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Hydrodynamic Description and Lattice Models

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Lattice Models for Fluctuating Hydrodynamics in Granular and Active Matter

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

Abstract

When a fluid is flowing, for instance under the action of gravity or a pressure gradient, its motion can be described introducing the continuous hydrodynamic fields such as density \(\rho (\mathbf {x},t)\), velocity \(\mathbf {u}(\mathbf {x},t)\) and temperature \(T(\mathbf {x},t)\), which measure the local mechanical and thermodynamical properties of the fluid at the position \(\mathbf {x}\) and time t. Hydrodynamic fields generally depend on time and space, and hydrodynamic equations aim at describing their time evolution, affected by spatial gradients and external forces.

Quei giorni perduti a rincorrere il vento

(F. de André)

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Notes

  1. 1.

    From now on we call \(\epsilon \) the Knudsen number to be consistent with literature.

References

  1. K. Huang, Statistical Mechanics, 2nd edn. (Wiley, USA, 1987)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. S. Chapman, T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd edn. (Cambridge University Press, Cambridge, 1990)

    MATH  Google Scholar 

  4. J.H. Ferziger, H.G. Kaper, Mathematical Theory of Transport Processes in Gases (North Holland, 1972)

    Google Scholar 

  5. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer, Berlin, 1999)

    Book  Google Scholar 

  6. J.J. Brey, J.W. Dufty, C.S. Kim, A. Santos, Hydrodynamics for granular flow at low density. Phys. Rev. E 58, 4638–4653 (1998). https://doi.org/10.1103/PhysRevE.58.4638

    Article  ADS  Google Scholar 

  7. M. Cencini, F. Cecconi, A. Vulpiani. Chaos: From Simple Models to Complex Systems (World Scientific, 2010)

    Google Scholar 

  8. L. Landau, E. Lifshitz, Fluid Mechanics. Course of Theoretical Physics, vol. 6 (Pergamon Press, 1987)

    Google Scholar 

  9. P.G. Drazin, W.H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 2004)

    Book  Google Scholar 

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

    Article  ADS  Google Scholar 

  11. M.C. Marchetti, J.E. 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

    Article  ADS  Google Scholar 

  12. E. Bertin, M. Droz, G. Grégoire, Boltzmann and hydrodynamic description for self-propelled particles. Phys. Rev. E 74, 022101 (2006). https://doi.org/10.1103/PhysRevE.74.022101

    Article  ADS  Google Scholar 

  13. E. Bertin, M. Droz, G. Grégoire, Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis. J. Phys. A. Math. Theor. 42(44), 445001 (2009). https://doi.org/10.1088/1751-8113/42/44/445001

    Article  MATH  Google Scholar 

  14. A. Peshkov, E. Bertin, F. Ginelli, H. Chaté, Boltzmann-Ginzburg-Landau approach for continuous descriptions of generic Vicsek-like models. Eur. Phys. J. Spec. Top. 223(7), 1315–1344 (2014). https://doi.org/10.1140/epjst/e2014-02193-y

    Article  Google Scholar 

  15. J. Toner, Y. Tu, Long-range order in a two-dimensional dynamical XY model: how birds fly together. Phys. Rev. Lett. 75, 4326–4329 (1995). https://doi.org/10.1103/PhysRevLett.75.4326

  16. J. Toner, Y. Tu, Flocks, herds, and schools: a quantitative theory of flocking. Phys. Rev. E 58, 4828–4858 (1998). https://doi.org/10.1103/PhysRevE.58.4828

    Article  ADS  MathSciNet  Google Scholar 

  17. 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

  18. J. Toner, Y. Tu, S. Ramaswamy, Hydrodynamics and phases of flocks. Adv. Phys. 318(1):170–244 (2005) [Special Issue]. http://www.sciencedirect.com/science/article/pii/S0003491605000540, https://doi.org/10.1016/j.aop.2005.04.011

  19. T. Ihle, Kinetic theory of flocking: derivation of hydrodynamic equations. Phys. Rev. E 83, 030901 (2011). https://doi.org/10.1103/PhysRevE.83.030901

    Article  ADS  Google Scholar 

  20. A. Baskaran, M.C. Marchetti, Hydrodynamics of self-propelled hard rods. Phys. Rev. E 77, 011920 (2008). https://doi.org/10.1103/PhysRevE.77.011920

    Article  ADS  MathSciNet  Google Scholar 

  21. 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 

  22. B. Hancock, A. Baskaran, Statistical mechanics and hydrodynamics of self-propelled hard spheres. J. Stat. Mech. (Theor. Exp.) 2017(3), 033205 (2017). http://stacks.iop.org/1742-5468/2017/i=3/a=033205, https://doi.org/10.1088/1742-5468/aa5ed1

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

    Google Scholar 

  24. U. Marini Bettolo Marconi, A. Puglisi, L. Rondoni, A. Vulpiani, Fluctuation–dissipation: response theory in statistical physics. Phys. Rep. 461(4–6), 111 – 195 (2008). http://www.sciencedirect.com/science/article/pii/S0370157308000768, https://doi.org/10.1016/j.physrep.2008.02.002

  25. R. Kubo, M. Toda, N. Hashitsume, Statistical Physics Ii: Nonequilibrium Statistical Mechanics (Springer Science & Business Media, Berlin, 2012)

    MATH  Google Scholar 

  26. M. Bixon, R. Zwanzig, Boltzmann-Langevin equation and hydrodynamic fluctuations. Phys. Rev. 187, 267–272 (1969). https://doi.org/10.1103/PhysRev.187.267

    Article  ADS  MathSciNet  Google Scholar 

  27. J.J. Brey, P. Maynar, M.I. García de Soria, Fluctuating hydrodynamics for dilute granular gases. Phys. Rev. E 79, 051305 (2009). https://doi.org/10.1103/PhysRevE.79.051305

    Article  ADS  Google Scholar 

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

    Google Scholar 

  29. T.P.C. van Noije, M.H. Ernst, R. Brito, J.A.C. Orza, Mesoscopic theory of granular fluids. Phys. Rev. Lett. 79, 411–414 (1997). https://doi.org/10.1103/PhysRevLett.79.411

    Article  ADS  Google Scholar 

  30. T.P.C. van Noije, M.H. Ernst, E. Trizac, I. Pagonabarraga, Randomly driven granular fluids: large-scale structure. Phys. Rev. E 59, 4326–4341 (1999). https://doi.org/10.1103/PhysRevE.59.4326

    Article  ADS  Google Scholar 

  31. G. Costantini, A. Puglisi, Fluctuating hydrodynamics in a vertically vibrated granular fluid with gravity. Phys. Rev. E 84, 031307 (2011). https://doi.org/10.1103/PhysRevE.84.031307

    Article  ADS  Google Scholar 

  32. A. Puglisi, V. Loreto, U. Marini, Bettolo 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

    Article  ADS  Google Scholar 

  33. G. Gradenigo, A. Sarracino, D. Villamaina, A. Puglisi, Fluctuating hydrodynamics and correlation lengths in a driven granular fluid. J. Stat. Mech. (Theor. Exp.) 2011(08), P08017 (2011). https://doi.org/10.1088/1742-5468/2011/08/P08017

  34. G. Gradenigo, A. Sarracino, D. Villamaina, A. Puglisi, Non-equilibrium length in granular fluids: From experiment to fluctuating hydrodynamics. EPL (Europhys. Lett.) 96(1), 14004 (2011). https://doi.org/10.1209/0295-5075/96/14004

  35. L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim, Macroscopic fluctuation theory for stationary non-equilibrium states. J. Stat. Phys. 107(3), 635–675 (2002). https://doi.org/10.1023/A:1014525911391

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim, Macroscopic fluctuation theory. Rev. Mod. Phys. 87, 593–636 (2015). https://doi.org/10.1103/RevModPhys.87.593

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. A. Vulpiani, F. Cecconi, M. Cencini, A. Puglisi, D. Vergni, Large Deviations in Physics (Springer, Berlin, 2014)

    Book  Google Scholar 

  38. G. Gallavotti, E.G.D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694–2697 (1995). https://doi.org/10.1103/PhysRevLett.74.2694

    Article  ADS  Google Scholar 

  39. C. Jarzynski, Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690–2693 (1997). https://doi.org/10.1103/PhysRevLett.78.2690

    Article  ADS  Google Scholar 

  40. J.L. Lebowitz, H. Spohn, A Gallavotti-cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95(1), 333–365 (1999). https://doi.org/10.1023/A:1004589714161

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. 75(12), 126001 (2012). https://doi.org/10.1088/0034-4885/75/12/126001

    Article  ADS  MathSciNet  Google Scholar 

  42. A.G. Thompson, J. Tailleur, M.E. Cates, R.A. Blythe, Lattice models of nonequilibrium bacterial dynamics. J. Stat. Mech. (Theor. Exp.) 2011(02), P02029 (2011). https://doi.org/10.1088/1742-5468/2011/02/P02029

  43. B. Derrida, Non-equilibrium steady states: fluctuations and large deviations of the density and of the current. J. Stat. Mech. (Theor. Exp.) 2007(07), P07023 (2007). https://doi.org/10.1088/1742-5468/2007/07/P07023

  44. J. Marro, R. Dickman, Nonequilibrium Phase Transitions in Lattice Models (Cambridge University Press, Cambridge, 2005)

    MATH  Google Scholar 

  45. C. Kipnis, C. Marchioro, E. Presutti, Heat flow in an exactly solvable model. J. Stat. Phys. 27(1), 65–74 (1982). https://doi.org/10.1007/BF01011740

    Article  ADS  MathSciNet  Google Scholar 

  46. S. Alexander, J. Bernasconi, W.R. Schneider, R. Orbach, Excitation dynamics in random one-dimensional systems. Rev. Mod. Phys. 53, 175–198 (1981). https://doi.org/10.1103/RevModPhys.53.175

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. B. Derrida, Y. Pomeau, Classical diffusion on a random chain. Phys. Rev. Lett. 48, 627–630 (1982). https://doi.org/10.1103/PhysRevLett.48.627

    Article  ADS  Google Scholar 

  48. B. Derrida, Velocity and diffusion constant of a periodic one-dimensional hopping model. J. Stat. Phys. 31(3), 433–450 (1983). https://doi.org/10.1007/BF01019492

    Article  ADS  MathSciNet  Google Scholar 

  49. A. Baldassarri, U. Marini, Bettolo Marconi, A. Puglisi, Cooling of a lattice granular fluid as an ordering process. Phys. Rev. E 65, 051301 (2002). https://doi.org/10.1103/PhysRevE.65.051301

    Article  ADS  Google Scholar 

  50. A. Baldassarri, A. Puglisi, U. Marini, Bettolo Marconi, Kinetic models of inelastic gases. Math. Models Methods Appl. Sci. 12(07), 965–983 (2002). https://doi.org/10.1142/S0218202502001982

    Article  MathSciNet  MATH  Google Scholar 

  51. A. Prados, A. Lasanta, P.I. Hurtado, Large fluctuations in driven dissipative media. Phys. Rev. Lett. 107, 140601 (2011). https://doi.org/10.1103/PhysRevLett.107.140601

    Article  ADS  Google Scholar 

  52. A. Prados, A. Lasanta, P.I. Hurtado, Nonlinear driven diffusive systems with dissipation: fluctuating hydrodynamics. Phys. Rev. E 86, 031134 (2012). https://doi.org/10.1103/PhysRevE.86.031134

    Article  ADS  Google Scholar 

  53. A. Lasanta, Algunas propiedades de los estados estacionarios de sistemas disipativos sencillos, Ph.D. thesis, Universidad de Granada, 2014

    Google Scholar 

  54. Z. Csahók, T. Vicsek, Lattice-gas model for collective biological motion. Phys. Rev. E 52, 5297–5303 (1995). https://doi.org/10.1103/PhysRevE.52.5297

    Article  ADS  Google Scholar 

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

    Alessandro Manacorda

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Manacorda, A. (2018). Hydrodynamic Description and Lattice Models. 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_3

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