Skip to main content
SpringerLink
Log in

Models of Particles Driven Out of Equilibrium

  • Chapter
  • First Online:
Statistical Physics of Complex Systems

Part of the book series: Springer Series in Synergetics ((SSSYN))

  • 1225 Accesses

Abstract

Systems of interacting particles can be driven into nonequilibrium stationary states in different ways, e.g., through some external vibration like in the case of granular matter, or through some self-propulsion mechanisms like for some types of bacteria. Particles may also be created or annihilated like in chemical reactions. This chapter provides the reader with examples of models of interacting particles that reach a nonequilibrium steady state. Different methods allowing for the description of the corresponding steady state are introduced.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 79.99
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.

    The Laplacian operator is defined as \(\Delta =\partial ^2/\partial x^2\) in one dimension, \(\Delta =\partial ^2/\partial x^2+\partial ^2/\partial y^2\) in two dimensions, and \(\Delta =\partial ^2/\partial x^2+\partial ^2/\partial y^2+\partial ^2/\partial z^2\) in three dimensions.

  2. 2.

    We consider here for simplicity the ring geometry, but the ZRP can actually be defined on an arbitrary graph [25].

  3. 3.

    However, note that temperature may play a role in the mechanics of grains either in the small frictional contact areas between grains, or by dilating or contracting the grains if the temperature slightly fluctuates. But the resulting displacements generally remain much smaller than the grain diameter.

  4. 4.

    More explicitly, Eq. (3.94) reads

    figure a

    where \((u_x,u_y)\) are the components of the vector \(\mathbf {u}\).

  5. 5.

    Here, i is a complex number such that \(i^2=-1\).

References

  1. Aranson, I.S., Kramer, L.: The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 99 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  2. Barrat, A., Kurchan, J., Loreto, V., Sellitto, M.: Edwards’ measures for powders and glasses. Phys. Rev. Lett. 85, 5034 (2000)

    Article  ADS  Google Scholar 

  3. Bechinger, C., Di Leonardo, R., Löwen, H., Reichhardt, C., Volpe, G., Volpe, G.: Active particles in complex and crowded environments. Rev. Mod. Phys. 88, 045006 (2016)

    Google Scholar 

  4. Bertin, E.: Theoretical approaches to the steady-state statistical physics of interacting dissipative units. J. Phys. A: Math. Theor. 50, 083001 (2017)

    Google Scholar 

  5. Bertin, E., Chaté, H., Ginelli, F., Mishra, S., Peshkov, A., Ramaswamy, S.: Mesoscopic theory for fluctuating active nematics. New J. Phys. 15, 085032 (2013)

    Google Scholar 

  6. Bertin, E., Dauchot, O.: Far-from-equilibrium state in a weakly dissipative model. Phys. Rev. Lett. 102, 160601 (2009)

    Google Scholar 

  7. Bertin, E., Droz, M., Grégoire, G.: Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis. J. Phys. A Math. Theor. 42, 445001 (2009)

    Google Scholar 

  8. Bi, D.P., Henkes, S., Daniels, K.E., Chakraborty, B.: The statistical physics of athermal materials. Annu. Rev. Condens. Matter Phys. 6, 63 (2015)

    Article  ADS  Google Scholar 

  9. Blythe, R.A., Evans, M.R.: Nonequilibrium steady states of matrix-product form: a solver’s guide. J. Phys. A Math. Theor. 40, R333 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  10. Blythe, R.A., Evans, M.R., Colaiori, F., Essler, F.H.L.: Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra. J. Phys. A Math. Gen. 33, 2313 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  11. Boudaoud, A., Cadot, O., Odille, B., Touzé, C.: Observation of wave turbulence in vibrating plates. Phys. Rev. Lett. 100, 234504 (2008)

    Google Scholar 

  12. Bricard, A., Caussin, J.B., Desreumaux, N., Dauchot, O., Bartolo, D.: Emergence of macroscopic directed motion in populations of motile colloids. Nature 503, 95 (2013)

    Article  ADS  Google Scholar 

  13. Buttinoni, I., Bialké, J., Kümmel, F., Löwen, H., Bechinger, C., Speck, T.: Dynamical clustering and phase separation in suspensions of self-propelled colloidal particles. Phys. Rev. Lett. 110, 238301 (2013)

    Google Scholar 

  14. Cates, M.E., Tailleur, J.: When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-induced phase separation. EPL 101, 20010 (2013)

    Article  ADS  Google Scholar 

  15. Cates, M.E., Tailleur, J.: Motility-induced phase separation. Annu. Rev. Condens. Matter Phys. 6, 219 (2015)

    Article  ADS  Google Scholar 

  16. Derrida, B.: Non-equilibrium steady states: fluctuations and large deviations of the density and of the current. J. Stat. Mech. P07023 (2007)

    Google Scholar 

  17. Derrida, B., Evans, M.R., Hakim, V., Pasquier, V.: Exact solution of a 1d asymmetric exclusion model using a matrix formulation. J. Phys. A Math. Gen. 26, 1493 (1993)

    Article  ADS  Google Scholar 

  18. Deseigne, J., Dauchot, O., Chaté, H.: Collective motion of vibrated polar disks. Phys. Rev. Lett. 105, 098001 (2010)

    Google Scholar 

  19. Düring, G., Josserand, C., Rica, S.: Weak turbulence for a vibrating plate: can one hear a Kolmogorov spectrum? Phys. Rev. Lett. 97, 025503 (2006)

    Google Scholar 

  20. Edwards, S.F., Grinev, D.V.: Statistical mechanics of vibration-induced compaction of powders. Phys. Rev. E 58, 4758 (1998)

    Article  ADS  Google Scholar 

  21. Edwards, S.F., Mounfield, C.C.: The statistical mechanics of granular systems composed of elongated grains. Physica A 210, 279 (1994)

    Article  ADS  Google Scholar 

  22. Edwards, S.F., Oakeshott, R.B.S.: Theory of powders. Physica A 157, 1080 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  23. Essler, F.H.L., Rittenberg, V.: Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries. J. Phys. A Math. Gen. 29, 3375 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  24. Evans, M.R., Hanney, T.: Nonequilibrium statistical mechanics of the zero-range process and related models. J. Phys. A Math. Gen. 38, R195 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  25. Evans, M.R., Majumdar, S.N., Zia, R.K.P.: Factorized steady states in mass transport models on an arbitrary graph. J. Phys. A Math. Gen. 39, 4859 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  26. Frisch, U.: Turbulence. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  27. Gomez-Solano, J.R., Blokhuis, A., Bechinger, C.: Dynamics of self-propelled Janus particles in viscoelastic fluids. Phys. Rev. Lett. 116, 138301 (2016)

    Google Scholar 

  28. Gradenigo, G., Ferrero, E.E., Bertin, E., Barrat, J.L.: Edwards thermodynamics for a driven athermal system with dry friction. Phys. Rev. Lett. 115, 140601 (2015)

    Google Scholar 

  29. Grégoire, G., Chaté, H.: Onset of collective and cohesive motion. Phys. Rev. Lett. 92, 025702 (2004)

    Google Scholar 

  30. Guioth, J., Bertin, E.: Lack of an equation of state for the nonequilibrium chemical potential of gases of active particles in contact. J. Chem. Phys. 150, 094108 (2019)

    Google Scholar 

  31. Hinrichsen, H.: Nonequilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys. 49, 815 (2000)

    Article  ADS  Google Scholar 

  32. Krebs, K., Sandow, S.: Matrix product eigenstates for one-dimensional stochastic models and quantum spin chains. J. Phys. A Math. Gen. 30, 3165 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  33. Kudrolli, A., Lumay, G., Volfson, D., Tsimring, L.S.: Swarming and swirling in self-propelled polar granular rods. Phys. Rev. Lett. 100, 058001 (2008)

    Google Scholar 

  34. Le Bellac, M.: Quantum and Statistical Field Theory. Oxford Science Publications, Oxford (1992)

    Google Scholar 

  35. Mallick, K., Sandow, S.: Finite-dimensional representations of the quadratic algebra: applications to the exclusion process. J. Phys. A Math. Gen. 30, 4513 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  36. Marchetti, M.C., Joanny, J.F., Ramaswamy, S., Liverpool, T.B., Prost, J., Rao, M., Simha, R.A.: Soft active matter. Rev. Mod. Phys. 85, 1143 (2013)

    Article  ADS  Google Scholar 

  37. Mehta, A., Edwards, S.F.: Statistical mechanics of powder mixtures. Physica A 157, 1091 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  38. Narayan, V., Menon, N., Ramaswamy, S.: Nonequilibrium steady states in a vibrated-rod monolayer: tetratic, nematic, and smectic correlations. J. Stat. Mech. (2006)

    Google Scholar 

  39. Nicolis, G., Prigogine, I.: Self-organization in nonequilibrium systems: from dissipative structures to order through fluctuations. J. Wiley, New York (1977)

    Google Scholar 

  40. Palacci, J., Cottin-Bizonne, C., Ybert, C., Bocquet, L.: Sedimentation and effective temperature of active colloidal suspensions. Phys. Rev. Lett. 105, 088304 (2010)

    Google Scholar 

  41. Palacci, J., Sacanna, S., Steinberg, A.P., Pine, D.J., Chaikin, P.M.: Living crystals of light-activated colloidal surfers. Science 339, 936 (2013)

    Article  ADS  Google Scholar 

  42. Peruani, F., Starruss, J., Jakovljevic, V., Søgaard-Andersen, L., Deutsch, A., Bär, M.: Collective motion and nonequilibrium cluster formation in colonies of gliding bacteria. Phys. Rev. Lett. 108, 098102 (2012)

    Google Scholar 

  43. Peshkov, A., Bertin, E., Ginelli, F., Chaté, H.: Boltzmann-Ginzburg-Landau approach for continuous descriptions of generic Vicsek-like models. Eur. Phys. J. Spec. Top. 223, 1315 (2014)

    Article  Google Scholar 

  44. Sandow, S.: Partially asymmetric exclusion process with open boundaries. Phys. Rev. E 50, 2660 (1994)

    Article  ADS  Google Scholar 

  45. Seifert, U.: Stochastic thermodynamics, fluctuations theorems and molecular machines. Rep. Prog. Phys. 75, 126001 (2012)

    Google Scholar 

  46. Theurkauff, I., Cottin-Bizonne, C., Palacci, J., Ybert, C., Bocquet, L.: Dynamic clustering in active colloidal suspensions with chemical signaling. Phys. Rev. Lett. 108, 268303 (2012)

    Google Scholar 

  47. Toner, J., Tu, Y.: Long-range order in a two-dimensional dynamical XY model: how birds fly together. Phys. Rev. Lett. 75, 4326 (1995)

    Article  ADS  Google Scholar 

  48. Toner, J., Tu, Y., Ramaswamy, S.: Hydrodynamics and phases of flocks. Ann. Phys. (N.Y.) 318, 170 (2005)

    Google Scholar 

  49. Touchette, H.: The large deviation approach to statistical mechanics. Phys. Rep. 478, 1 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  50. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  51. Zakharov, V.E., L’vov, V.S., Falkovich, G.: Kolmogorov Spectra of Turbulence I: Wave Turbulence. Springer, Berlin (1992)

    Google Scholar 

  52. Zhou, S., Sokolov, A., Lavrentovich, O.D., Aranson, I.S.: Living liquid crystals. Proc. Natl. Acad. Sci. USA 111, 1265 (2014)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LIPhy, CNRS and Université Grenoble Alpes, Grenoble, France

    Eric Bertin

Authors
  1. Eric Bertin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric Bertin .

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bertin, E. (2021). Models of Particles Driven Out of Equilibrium. In: Statistical Physics of Complex Systems. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-030-79949-6_3

Download citation

Publish with us

Policies and ethics

Search

Navigation