Role of Molecular Chaos in Granular Fluctuating Hydrodynamics

G. Costantini; A. Puglisi

Mathematical Modelling of Natural Phenomena (2011)

  • Volume: 6, Issue: 4, page 2-18
  • ISSN: 0973-5348

Abstract

top
We perform a numerical study of the fluctuations of the rescaled hydrodynamic transverse velocity field during the cooling state of a homogeneous granular gas. We are interested in the role of Molecular Chaos for the amplitude of the hydrodynamic noise and its relaxation in time. For this purpose we compare the results of Molecular Dynamics (MD, deterministic dynamics) with those from Direct Simulation Monte Carlo (DSMC, random process), where Molecular Chaos can be directly controlled. It is seen that the large time decay of the fluctuation’s autocorrelation is always dictated by the viscosity coefficient predicted by granular hydrodynamics, independently of the numerical scheme (MD or DSMC). On the other side, the noise amplitude in Molecular Dynamics, which is known to violate the equilibrium Fluctuation-Dissipation relation, is not always accurately reproduced in a DSMC scheme. The agreement between the two models improves if the probability of recollision (controlling Molecular Chaos) is reduced by increasing the number of virtual particles per cells in the DSMC. This result suggests that DSMC is not necessarily more efficient than MD, if the real number of particles is small (~103 ± 104) and if one is interested in accurately reproduce fluctuations. An open question remains about the small-times behavior of the autocorrelation function in the DSMC, which in MD and in kinetic theory predictions is not a straight exponential.

How to cite

top

Costantini, G., and Puglisi, A.. "Role of Molecular Chaos in Granular Fluctuating Hydrodynamics." Mathematical Modelling of Natural Phenomena 6.4 (2011): 2-18. <http://eudml.org/doc/222326>.

@article{Costantini2011,
abstract = {We perform a numerical study of the fluctuations of the rescaled hydrodynamic transverse velocity field during the cooling state of a homogeneous granular gas. We are interested in the role of Molecular Chaos for the amplitude of the hydrodynamic noise and its relaxation in time. For this purpose we compare the results of Molecular Dynamics (MD, deterministic dynamics) with those from Direct Simulation Monte Carlo (DSMC, random process), where Molecular Chaos can be directly controlled. It is seen that the large time decay of the fluctuation’s autocorrelation is always dictated by the viscosity coefficient predicted by granular hydrodynamics, independently of the numerical scheme (MD or DSMC). On the other side, the noise amplitude in Molecular Dynamics, which is known to violate the equilibrium Fluctuation-Dissipation relation, is not always accurately reproduced in a DSMC scheme. The agreement between the two models improves if the probability of recollision (controlling Molecular Chaos) is reduced by increasing the number of virtual particles per cells in the DSMC. This result suggests that DSMC is not necessarily more efficient than MD, if the real number of particles is small (~103 ± 104) and if one is interested in accurately reproduce fluctuations. An open question remains about the small-times behavior of the autocorrelation function in the DSMC, which in MD and in kinetic theory predictions is not a straight exponential. },
author = {Costantini, G., Puglisi, A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {granular gas; fluctuations; hydrodynamics; numerical simulations},
language = {eng},
month = {7},
number = {4},
pages = {2-18},
publisher = {EDP Sciences},
title = {Role of Molecular Chaos in Granular Fluctuating Hydrodynamics},
url = {http://eudml.org/doc/222326},
volume = {6},
year = {2011},
}

TY - JOUR
AU - Costantini, G.
AU - Puglisi, A.
TI - Role of Molecular Chaos in Granular Fluctuating Hydrodynamics
JO - Mathematical Modelling of Natural Phenomena
DA - 2011/7//
PB - EDP Sciences
VL - 6
IS - 4
SP - 2
EP - 18
AB - We perform a numerical study of the fluctuations of the rescaled hydrodynamic transverse velocity field during the cooling state of a homogeneous granular gas. We are interested in the role of Molecular Chaos for the amplitude of the hydrodynamic noise and its relaxation in time. For this purpose we compare the results of Molecular Dynamics (MD, deterministic dynamics) with those from Direct Simulation Monte Carlo (DSMC, random process), where Molecular Chaos can be directly controlled. It is seen that the large time decay of the fluctuation’s autocorrelation is always dictated by the viscosity coefficient predicted by granular hydrodynamics, independently of the numerical scheme (MD or DSMC). On the other side, the noise amplitude in Molecular Dynamics, which is known to violate the equilibrium Fluctuation-Dissipation relation, is not always accurately reproduced in a DSMC scheme. The agreement between the two models improves if the probability of recollision (controlling Molecular Chaos) is reduced by increasing the number of virtual particles per cells in the DSMC. This result suggests that DSMC is not necessarily more efficient than MD, if the real number of particles is small (~103 ± 104) and if one is interested in accurately reproduce fluctuations. An open question remains about the small-times behavior of the autocorrelation function in the DSMC, which in MD and in kinetic theory predictions is not a straight exponential.
LA - eng
KW - granular gas; fluctuations; hydrodynamics; numerical simulations
UR - http://eudml.org/doc/222326
ER -

References

top
  1. A. Barrat, V. Loreto, A. Puglisi. Temperature probes in binary granular gases. Physica A, 334 (2004), No. 3-4, 513–523.  
  2. G. A. Bird. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon, Oxford, 1994.  
  3. J. J. Brey, M. I. Garcia de Soria, P. Maynar. Breakdown of the fluctuation-dissipation relations in granular gases. Europhys. Lett., 84 (2008), No. 2, 24002.  
  4. J. J. Brey, J. W. Dufty, C. S. Kim, A. Santos. Hydrodynamics for granular flow at low density. Phys. Rev. E, 58 (1998), No. 4, 4638–4653.  
  5. J. J. Brey, P. Maynar, M. I. Garcia de Soria. Fluctuating hydrodynamics for dilute granular gases. Phys. Rev. E, 79 (2009), No. 5, 051305.  
  6. J. J. Brey, M. J. Ruiz-Montero. Validity of the boltzmann equation to describe low-density granular systems. Phys. Rev. E, 69 (2004), No. 1, 011305.  
  7. J. J. Brey, M. J. Ruiz-Montero, F. Moreno. Instability and spatial correlations in a dilute granular gas. Phys. Fluids, 10 (1008), No. 11, 2976–2982.  
  8. J. J. Brey, M. J. Ruiz-Montero, F. Moreno. Steady-state representation of the homogeneous cooling state of a granular gas. Phys. Rev. E, 69 (2004), No. 051303–.  
  9. J.J. Brey, M.I.G. de Soria, P. Maynar, M.J. Ruiz-Montero. Energy fluctuations in the homogeneous cooling state of granular gases. Phys. Rev. E, 70 (2004), No. 1, 011302.  
  10. J.J. Brey, M.J. Ruiz-Montero. Average energy and fluctuations of a granular gas at the threshold of the clustering instability. Granular Matter, 10 (2007), No. 1, 53–59.  Zbl1200.76202
  11. G. Costantini, A. Puglisi. Fluctuating hydrodynamics for dilute granular gases: a Monte Carlo study. Phys. Rev. E, 82 (2010), No. 1, 011305.  
  12. G. Costantini, A. Puglisi, U. Marini Bettolo Marconi. Granular Brownian ratchet model. Phys. Rev. E, 75 (2007), No. 6, 061124–.  
  13. G. Costantini, A. Puglisi, U. Marini Bettolo Marconi. Velocity fluctuations in a one dimensional inelastic Maxwell model. J. Stat. Mech., (2008), P08031.  
  14. J. W. Dufty, J. J. Brey. Green-Kubo expressions for a granular gas. J. Stat. Phys., 109 (2002), No. 3-4, 433–448.  Zbl1015.82032
  15. J. Eggers. Sand as Maxwell’s demon. Phys. Rev. Lett., 83 (1999), No. 25, 5322–5325.  
  16. K. Feitosa, N. Menon. Breakdown of energy equipartition in a 2d binary vibrated granular gas. Phys. Rev. Lett., 88 (2002), No. 19, 198301.  
  17. A. L. Garcia, M. Malek Mansour, G. C. Lie, M. Mareschal, E. Clementi. Hydrodynamic fluctuations in a dilute gas under shear. Phys. Rev. A, 36 (1987), No. 9, 4348–4355.  
  18. I. Goldhirsch. Scales and kinetics of granular flows. Chaos, 9 (1999), No. 3, 659–672.  Zbl1055.76569
  19. I. Goldhirsch, G. Zanetti. Clustering instability in dissipative gases. Phys. Rev. Lett., 70 (1993), No. 11, 1619–1622.  
  20. R. Kubo, M. Toda, N. Hashitsume. Statistical physics II: Nonequilibrium stastical mechanics. Springer, Berlin, 1991.  Zbl0757.60109
  21. L. D. Landau, E. M. Lifchitz. Physique Statistique. Éditions MIR, Moscow, 1967.  
  22. J. F. Lutsko. Molecular chaos, pair correlations, and shear-induced ordering of hard spheres. Phys. Rev. Lett., 77 (1996), No. 11, 2225–2228.  
  23. J. F. Lutsko. A model for the atomic-scale structure of the homogeneous cooling state of granular fluids. Phys. Rev. E, 63 (2001), No. 6, 061211.  
  24. M. Mansour Malek, A. L. Garcia, G. C. Lie, E. Clementi. Fluctuating hydrodynamics in a dilute gas. Phys. Rev. Lett., 58 (1987), No. 9, 874–877.  
  25. U. Marini Bettolo Marconi, A. Puglisi. Mean-field model of free-cooling inelastic mixtures. Phys. Rev. E, 65 (2002), No. 5, 051305.  Zbl1174.82326
  26. U. Marini Bettolo Marconi, A. Puglisi, L. Rondoni, A. Vulpiani. Fluctuation-dissipation: Response theory in statistical physics. Phys. Rep., 461 (2008), No. 4-6, 111–195.  
  27. P. Maynar, M. I. G. de Soria, E. Trizac. Fluctuating hydrodynamics for driven granular gases. Eur. Phys. J. Special Topics, 170 (2009), No. 1, 123–139.  
  28. R. Pagnani, U. Marini Bettolo Marconi, A. Puglisi. Driven low density granular mixtures. Phys. Rev. E, 66 (2002), No. 5, 051304.  
  29. T. Pöschel, N. Brilliantov, editors. Granular Gas Dynamics. Lecture Notes in Physics 624. Springer, Berlin, 2003.  Zbl1028.82534
  30. T. Pöschel, S. Luding, editors.Granular Gases. Lecture Notes in Physics 564. Springer, Berlin, 2001.  
  31. A. Puglisi, A. Baldassarri, V. Loreto. Fluctuation-dissipation relations in driven granular gases. Physical Review E, 66 (2002), No. 6, 061305.  
  32. A. Puglisi, A. Baldassarri, A. Vulpiani. Violations of the Einstein relation in granular fluids: the role of correlations. J. Stat. Mech., (2007), P08016.  
  33. A. Sarracino, D. Villamaina, G. Costantini, A. Puglisi. Granular brownian motion. J. Stat. Mech., (2010) P04013.  
  34. A. Sarracino, D. Villamaina, G. Gradenigo, A. Puglisi. Irreversible dynamics of a massive intruder in dense granular fluids. Europhys. Lett., 92 (2010), No. 3, 34001.  Zbl1267.82093
  35. T. C. P. van Noije, M. H. Ernst, R. Brito, J. A. G. Orza. Mesoscopic theory of granular fluids. Phys. Rev. Lett., 79 (1007), No. 3, 411–414.  
  36. D. Villamaina, A. Puglisi, A. Vulpiani. The fluctuation-dissipation relation in sub-diffusive systems: the case of granular single-file diffusion. J. Stat. Mech., (2008), L10001.  
  37. P. Visco, A. Puglisi, A. Barrat, F. van Wijland, E. Trizac. Energy fluctuations in vibrated and driven granular gases. Eur. Phys. J. B, 51 (2006), No. 3, 377–387. Zbl1107.82369

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.