Skip to main content
SpringerLink
Log in

Theoretical Models of Granular and Active Matter

  • Chapter
  • First Online:
Lattice Models for Fluctuating Hydrodynamics in Granular and Active Matter

Part of the book series: Springer Theses ((Springer Theses))

  • 358 Accesses

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)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    From Bogoliubov, Born, Green, Kirkwood and Yvon [2].

References

  1. C. Cercignani, R. Illner, M. Pulvirenti, in The Mathematical Theory of Dilute Gases (Springer Science & Business Media, Berlin, 2013)

    MATH  Google Scholar 

  2. K. Huang, in Statistical Mechanics, 2nd edn. (Wiley, New York, 1987)

    MATH  Google Scholar 

  3. T. Pöschel, N.V. Brilliantov, in Granular Gas Dynamics (Springer, Berlin, 2003)

    Book  Google Scholar 

  4. N. Brilliantov, T. Pöschel (eds.), in Kinetic Theory of Granular Gases (Oxford University Press, Oxford, 2004)

    MATH  Google Scholar 

  5. A. Puglisi, in Transport and Fluctuations in Granular Fluids (Springer, Berlin, 2014)

    Google Scholar 

  6. 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

  7. 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

    Article  Google Scholar 

  8. 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

    Article  ADS  Google Scholar 

  9. 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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. 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

    Article  ADS  Google Scholar 

  11. 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

    Article  ADS  Google Scholar 

  12. 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

    Article  Google Scholar 

  13. P.K. Haff, Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430 (1983). https://doi.org/10.1017/S0022112083003419

    Article  ADS  MATH  Google Scholar 

  14. I. Goldhirsch, G. Zanetti, Clustering instability in dissipative gases. Phys. Rev. Lett. 70, 1619–1622 (1993). https://doi.org/10.1103/PhysRevLett.70.1619

  15. 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

    Article  ADS  Google Scholar 

  16. 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

  17. 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

    Article  ADS  Google Scholar 

  18. 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

    Article  ADS  MathSciNet  Google Scholar 

  19. 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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. 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

    Article  ADS  Google Scholar 

  21. 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

    Article  ADS  Google Scholar 

  22. 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

  23. 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

  24. 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

  25. 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

    Article  ADS  MathSciNet  Google Scholar 

  26. 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

    Article  ADS  Google Scholar 

  27. 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

  28. 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

  29. 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

  30. 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

  31. 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

    Article  Google Scholar 

  32. 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

    Article  ADS  Google Scholar 

  33. 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

    Article  Google Scholar 

  34. 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

    Article  ADS  Google Scholar 

  35. F. Ginelli, H. Chaté, Relevance of metric-free interactions in flocking phenomena. 105, 168103 (2010). https://doi.org/10.1103/PhysRevLett.105.168103

  36. 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

  37. 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

    Article  Google Scholar 

  38. 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

  39. 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

  40. Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators (Springer, Berlin, 1975), pp. 420–422. https://doi.org/10.1007/BFb0013365

  41. Y. Kuramoto, in Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984)

    Book  Google Scholar 

  42. 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

  43. E. Bertin, in Statistical Physics of Complex Systems (Springer, Berlin, 2016)

    Book  Google Scholar 

  44. 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

  45. 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

  46. 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

  47. 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

    Article  ADS  Google Scholar 

  48. 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

    Article  ADS  Google Scholar 

  49. 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

    Article  Google Scholar 

  50. 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

    Article  Google Scholar 

  51. C. Gardiner, Stochastic Methods. A Handbook for the Natural and Social Sciences., 4th edn. (Springer, Berlin, 2009)

    Google Scholar 

  52. Phys. Rev. E Theory of continuum random walks and application to chemotaxis. 48, 2553–2568 (1993). https://doi.org/10.1103/PhysRevE.48.2553

  53. 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

  54. 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

    Article  ADS  Google Scholar 

  55. 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

    Article  ADS  Google Scholar 

  56. 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

    Article  ADS  Google Scholar 

  57. 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

    Article  ADS  Google Scholar 

  58. 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

  59. 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

  60. 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

  61. 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

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Sapienza, Rome, Italy

    Alessandro Manacorda

Authors
  1. Alessandro Manacorda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alessandro Manacorda .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

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

Publish with us

Policies and ethics

Search

Navigation