Self-diffusion in granular gases

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 61, Issue 2, 2000, Pages 1716-1721

  Brilliantov, Nikolai V ^a,b   Pöschel, Thorsten ^b  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

^b HUMBOLDT UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

GRANULAR MATERIALS;

FREELY COOLING GRANULAR GAS; GRANULAR GAS; POWER-LAW TIME DEPENDENCES; RESTITUTION COEFFICIENT; SELF-DIFFUSION; TIME DEPENDENCE; VELOCITY DEPENDENCE; VELOCITY-DEPENDENT;

GASES;

EID: 0001640547     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.61.1716     Document Type: Article

References (37)

1. I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619 (1993).
2. S. McNamara and W. R. Young, Phys. Rev. E 50, R28 (1993).
3. P. Deltour and J. L. Barrat, J. Phys. I 7, 137 (1997)
4. F. Spahn, U. Schwarz, and J. Kurths, Phys. Rev. Lett. 78, 1596 (1997)
5. T. Aspelmeier, G. Giese, and A. Zippelius, Phys. Rev. E 57, 857 (1997).
6. R. M. Brach, J. Appl. Mech. 56, 133 (1989)
7. S. Wall, W. John, H. C. Wang, and S. L. Goren, Aerosol. Sci. Technol. 12, 926 (1990)
8. W. Goldsmith, Impact: The Theory and Physical Behaviour of Colliding Solids (Edward Arnold, London, 1960)
9. P. F. Luckham, Powder Technol. 58, 75 (1989)
10. F. G. Bridges, A. Hatzes, and D. N. C. Lin, Nature (London) 309, 333 (1984).
11. N. Brilliantov, F. Spahn, J.-M. Hertzsch, and T. Pöschel, Phys. Rev. E 53, 5382 (1996)
12. Phys. Rev. EW. A. M. Morgado and I. Oppenheim, 55, 1940 (1997)
13. G. Kuwabara and K. Kono, Jpn. J. Appl. Phys., Part 1 26, 1230 (1987).
14. T. Schwager and T. Pöschel, Phys. Rev. E 57, 650 (1998).
15. R. Ramírez, T. Pöschel, N. V. Brilliantov, and T. Schwager, Phys. Rev. E 60, 4465 (1999).
16. These conditions may be satisfied, e.g., for ice at very low temperature (in the context of planetary ring dynamics). The experimental results by Bridges et al. (Fig. 1 of the last reference in 4) can be fitted with good accuracy with the coefficient of restitution of the viscoelastic model (Fig. 1 of the first reference in 5).
17. S. E. Esipov and T. Pöschel, J. Stat. Phys. 86, 1385 (1997).
18. T. P. C. van Noije and M. H. Ernst, Physica A 251, 266 (1998).
19. T. P. C. van Noije and M. H. Ernst, Granular Matter 1, 57 (1998).
20. References 91011 investigate the velocity distribution function for constant restitution coefficient and find that the Maxwell distribution is a good approximation. For the velocity-dependent restitution coefficient we expect the velocity distribution is close to the Maxwellian too.
21. The hydrodynamic contribution to the self-diffusion coefficient grows with time (as a time integral of the time correlation function). For granular materials it may be much more important than for fluids due to long-ranged spatial correlations in the velocity field 14. This problem is beyond the scope of the present study.
22. T. P. C. van Noije and M. H. Ernst, R. Brito, and J. A. G. Orza, Phys. Rev. Lett. 79, 411 (1997).
23. The term “pseudo” was initially used to refer to the dynamics of systems with singular hard-core potential 1617.
24. M. H. Ernst, J. R. Dorfman, W. R. Hoegy, and J. M. J. van Leeuwen, Physica A 45, 127 (1969).
25. P. Resibois and M. de Leener, Classical Kinetic Theory of Fluids (Wiley, New York, 1977).
26. For application to “ordinary” fluids, see 19, and to granular systems 1020. A rigorous definition of (Formula presented) includes a prefactor, preventing successive collisions of the same pair of particles 1617, which, however, does not affect the present analysis.
27. D. Chandler, J. Chem. Phys. 60, 3500 (1974)
28. J. Chem. Phys.B. J. Berne, 66, 2821 (1977)
29. N. V. Brilliantov and O. P. Revokatov, Chem. Phys. Lett. 104, 444 (1984).
30. M. Huthmann and A. Zippelius, Phys. Rev. E 56, R6275 (1997)
31. Phys. Rev. ES. Luding, M. Huthmann, S. McNamara, and A. Zippelius, 58, 3416 (1998).
32. Note that Eq. (33) is an expansion in terms of (Formula presented). Thus, it restricts the temperature T to be small as compared with (Formula presented).
33. The equations for (Formula presented)–(Formula presented) are (Formula presented) (Formula presented) (Formula presented) (Formula presented) (Formula presented) –(Formula presented).
34. To neglect terms of the order of (Formula presented) in Eq. (41) the following condition is required: (Formula presented)
35. R. Brito and M. H. Ernst, Europhys. Lett. 43, 497 (1998).
36. Another problem of using the number of collision as an inherent time is connected with the phenomenon of inelastic collapse 2, where an infinite number of collisions occurs in finite time. Suppose three particles of a (finite) system undergo an inelastic collapse at time (Formula presented), hence, (Formula presented). This situation may occur in a homogeneously cooling system. If we choose (Formula presented) to be the system inherent time, this time scale does not allow us to describe the evolution of the system for times larger than (Formula presented), whereas the temperature-based time scale (Formula presented) is unaffected by the collapse.
37. S. McNamara and W. R. Young, Phys. Rev. E 53, 5089 (1996).