Abstract
A mixture of dissipative hard grains generically exhibits a breakdown of kinetic energy equipartition. The undriven and thus freely cooling binary problem, in the tracer limit where the density of one species becomes minute, may exhibit an extreme form of this breakdown, with the minority species carrying a finite fraction of the total kinetic energy of the system. We investigate the fingerprint of this non-equilibrium phase transition, akin to an ordering process, on transport properties. The analysis, performed by solving the Boltzmann kinetic equation from a combination of analytical and Monte Carlo techniques, hints at the possible failure of hydrodynamics in the ordered region. As a relevant byproduct of the study, the behaviour of the second- and fourth-degree velocity moments is also worked out.
Graphical abstract
Similar content being viewed by others
References
N. Brilliantov, T. Pöschel, Kinetic Theory of Granular Gases (Oxford University Press, Oxford, 2004).
I. Goldhirsch, Annu. Rev. Fluid Mech. 35, 267 (2003).
A. Goldshtein, M. Shapiro, J. Fluid Mech. 282, 41 (1995).
J.J. Brey, J.W. Dufty, A. Santos, J. Stat. Phys. 87, 1051 (1997).
J.J. Brey, J.W. Dufty, A. Santos, J. Stat. Phys. 97, 2811 (1999).
J.W. Dufty, A. Baskaran, L. Zogaib, Phys. Rev. E 69, 051301 (2004).
F. Vega Reyes, V. Garzó, A. Santos, Phys. Rev. E 75, 061306 (2007).
J.J. Brey, J.W. Dufty, C.S. Kim, A. Santos, Phys. Rev. E 58, 4638 (1998).
V. Garzó, J.W. Dufty, Phys. Rev. E 59, 5895 (1999).
V. Garzó, J.W. Dufty, Phys. Fluids 14, 1476 (2002).
V. Garzó, J.W. Dufty, C.M. Hrenya, Phys. Rev. E 76, 031303 (2007).
V. Garzó, C.M. Hrenya, J.W. Dufty, Phys. Rev. E 76, 031304 (2007).
V. Garzó, A. Santos, Kinetic Theory of Gases in Shear Flows. Nonlinear Transport (Kluwer Academic, Dordrecht, 2003).
A.V. Bobylev, J.A. Carrillo, I.M. Gamba, J. Stat. Phys. 98, 743 (2000).
J.A. Carrillo, C. Cercignani, I.M. Gamba, Phys. Rev. E 62, 7700 (2000).
E. Ben-Naim, P.L. Krapivsky, in Granular Gas Dynamics, edited by T. Pöschel, S. Luding, Lect. Notes in Phys. Vol. 624 (Springer, Berlin, 2003), pp. 65--94.
M.H. Ernst, E. Trizac, A. Barrat, J. Stat. Phys. 124, 549 (2006).
M.H. Ernst, E. Trizac, A. Barrat, Europhys. Lett. 76, 56 (2006).
E. Trizac, P. Krapivsky, Phys. Rev. Lett. 91, 218302 (2003).
V. Garzó, A. Santos, Math. Model. Nat. Phenom. 6, 37 (2011).
V. Garzó, J. Stat. Phys. 112, 657 (2003).
K. Kohlstedt, A. Snezhko, M.V. Sakozhnikov, I.S. Aranson, J.S. Olafson, E. Ben Naim, Phys. Rev. Lett. 95, 068001 (2005).
V. Garzó, E. Trizac, EPL 94, 50009 (2011).
V. Garzó, E. Trizac, Phys. Rev. E 85, 011302 (2012).
V. Garzó, E. Trizac, Granular Matter 14, 99 (2012).
S. Chapman, T.G. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge University Press, Cambridge, 1970).
V. Garzó, A. Astillero, J. Stat. Phys. 118, 935 (2005).
G.I. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Oxford, Clarendon, 1994).
J.M. Montanero, V. Garzó, Granular Matter 4, 17 (2002).
A. Barrat, E. Trizac, Granular Matter 4, 57 (2002).
S. Dahl, C. Hrenya, V. Garzó, J.W. Dufty, Phys. Rev. E 66, 041301 (2002).
A. Barrat, E. Trizac, Phys. Rev. E 66, 051303 (2002).
A. Barrat, E. Trizac, Granular Matter 4, 57 (2002).
P.E. Krouskop, J. Talbot, Phys. Rev. E 68, 021304 (2003).
Hong-qiang Wang, Guo-jun Jin, Yu-qiang Ma, Phys. Rev. E 68, 031301 (2003).
R.D. Wildman, D.J. Parker, Phys. Rev. Lett. 88, 064301 (2002).
K. Feitosa, N. Menon, Phys. Rev. Lett. 88, 198301 (2002).
P.A. Martin, J. Piasecki, Europhys. Lett. 46, 613 (1999).
V. Garzó, J.W. Dufty, Phys. Rev. E 60, 5706 (1999).
C. Truesdell, R.G. Muncaster, Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas (New York, Academic Press, 1980).
V. Garzó, A. Santos, J. Phys. A: Math. Theor. 40, 14927 (2007).
V. Garzó, E. Trizac, J. Non-Newtonian Fluid Mech. 165, 932 (2010).
U.M.B. Marconi, A. Puglisi, Phys. Rev. E 65, 051305 (2002).
U.M.B. Marconi, A. Puglisi, Phys. Rev. E 66, 011301 (2002).
E. Ben-Naim, P.L. Krapivsky, Eur. Phys. J. E 8, 507 (2002).
G. Constantini, U.M.B. Marconi, A. Puglisi, J. Stat. Mech. P08031 (2007).
A. Santos, J.W. Dufty, Phys. Rev. Lett. 86, 4823 (2001).
E. Ben-Naim, P.L. Krapivsky, Phys. Rev. E 66, 011309 (2002).
M.H. Ernst, R. Brito, Europhys. Lett. 58, 182 (2002).
E. Ben-Naim, P.L. Krapivsky, J. Phys. A: Math. Gen. 35, L147 (2002).
M.H. Ernst, R. Brito, J. Stat. Phys. 109, 407 (2002).
V. Garzó, J.M. Montanero, J. Stat. Phys. 129, 27 (2007).
A. Santos, Physica A 321, 442 (2003).
V. Garzó, in Rarefied Gas Dynamics 24, edited by M. Capitelli, AIP Conf. Proc., Vol. 762 (AIP, 2005) p. 803.
J.J. Brey, M.J. Ruiz-Montero, D. Cubero, R. García-Rojo, Phys. Fluids 12, 876 (2000).
V. Garzó, J.M. Montanero, Phys. Rev. E 68, 021301 (2004).
For a review, see E.H. Hauge, in Transport Phenomena, edited by G. Kirczenow, J. Marro (Springer, Berlin, 1974).
J.J. Brey, M.I. García de Soria, P. Maynar, Phys. Rev. E 82, 021303 (2010).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Garzó, V., Khalil, N. & Trizac, E. Anomalous transport of impurities in inelastic Maxwell gases. Eur. Phys. J. E 38, 16 (2015). https://doi.org/10.1140/epje/i2015-15016-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1140/epje/i2015-15016-5