Abstract
This chapter introduces the main tehoretical models used to describe and reproduce the behavior of granular and active matter. The first Sect. 2.1 is dedicated to kinetic theory: established for the study of elastic gases, its aim is to describe a gas in term of mechanical coordinates of all its particles to derive its macroscopic properties such as pressure, energy and entropy through the statistical properties of the microscopic variables. This method, which was derived for elastic gases, can apply also for granular materials. The second Sect. 2.2 reviews the most important physical models of active matter, focusing on the essential ingredients to produce the typical interactions and self-propulsion discussed in Chap. 1. The last Sect. 2.3 investigates a possible theoretical comparison and symmetry between granular and active matter.
Walk on, through the wind
Walk on, through the rain
Though your dreams be tossed and blown
Walk on, walk on
With hope in your heart
And you’ll never walk alone
(R. Rodgers, O. Hammerstein)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
From Bogoliubov, Born, Green, Kirkwood and Yvon [2].
References
C. Cercignani, R. Illner, M. Pulvirenti, in The Mathematical Theory of Dilute Gases (Springer Science & Business Media, Berlin, 2013)
K. Huang, in Statistical Mechanics, 2nd edn. (Wiley, New York, 1987)
T. Pöschel, N.V. Brilliantov, in Granular Gas Dynamics (Springer, Berlin, 2003)
N. Brilliantov, T. Pöschel (eds.), in Kinetic Theory of Granular Gases (Oxford University Press, Oxford, 2004)
A. Puglisi, in Transport and Fluctuations in Granular Fluids (Springer, Berlin, 2014)
M. Ernst, J. Dorfman, W. Hoegy, J.V. Leeuwen, Hard-sphere dynamics and binary-collision operators. Physica 45(1), 127–146 (1969). http://www.sciencedirect.com/science/article/pii/0031891469900676, https://doi.org/10.1016/0031-8914(69)90067-6
J.-M. Hertzsch, F. Spahn, N.V. Brilliantov, On low-velocity collisions of viscoelastic particles. J. Phys. II Fr. 5(11), 1725–1738 (1995). https://doi.org/10.1051/jp2:1995210
N.V. Brilliantov, F. Spahn, J.-M. Hertzsch, T. Pöschel, Model for collisions in granular gases. Phys. Rev. E 53, 5382–5392 (1996). https://doi.org/10.1103/PhysRevE.53.5382
A. Goldshtein, M. Shapiro, Mechanics of collisional motion of granular materials. part 1. general hydrodynamic equations. J. Fluid Mech. 282, 75–114 (1995). https://doi.org/10.1017/S0022112095000048
S. McNamara, W.R. Young, Inelastic collapse and clumping in a one-dimensional granular medium. Phys. Fluids A Fluid Dyn. 4(3), 496–504 (1992). https://doi.org/10.1063/1.858323
S. McNamara, W.R. Young, Dynamics of a freely evolving, two-dimensional granular medium. Phys. Rev. E 53, 5089–5100 (1996). https://doi.org/10.1103/PhysRevE.53.5089
T. van Noije, M. Ernst, Velocity distributions in homogeneous granular fluids: the free and the heated case. Granul. Matter 1(2), 57–64 (1998). https://doi.org/10.1007/s100350050009
P.K. Haff, Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430 (1983). https://doi.org/10.1017/S0022112083003419
I. Goldhirsch, G. Zanetti, Clustering instability in dissipative gases. Phys. Rev. Lett. 70, 1619–1622 (1993). https://doi.org/10.1103/PhysRevLett.70.1619
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, 051303 (2004). https://doi.org/10.1103/PhysRevE.69.051303
A. Puglisi, V. Loreto, U.M.B. Marconi, A. Petri, A. Vulpiani, Clustering and non-Gaussian behavior in granular matter. Phys. Rev. Lett. 81, 3848–3851 (1998). https://doi.org/10.1103/PhysRevLett.81.3848
A. Puglisi, V. Loreto, U.M.B. Marconi, A. Vulpiani, Kinetic approach to granular gases. Phys. Rev. E 59, 5582–5595 (1999). https://doi.org/10.1103/PhysRevE.59.5582
M. Ernst, Nonlinear model-Boltzmann equations and exact solutions. Phys. Rep. 78, 1–171 (1981). https://doi.org/10.1016/0370-1573(81)90002-8
D. Blackwell, R.D. Mauldin, Ulam’s redistribution of energy problem: collision transformations. Lett. Math. Phys. 10(2), 149–153 (1985). https://doi.org/10.1007/BF00398151
E. Ben-Naim, P.L. Krapivsky, Maxwell model of traffic flows. Phys. Rev. E 59, 88–97 (1999). https://doi.org/10.1103/PhysRevE.59.88
E. Ben-Naim, P.L. Krapivsky, Multiscaling in inelastic collisions. Phys. Rev. E 61, R5–R8 (2000). https://doi.org/10.1103/PhysRevE.61.R5
A. Baldassarri, U.M.B. Marconi, A. Puglisi, Influence of correlations on the velocity statistics of scalar granular gases. EPL (Europhys. Lett.) 58(1), 14 (2002). https://doi.org/10.1209/epl/i2002-00600-6
G. Costantini, U.M.B. Marconi, A. Puglisi, Velocity fluctuations in a one-dimensional inelastic Maxwell model. J. Stat. Mech. Theory Exp. 2007(08), P08031 (2007). http://stacks.iop.org/1742-5468/2007/i=08/a=P08031, https://doi.org/10.1088/1742-5468/2007/08/P08031
T. Vicsek, A. Zafeiris, Collective motion. Phys. Rep. 517(3–4), 71–140 (2012). http://www.sciencedirect.com/science/article/pii/S0370157312000968, https://doi.org/10.1016/j.physrep.2012.03.004
J. Elgeti, R.G. Winkler, G. Gompper, Physics of microswimmers–single particle motion and collective behavior: a review. Rep. Prog. Phys. 78(5), 056601 (2015). https://doi.org/10.1088/0034-4885/78/5/056601
A. Baskaran, M.C. Marchetti, Statistical mechanics and hydrodynamics of bacterial suspensions. Proc. Natl. Acad. Sci. 106(37), 15567–15572 (2009). https://doi.org/10.1073/pnas.0906586106
M.C. Marchetti, J.F. Joanny, S. Ramaswamy, T.B. Liverpool, J. Prost, M. Rao, R.A. Simha, Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143–1189 (2013). https://doi.org/10.1103/RevModPhys.85.1143
C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, G. Volpe, Active particles in complex and crowded environments. Rev. Mod. Phys. 88, 045006 (2016). https://doi.org/10.1103/RevModPhys.88.045006
J. Tailleur, M.E. Cates, Statistical mechanics of interacting run-and-tumble bacteria. Phys. Rev. Lett. 100, 218103 (2008). https://doi.org/10.1103/PhysRevLett.100.218103
W. Bialek, A. Cavagna, I. Giardina, T. Mora, E. Silvestri, M. Viale, A.M. Walczak, Statistical mechanics for natural flocks of birds. Proc. Natl. Acad. Sci. 4791(13), 109–4786 (2012). http://www.pnas.org/content/109/13/4786.abstract, arXiv:http://www.pnas.org/content/109/13/4786.full.pdf, https://doi.org/10.1073/pnas.1118633109
A.P. Solon, M.E. Cates, J. Tailleur, Active Brownian particles and run-and-tumble particles: a comparative study. Eur. Phys. J. Spec. Top. 224(7), 1231–1262 (2015). https://doi.org/10.1140/epjst/e2015-02457-0
M. Paoluzzi, C. Maggi, U.M.B. Marconi, N. Gnan, Critical phenomena in active matter. Phys. Rev. E 94, 052602 (2016). https://doi.org/10.1103/PhysRevE.94.052602
A. Czirók, T. Vicsek, Collective behavior of interacting self-propelled particles. Phys. A Stat. Mech. Appl. 281(1–4), 17–29 (2000). https://doi.org/10.1016/S0378-4371(00)00013-3
H. Chaté, F. Ginelli, G. Grégoire, F. Peruani, F. Raynaud, Modeling collective motion: variations on the Vicsek model. Eur. Phys. J. B 64, 451–456 (2008). https://doi.org/10.1140/epjb/e2008-00275-9
F. Ginelli, H. Chaté, Relevance of metric-free interactions in flocking phenomena. 105, 168103 (2010). https://doi.org/10.1103/PhysRevLett.105.168103
A.P. Solon, J. Tailleur, Revisiting the flocking transition using active spins. Phys. Rev. Lett. 111, 078101 (2013). https://doi.org/10.1103/PhysRevLett.111.078101
T. Mora, A.M. Walczak, L. Del Castello, F. Ginelli, S. Melillo, L. Parisi, M. Viale, A. Cavagna, I. Giardina, Local equilibrium in bird flocks. Nat. Phys. 12(12), 1153–1157 (2016). https://doi.org/10.1038/nphys3846
E. Fodor, C. Nardini, M.E. Cates, J. Tailleur, P. Visco, F. van Wijland, How far from equilibrium is active matter? Phys. Rev. Lett. 117, 038103 (2016). https://doi.org/10.1103/PhysRevLett.117.038103
U.M.B. Marconi, A. Puglisi, C. Maggi, Heat, temperature and clausius inequality in a model for active Brownian particles. Sci. Rep. 7 (2017). https://doi.org/10.1038/srep46496
Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators (Springer, Berlin, 1975), pp. 420–422. https://doi.org/10.1007/BFb0013365
Y. Kuramoto, in Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984)
J.A. Acebrón, L.L. Bonilla, C.J. Pérez Vicente, F. Ritort, R. Spigler, The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005). https://doi.org/10.1103/RevModPhys.77.137
E. Bertin, in Statistical Physics of Complex Systems (Springer, Berlin, 2016)
E. Bertin, Theoretical approaches to the steady-state statistical physics of interacting dissipative units. J. Phys. A Math. Theory 50(8), 083001 (2017). http://stacks.iop.org/1751-8121/50/i=8/a=083001, https://doi.org/10.1088/1751-8121/aa546b
S.H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D Nonlinear Phenom. 143(1), 1–20 (2000). http://www.sciencedirect.com/science/article/pii/S0167278900000944, https://doi.org/10.1016/S0167-2789(00)00094-4
T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet, Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995). https://doi.org/10.1103/PhysRevLett.75.1226
A. Czirók, H.E. Stanley, T. Vicsek, Spontaneously ordered motion of self-propelled particles. J. Phys. A Math. Gen. 30(5), 1375 (1997). https://doi.org/10.1088/0305-4470/30/5/009
A. Czirók, E. Ben-Jacob, I. Cohen, T. Vicsek, Formation of complex bacterial colonies via self-generated vortices. Phys. Rev. E 54, 1791–1801 (1996). https://doi.org/10.1103/PhysRevE.54.1791
A. Chepizhko, V. Kulinskii, On the relation between Vicsek and Kuramoto models of spontaneous synchronization. Phys. A Stat. Mech. Appl. 389(23), 5347–5352 (2010). https://doi.org/10.1016/j.physa.2010.08.016
P. Romanczuk, M. Bär, W. Ebeling, B. Lindner, L. Schimansky-Geier, Active Brownian particles. Eur. Phys. J. Spec. Top. 202(1), 1–162 (2012). https://doi.org/10.1140/epjst/e2012-01529-y
C. Gardiner, Stochastic Methods. A Handbook for the Natural and Social Sciences., 4th edn. (Springer, Berlin, 2009)
Phys. Rev. E Theory of continuum random walks and application to chemotaxis. 48, 2553–2568 (1993). https://doi.org/10.1103/PhysRevE.48.2553
M.E. Cates, J. Tailleur, When are active Brownian particles and run-and-tumble particles equivalent? consequences for motility-induced phase separation. EPL (Europhys. Lett.) 101(2), 20010 (2013). https://doi.org/10.1209/0295-5075/101/20010
N. Koumakis, C. Maggi, R. Di Leonardo, Directed transport of active particles over asymmetric energy barriers. Soft Matter 10, 5695–5701 (2014). https://doi.org/10.1039/C4SM00665H
T.F.F. Farage, P. Krinninger, J.M. Brader, Effective interactions in active Brownian suspensions. Phys. Rev. E 91, 042310 (2015). https://doi.org/10.1103/PhysRevE.91.042310
C. Maggi, U.M.B. Marconi, N. Gnan, R. Di Leonardo, Multidimensional stationary probability distribution for interacting active particles. Sci. Rep. 5, 10742 (2015). https://doi.org/10.1038/srep10742
U.M.B. Marconi, C. Maggi, Towards a statistical mechanical theory of active fluids. Soft Matter 11, 8768–8781 (2015). https://doi.org/10.1039/C5SM01718A
D. Grossman, I.S. Aranson, E.B. Jacob, Emergence of agent swarm migration and vortex formation through inelastic collisions. New J. Phys. 10(2), 023036 (2008). http://stacks.iop.org/1367-2630/10/i=2/a=023036, https://doi.org/10.1088/1367-2630/10/2/023036
C.A. Weber, T. Hanke, J. Deseigne, S. Léonard, O. Dauchot, E. Frey, H. Chaté, Long-range ordering of vibrated polar disks. Phys. Rev. Lett. 110, 208001 (2013). https://doi.org/10.1103/PhysRevLett.110.208001
J. Deseigne, O. Dauchot, H. Chaté, Collective motion of vibrated polar disks. Phys. Rev. Lett. 105, 098001 (2010). https://doi.org/10.1103/PhysRevLett.105.098001
N. Kumar, H. Soni, S. Ramaswamy, A. Sood, Flocking at a distance in active granular matter. Nat. Commun. 5 (2014). https://doi.org/10.1038/ncomms5688
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Manacorda, A. (2018). Theoretical Models of Granular and Active Matter. In: Lattice Models for Fluctuating Hydrodynamics in Granular and Active Matter. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-95080-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-95080-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-95079-2
Online ISBN: 978-3-319-95080-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)