorrelation of the Branching Fractal Vasculature." Acoustics 1, no. 2 (April 9, 2019): 369–81. http://dx.doi.org/10.3390/acoustics1020020.

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The fractal branching vasculature within soft tissues and the mathematical properties of the branching system influence a wide range of important phenomena from blood velocity to ultrasound backscatter. Among the mathematical descriptors of branching networks, the spatial autocorrelation function plays an important role in statistical measures of the tissue and of wave propagation through the tissue. However, there are open questions about analytic models of the 3D autocorrelation function for the branching vasculature and few experimental validations for soft vascularized tissue. To address this, high resolution computed tomography scans of a highly vascularized placenta perfused with radiopaque contrast through the umbilical artery were examined. The spatial autocorrelation function was found to be consistent with a power law, which then, in theory, predicts the specific power law behavior of other related functions, including the backscatter of ultrasound.
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32

Rudyak, V. Ya, G. V. Kharlamov, and A. A. Belkin. "The velocity autocorrelation function of nanoparticles in a hard-sphere molecular system." Technical Physics Letters 26, no. 7 (July 2000): 553–56. http://dx.doi.org/10.1134/1.1262909.

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33

Cichocki, B., and B. U. Felderhof. "Comment on “Response to ‘Rotational velocity autocorrelation function of interacting Brownian particles’”." Physica A: Statistical Mechanics and its Applications 297, no. 1-2 (August 2001): 115–16. http://dx.doi.org/10.1016/s0378-4371(01)00224-2.

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34

Tsekov, R., and B. Radoev. "Velocity autocorrelation function in fluctuating hydrodynamics: frequency dependence of the kinematic viscosity." Journal of Physics: Condensed Matter 4, no. 20 (May 18, 1992): L303—L305. http://dx.doi.org/10.1088/0953-8984/4/20/001.

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35

Brey, J. J., J. Gómez Ordóñez, and A. Santos. "Simulation results for the velocity autocorrelation function in a bond percolation model." Physics Letters A 136, no. 1-2 (March 1989): 26–29. http://dx.doi.org/10.1016/0375-9601(89)90669-5.

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36

Chimoto, Kosuke, and Hiroaki Yamanaka. "Tuning S-Wave Velocity Structure of Deep Sedimentary Layers in the Shimousa Region of the Kanto Basin, Japan, Using Autocorrelation of Strong-Motion Records." Bulletin of the Seismological Society of America 110, no. 6 (July 28, 2020): 2882–91. http://dx.doi.org/10.1785/0120200156.

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ABSTRACT The autocorrelation of ambient noise is used to capture reflected waves for crustal and sedimentary structures. We applied autocorrelation to strong-motion records to capture the reflected waves from sedimentary layers and used them for tuning the S-wave velocity structure of these layers. Because a sedimentary-layered structure is complicated and generates many reflected waves, it is important to identify the boundary layer from which the waves reflected. We used spectral whitening during autocorrelation analysis to capture the reflected waves from the seismic bedrock with an appropriate smoothing band, which controls the wave arrival from the desired layer boundary. The effect of whitening was confirmed by the undulation frequency observed in the transfer function of the sedimentary layers. After careful determination of parameters for spectral whitening, we applied data processing to the strong-motion records observed at the stations in the Shimousa region of the Kanto Basin, Japan, to estimate the arrival times of the reflected waves. The arrival times of the reflected waves were found to be fast in the northern part of the Shimousa region and slow in the western and southern parts. These arrival times are consistent with those obtained using existing models. Because we observed a slight difference in the arrival times, the autocorrelation function at each station was used for tuning the S-wave velocity structure model of the sedimentary layers using the inversion technique. The tuned models perfectly match the autocorrelation functions in terms of the arrival time of the reflected waves from the seismic bedrock.
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37

Stodolka¹, Jacek, Marian Golema², and Juliusz Migasiewicz. "Balance Maintenance in the Upright Body Position: Analysis of Autocorrelation." Journal of Human Kinetics 50, no. 1 (April 1, 2016): 45–52. http://dx.doi.org/10.1515/hukin-2015-0140.

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Abstract The present research aimed to analyze values of the autocorrelation function measured for different time values of ground reaction forces during stable upright standing. It was hypothesized that if recording of force in time depended on the quality and way of regulating force by the central nervous system (as a regulator), then the application of autocorrelation for time series in the analysis of force changes in time function would allow to determine regulator properties and its functioning. The study was performed on 82 subjects (students, athletes, senior and junior soccer players and subjects who suffered from lower limb injuries). The research was conducted with the use of two Kistler force plates and was based on measurements of ground reaction forces taken during a 15 s period of standing upright while relaxed. The results of the autocorrelation function were statistically analyzed. The research revealed a significant correlation between a derivative extreme and velocity of reaching the extreme by the autocorrelation function, described as gradient strength. Low correlation values (all statistically significant) were observed between time of the autocorrelation curve passing through 0 axis and time of reaching the first peak by the said function. Parameters computed on the basis of the autocorrelation function are a reliable means to evaluate the process of flow of stimuli in the nervous system. Significant correlations observed between the parameters of the autocorrelation function indicate that individual parameters provide similar properties of the central nervous system.
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38

Gwizdała, W., A. Dawid, and Z. Gburski. "Molecular Dynamics Simulation Study of the Liquid Crystal Phase in a Small Mesogene Cluster (5CB)22." Solid State Phenomena 140 (October 2008): 89–96. http://dx.doi.org/10.4028/www.scientific.net/ssp.140.89.

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The molecular dynamics (MD) technique was used to investigate the nano droplet composed of twenty mesogene molecules 4-cyano-4-n-pentylbiphenyl (5CB). The 5CB molecules were treated as rigid bodies, the intermolecular interaction was taken to be the full site-site pairwise additive Lennard-Jones (LJ) potential plus a Coulomb interaction. The radial distribution functions in the temperature range from 150 to 400 K, were calculated as well as the linear and angular velocity autocorrelation functions. In addition the total dipole moment autocorrelation function and dielectric loss of (5CB)22 mesogene cluster were calculated and the liquid crystal ordering in the nanoscale system was studied up to its vaporization temperature.
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39

Jameson, A. R., and A. B. Kostinski. "On the Enhanced Temporal Coherency of Radar Observations in Precipitation." Journal of Applied Meteorology and Climatology 49, no. 8 (August 1, 2010): 1794–804. http://dx.doi.org/10.1175/2010jamc2403.1.

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Abstract In this work, the authors present observations of enhanced temporal coherency beyond that expected using the observations of the standard deviation of the Doppler velocities and the assumption of a family of exponentially decaying autocorrelation functions. The purpose of this paper is to interpret these observations by developing the complex amplitude autocorrelation function when both incoherent and coherent backscatter are present. Using this expression, it is then shown that when coherent scatter is present, the temporal coherency increases as observed. Data are analyzed in snow and in rain. The results agree with the theoretical expectations, and the authors interpret this agreement as an indication that coherent scatter is the likely explanation for the observed enhanced temporal coherency. This finding does not affect decorrelation times measured using time series. However, when the time series is not available (as in theoretical studies), the times to decorrelation are often computed based upon the assumptions that the autocorrelation function is a member of the family of exponentially decaying autocorrelation functions and that the signal decorrelation is due solely to the Doppler velocity fluctuations associated with incoherent scatter. Such an approach, at times, may significantly underestimate the true required times to decorrelation thus leading to overestimates of statistical reliability of parameters.
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40

Marro, J., and J. Masoliver. "Long-time tails in the velocity autocorrelation function of hard-rod binary mixtures." Physical Review Letters 54, no. 8 (February 25, 1985): 731–34. http://dx.doi.org/10.1103/physrevlett.54.731.

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41

Kirkpatrick, T. R., and J. C. Nieuwoudt. "Mode-coupling theory of the intermediate-time behavior of the velocity autocorrelation function." Physical Review A 33, no. 4 (April 1, 1986): 2658–62. http://dx.doi.org/10.1103/physreva.33.2658.

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42

Scheunders, P., and J. Naudts. "Long-time tail of the velocity-autocorrelation function in the Lorentz lattice gas." Physical Review A 41, no. 6 (March 1, 1990): 3415–18. http://dx.doi.org/10.1103/physreva.41.3415.

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43

Macucci, M., B. Pellegrini, P. Terreni, and L. Reggiani. "Non-equilibrium velocity autocorrelation function for semiconductors in the presence of trapping phenomena." physica status solidi (b) 152, no. 2 (April 1, 1989): 601–16. http://dx.doi.org/10.1002/pssb.2221520222.

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44

Lowe, C. P., and A. J. Masters. "The long-time behaviour of the velocity autocorrelation function in a Lorentz gas." Physica A: Statistical Mechanics and its Applications 195, no. 1-2 (April 1993): 149–62. http://dx.doi.org/10.1016/0378-4371(93)90259-7.

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45

Wang, S. K., S. J. Lee, O. C. Jones, and R. T. Lahey. "Statistical Analysis of Turbulent Two-Phase Pipe Flow." Journal of Fluids Engineering 112, no. 1 (March 1, 1990): 89–95. http://dx.doi.org/10.1115/1.2909374.

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The statistical characteristics of turbulent two-phase pipe flow have been evaluated. In particular, the autocorrelation functions and the power spectral density functions of the axial turbulence fluctuations in the liquid phase were determined. The high frequency content of the power spectrum in bubbly two-phase pipe flow was found to be significantly larger than in single-phase pipe flow and, in agreement with previous studies of homogeneous two-phase flows (Lance et al., 1983), diminished asymptotically with a characteristic −8/3 slope at high frequency. The power spectrum and the autocorrelation functions in two-phase pipe flow, although distinctively different from those in single-phase pipe flow, were insensitive to the local void fraction and the mean liquid velocity when plotted against wave number and spatial separation, respectively. Finally, the dissipation scale, determined from the shape of the autocorrelation function, indicated that the turbulent dissipation rate in two-phase pipe flow was significantly greater than that in single-phase pipe flow.
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46

Suthar, P. H. "Dynamical Variable of Bi50Zn50 Bulk Metallic Glass at Various Temperature." Advanced Materials Research 1141 (August 2016): 166–70. http://dx.doi.org/10.4028/www.scientific.net/amr.1141.166.

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The theoretical study of dynamical variables: velocity autocorrelation function (VACF), power spectrum (PS) and mean square displacement (MSD) of Bi50Zn50 bulk metallic glass at T= 793K, 873K, 1023K and 1123K have been calculated based on the static harmonic well approximation. The effective interatomic potential for bulk metallic glass is computed using our established model potential with the exchange correlation functions due to Farid et al. It is observed that negative dip in velocity auto correlation function decreases as the temperature increases. In the power spectrum as temperature increases, the peak of power spectrum shifts toward lower ω. The obtained result of MSD concludes that the vibrating component in the atomic motion is decreases as increases the temperatures.
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47

VARLEY, RODNEY L. "THE BARE DIFFUSION COEFFICIENT AND THE PECULIAR VELOCITY AUTO-CORRELATION FUNCTION." Fluctuation and Noise Letters 06, no. 02 (June 2006): L179—L199. http://dx.doi.org/10.1142/s0219477506003288.

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The bare diffusion coefficient is given as the time integral of the peculiar velocity autocorrelation function or PVACF and this result is different from the well known Green-Kubo formula. The bare diffusion coefficient characterizes the diffusion process on a length scale lambda. The PVACF is given here for the first time in terms of the positions and velocities of the N particles of the system so the PVACF is in a form suitable for evaluation by molecular dynamics simulations. The computer simulations show that for the two dimensional hard disk system, the PVACF decays increasingly rapidly in time as lambda is reduced and this is probably a general characteristic. Finally, the Einstein formula for Brownian motion is a bit different for the bare diffusion coefficient.
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48

Tanaka, M. "Molecular Dynamics Study of Velocity Autocorrelation Function in a Model of Expanded Liquid Rubidium." Progress of Theoretical Physics Supplement 69 (May 14, 2013): 439–50. http://dx.doi.org/10.1143/ptp.69.439.

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49

Malomuzh, N. P., K. S. Shakun, and A. A. Kuznetsova. "New Possibilities Provided by the Analysis of the Molecular Velocity Autocorrelation Function in Liquids." Ukrainian Journal of Physics 63, no. 4 (June 18, 2018): 317. http://dx.doi.org/10.15407/ujpe63.4.317.

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Long-time tails of the molecular velocity autocorrelation function (VACF) in liquid argon at temperatures higher and lower than the spinodal temperature have been analyzed. By considering the time dependence of the VACF, the self-diffusion and shear viscosity coefficients, and the Maxwell relaxation time are determined, as well as their changes when crossing the spinodal. It is shown that the characteristic changes in the temperature dependences of the indicated kinetic coefficients allow the spinodal position to be determined with a high accuracy. A possibility toapply the proposed method to other low-molecular liquids is considered. As an example, nitrogen and oxygen are used, for which the averaged potential of intermolecular interaction has the Lennard-Jones form.
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HEYES, DAVID M., JACK G. POWLES, and GERALD RICKAYZEN. "The velocity autocorrelation function and self-diffusion coefficient of fluids with steeply repulsive potentials." Molecular Physics 100, no. 5 (March 10, 2002): 595–610. http://dx.doi.org/10.1080/00268970110096704.

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