Lattice Models for Granular-Like Velocity Fields: Hydrodynamic Description

Abstract

A recently introduced model describing—on a 1d lattice—the velocity field of a granular fluid is discussed in detail. The dynamics of the velocity field occurs through next-neighbours inelastic collisions which conserve momentum but dissipate energy. The dynamics is described through the corresponding Master Equation for the time evolution of the probability distribution. In the continuum limit, equations for the average velocity and temperature fields with fluctuating currents are derived, which are analogous to hydrodynamic equations of granular fluids when restricted to the shear modes. Therefore, the homogeneous cooling state, with its linear instability, and other relevant regimes such as the uniform shear flow and the Couette flow states are described. The evolution in time and space of the single particle probability distribution, in all those regimes, is also discussed, showing that the local equilibrium is not valid in general. The noise for the momentum and energy currents, which are correlated, are white and Gaussian. The same is true for the noise of the energy sink, which is usually negligible.

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Notes

  1. 1.

    For the usual choice of an initial Gaussian distribution, see Sect. 4.1.

  2. 2.

    Note that nearest-neighbours velocity correlations go from negative values for \(\nu =0\) (elastic limit, due to momentum conservation, where all \(\langle v_iv_{i+j}\rangle =-1/L\)) to positive values for high \(\nu \), since the granular collision rule tends to paralelise the velocities. This explains at an intuitive level why there is a value of \(\nu \) where the above correlations vanish [26, 51].

References

  1. 1.

    Jaeger, H.M., Nagel, S.R., Behringer, R.P.: Granular solids, liquids, and gases. Rev. Mod. Phys. 68(4), 1259 (1996)

    ADS  Article  Google Scholar 

  2. 2.

    Puglisi, A.: Transport and fluctuations in granular fluids. Springer, Berlin (2014)

    Google Scholar 

  3. 3.

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

  4. 4.

    van Noije, T.P.C., Ernst, M.H.: Velocity distributions in homogeneous granular fluids: the free and the heated case. Gran. Matt. 1, 57 (1998)

    Article  Google Scholar 

  5. 5.

    Lun, C.K.K., Savage, S.B., Jeffrey, D.J., Chepurniy, N.: Kinetic theories for granular flow: inelastic particles in couette flow and slightly inelastic particles in a general flowfield. J. Fluid. Mech. 140, 223 (1984)

    ADS  Article  MATH  Google Scholar 

  6. 6.

    Brey, J.J., Dufty, J.W., Kim, C.S., Santos, A.: Hydrodynamics for granular flow at low density. Phys. Rev. E 58(4), 4638 (1998)

    ADS  Article  Google Scholar 

  7. 7.

    Goldhirsch, I.: Scales and kinetics of granular. Chaos 9, 659 (1999)

    ADS  Article  MATH  Google Scholar 

  8. 8.

    Kadanoff, L.P.: Built upon sand: Theoretical ideas inspired by granular flows. Rev. Mod. Phys. 71(1), 435–444 (1999)

    ADS  Article  Google Scholar 

  9. 9.

    van Noije, T.P.C., Ernst, M.H.: Cahn-hilliard theory for unstable granular fluids. Phys. Rev. E 61, 1765 (2000)

    ADS  Article  Google Scholar 

  10. 10.

    Einstein, A.: Zur allgemeinen molekularen theorie der wärme. Ann. Phys. 319(7), 354–362 (1904)

    Article  MATH  Google Scholar 

  11. 11.

    Onsager, L., Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. 91(6), 1505 (1953)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Landau, L.D., Lifshitz, E.M.: Statistical Physics 3rd edition Course of Theoretical Physics, vol. 5. Pergamon Press, Oxford (1980)

    Google Scholar 

  13. 13.

    Brey, J.J., Maynar, P., de Soria, M.I.G.: Fluctuating hydrodynamics for dilute granular gases. Phys. Rev. E 79, 051305 (2009)

    ADS  Article  Google Scholar 

  14. 14.

    Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Fluctuations in stationary nonequilibrium states of irreversible processes. Phys. Rev. Lett. 87(4), 040601 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Springer, New York (1999)

    Google Scholar 

  16. 16.

    Kipnis, C., Marchioro, C., Presutti, E.: Heat flow in an exactly solvable model. J. Stat. Phys. 27(1), 65–74 (1982)

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Hurtado, P.I., Garrido, P.L.: Test of the additivity principle for current fluctuations in a model of heat conduction. Phys. Rev. Lett. 102(25), 250601 (2009)

    ADS  Article  Google Scholar 

  18. 18.

    Hurtado, P.I., Garrido, P.L.: Large fluctuations of the macroscopic current in diffusive systems: A numerical test of the additivity principle. Phys. Rev. E 81(4), 041102 (2010)

    ADS  Article  Google Scholar 

  19. 19.

    Hurtado, P.I., Garrido, P.L.: Current fluctuations and statistics during a large deviation event in an exactly solvable transport model. J. Stat. Mech. (Theor. Exp.) 2009(02), P02032 (2009)

    Google Scholar 

  20. 20.

    Hurtado, P., Krapivsky, P.: Compact waves in microscopic nonlinear diffusion. Phys. Rev. E 85(6), 060103 (2012)

    ADS  Article  Google Scholar 

  21. 21.

    Srebro, Y., Levine, D.: Exactly solvable model for driven dissipative systems. Phys. Rev. Lett. 93, 240610 (2004)

    Article  Google Scholar 

  22. 22.

    Prados, A., Lasanta, A., Hurtado, P.I.: Nonlinear driven diffusive systems with dissipation: Fluctuating hydrodynamics. Phys. Rev. E 86(3), 031134 (2012)

    ADS  Article  Google Scholar 

  23. 23.

    Prados, A., Lasanta, A., Hurtado, P.I.: Large fluctuations in driven dissipative media. Phys. Rev. Lett. 107(14), 140601 (2011)

    ADS  Article  Google Scholar 

  24. 24.

    Hurtado, P.I., Lasanta, A., Prados, A.: Typical and rare fluctuations in nonlinear driven diffusive systems with dissipation. Phys. Rev. E 88(2), 022110 (2013)

    ADS  Article  Google Scholar 

  25. 25.

    Lasanta, A., Hurtado, P.I., Prados, A.: Statistics of the dissipated energy in driven diffusive systems. Eur. Phys. J. E 39(3), 35 (2016)

    Article  Google Scholar 

  26. 26.

    Lasanta, A., Manacorda, A., Prados, A., Puglisi, A.: Fluctuating hydrodynamics and mesoscopic effects of spatial correlations in dissipative systems with conserved momentum. New J. Phys. 17, 083039 (2015)

    ADS  Article  Google Scholar 

  27. 27.

    Spohn, H.: Long range correlations for stochastic lattice gases in a non-equilibrium steady state. J. Phys. A Math. Gen. 16, 4275 (1983)

    ADS  MathSciNet  Article  Google Scholar 

  28. 28.

    Grinstein, G., Lee, D.-H., Sachdev, S.: Conservation laws, anisotropy, and self-organized criticality in noisy nonequilibrium systems. Phys. Rev. Lett. 64(16), 1927 (1990)

    ADS  Article  Google Scholar 

  29. 29.

    Garrido, P.L., Lebowitz, J.L., Maes, C., Spohn, H.: Long-range correlations for conservative dynamics. Phys. Rev. A 42(4), 1954 (1990)

    ADS  MathSciNet  Article  Google Scholar 

  30. 30.

    Kundu, A., Hirschberg, O., Mukamel, D.: Long range correlations in stochastic transport with energy and momentum conservation

  31. 31.

    Ramaswamy, S.: The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys. 1, 323 (2010)

    ADS  Article  Google Scholar 

  32. 32.

    Kumar, N., Soni, H., Ramaswamy, S., Sood, A.K.: Flocking at a distance in active granular matter. Nat. Commun. 5, 4688 (2014)

    ADS  Article  Google Scholar 

  33. 33.

    Baskaran, A., Marchetti, M.C.: Enhanced diffusion and ordering of self-propelled rods. Phys. Rev. Lett. 101, 268101 (2008)

    ADS  Article  Google Scholar 

  34. 34.

    Marchetti, M., Joanny, J., Ramaswamy, S., Liverpool, T., Prost, J., Rao, M., Simha, R.A.: Hydrodynamics of soft active matter. Rev. Mod. Phys. 85(3), 1143 (2013)

    ADS  Article  Google Scholar 

  35. 35.

    Chaté, H., Ginelli, F., Montagne, R.: Simple model for active nematics: Quasi-long-range order and giant fluctuations. Phys. Rev. Lett. 96, 180602 (2006)

    ADS  Article  Google Scholar 

  36. 36.

    Raymond, J.R., Evans, M.R.: Flocking regimes in a simple lattice model. Phys. Rev. E 73, 036112 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  37. 37.

    Simha, R.A., Ramaswamy, S.: Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles. Phys. Rev. Lett. 89, 058101 (2002)

    ADS  Article  MATH  Google Scholar 

  38. 38.

    Brey, J.J., Cubero, D.: Hydrodynamic transport coefficients of granular gases. In: Pöschel, T., Luding, S. (eds.) Granular Gas. Springer, Berlin (2001)

  39. 39.

    Pöschel, T., Luding, S. (eds.): Granular Gases. Lecture Notes in Physics vol. 564. Springer, Berlin (2001)

  40. 40.

    Haff, P.: Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430 (1983)

    ADS  Article  MATH  Google Scholar 

  41. 41.

    Ernst, H.: Nonlinear model-Boltzmann equations and exact solutions. Phys. Rep. 78, 1–171 (1981)

    ADS  MathSciNet  Article  Google Scholar 

  42. 42.

    Brey, J.J., Ruiz-Montero, M., Cubero, D.: Homogeneous cooling state of a low-density granular flow. Phys. Rev. E 54(4), 3664 (1996)

    ADS  Article  Google Scholar 

  43. 43.

    Brey, J.J., Prados, A., de Soria, M.G., Maynar, P.: Scaling and aging in the homogeneous cooling state of a granular fluid of hard particles. J. Phys. A Math. Theor. 40(48), 14331 (2007)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Baldassarri, A., Marconi, U.M.B., Puglisi, A.: Influence of correlations on the velocity statistics of scalar granular gases. EPL (Europhysics Letters) 58(1), 14 (2002)

    ADS  Article  Google Scholar 

  45. 45.

    Ernst, M.H., Trizac, E., Barrat, A.: The rich behavior of the boltzmann equation for dissipative gases. Europhys. Lett. 76, 56 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  46. 46.

    Bortz, A.B., Kalos, M.H., Lebowitz, J.L.: A new algorithm for Monte Carlo simulation of Ising spin systems. J. Comput. Phys. 17(1), 10–18 (1975)

    ADS  Article  Google Scholar 

  47. 47.

    Prados, A., Brey, J.J., Sanchez-Rey, B.: A dynamical monte carlo algorithm for master equations with time-dependent transition rates. J. Stat. Phys. 89(3–4), 709–734 (1997)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  48. 48.

    Kampen, N.G.V.: Stochastic Processes in Physics and Chemistry. Norht-Holland, Amsterdam (1992)

    Google Scholar 

  49. 49.

    Marconi, U.M.B., Puglisi, A., Vulpiani, A.: About an H-theorem for systems with non-conservative interactions. J. Stat. Mech. 2013, P08003 (2013)

    MathSciNet  Article  Google Scholar 

  50. 50.

    de Soria, M.I.G., Maynar, P., Mischler, S., Mouhot, C., Rey, T., Trizac, E.: Towards an h-theorem for granular gases. J. Stat. Mech. Theory Exp. 2015(11), P11009 (2015)

    MathSciNet  Article  Google Scholar 

  51. 51.

    Plata, C. A., Manacorda, A., Lasanta, A., Puglisi, A., Prados, A.: Lattice models for granular-like velocity fields: finite size effects. arXiv:1606.09023

  52. 52.

    McNamara, S.: Hydrodynamic modes of a uniform granular medium. Phys. Fluids A 5, 3056 (1993)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  53. 53.

    van Noije, T.P.C., Ernst, M.H., Brito, R., Orza, J.A.G.: Mesoscopic theory of granular fluids. Phys. Rev. Lett. 79, 411 (1997)

    ADS  Article  Google Scholar 

  54. 54.

    García de Soria, M.I., Maynar, P., Schehr, G., Barrat, A., Trizac, E.: Dynamics of annihilation i. linearized boltzmann equation and hydrodynamics. Phys. Rev. E 77, 051127 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  55. 55.

    Lees, A.W., Edwards, S.F.: The computer study of transport processes under extreme conditions. J. Phys. C Solid State Phys. 5(15), 1921 (1972)

    ADS  Article  Google Scholar 

  56. 56.

    Santos, A., Garzó, V.: Simple shear flow in inelastic maxwell models. J. Stat. Mech. Theory Exp. 2007(08), P08021 (2007)

    Article  Google Scholar 

  57. 57.

    Santos, A., Garzó, V., Dufty, J.W.: Inherent rheology of a granular fluid in uniform shear flow. Phys. Rev. E 69, 061303 (2004)

    ADS  Article  Google Scholar 

  58. 58.

    Garzó, V.: Transport coefficients for an inelastic gas around uniform shear flow: Linear stability analysis. Phys. Rev. E 73, 021304 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  59. 59.

    Kubo, R., Toda, M., Hashitsume, N.: Statistical physics II: Nonequilibrium stastical mechanics. Springer, New York (1991)

    Google Scholar 

  60. 60.

    García de Soria, M.I., Maynar, P., Trizac, E.: Universal reference state in a driven homogeneous granular gas. Phys. Rev. E 85, 051301 (2012)

    ADS  Article  Google Scholar 

  61. 61.

    García de Soria, M.I., Maynar, P., Trizac, E.: Linear hydrodynamics for driven granular gases. Phys. Rev. E 87, 022201 (2013)

    ADS  Article  Google Scholar 

  62. 62.

    Marconi, U.M.B., Puglisi, A., Rondoni, L., Vulpiani, A.: Fluctuation–dissipation: Response theory in statistical physics. Phys. Rep. 461, 111 (2008)

    ADS  Article  Google Scholar 

  63. 63.

    Prados, A., Trizac, E.: Kovacs-like memory effect in driven granular gases. Phys. Rev. Lett. 112, 198001 (2014)

    ADS  Article  Google Scholar 

  64. 64.

    Trizac, E., Prados, A.: Memory effect in uniformly heated granular gases. Phys. Rev. E 90, 012204 (2014)

    ADS  Article  Google Scholar 

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Acknowledgments

We acknowledge Pablo Maynar for really helpful discussions. C. A. P. acknowledges the support from the FPU Fellowship Programme of Spanish Ministerio de Educación, Cultura y Deporte through Grant FPU14/00241. C. A. P. and A. Prados acknowledge the support of the Spanish Ministerio de Economía y Competitividad through Grant FIS2014-53808-P.

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Correspondence to Alessandro Manacorda.

Appendix: Gaussian Character of the Noises

Appendix: Gaussian Character of the Noises

In the large system size limit \(L\gg 1\), the current noise introduced in the Sect. (5.1) is white. We can introduce a new noise field \(\tilde{\xi }(x,t)\) by

$$\begin{aligned} \xi ^{(j)}(x,t)=L^{-1/2}\tilde{\xi }(x,t) \end{aligned}$$
(89)

and \(\tilde{\xi }(x,t)\) remains finite in the large system size limit \(L \rightarrow \infty \),

$$\begin{aligned} \langle \tilde{\xi }(x,t)\rangle =0, \, \quad \langle \tilde{\xi }(x,t)\tilde{\xi }(x',t')\rangle \sim 2\,T(x,t) \delta (x-x')\delta (t-t'). \end{aligned}$$
(90)

Here we show that all the higher-order cumulants of \(\tilde{\xi }(x,t)\) vanish in the thermodynamic limit as \(L \rightarrow \infty \). Let us consider a cumulant of order n of the microscopic noise \(\xi _{l,p}\) that is equal to the nth order moment of the \(\xi \) plus a sum of nonlinear products of lower moments of \(\xi \). A calculation analogous to the one carried out for the correlation \(\langle \xi ^{(j)}_{l,p}\xi ^{(j)}_{l',p'}\rangle \) shows that the leading behaviour of any moment is of the order of \(L^{-1}\), which is obtained when all the times are the same. Therefore, the moment \(\langle j_{l,p}j_{l',p'}\ldots j_{l^{(n)},p^{(n)}}\rangle \) gives the leading behaviour of the considered cumulant, which is thus of the order of \(L^{-1}\) for \(p=p'= \cdots =p^{(n)}\); any other contribution to the cumulant is at least of the order of \(L^{-2}\). We have that

$$\begin{aligned} \langle j_{l,p}j_{l',p'}\ldots j_{l^{(n)},p^{(n)}}\rangle \sim L^{-1}\langle C_{l,p}\rangle \delta _{l,l'}\delta _{l',l''}\delta _{l^{(n-1)},l^{(n)}}\cdots \delta _{p,p'}\delta _{p',p''}\delta _{p^{(n-1)},p^{(n)}}, \end{aligned}$$
(91)

where \(\langle C_{l,p}\rangle \) is certain average that remains finite in the large system size limit as \(L \rightarrow \infty \). In the continuous limit, each current introduces a factor \(L^{2}\) due to the scaling introduced in Sect. 5. Moreover, we take into account the relationship between Kronecker and Dirac \(\delta \)’s in the continuum limit to write the cumulants \(\langle \langle \cdots \rangle \rangle \) of the rescaled noise introduced in (89) as

$$\begin{aligned}&\langle \langle \tilde{\xi }(x,t)\tilde{\xi }(x',t')\cdots \tilde{\xi }(x^{(n)},t^{(n)})\rangle \rangle \sim {L^{3\left( 1-\frac{n}{2} \right) }}{\langle C(x,t)\rangle }\times \nonumber \\&\qquad \delta (x-x')\delta (x'-x'')\delta (x^{(n-1)}-x^{(n)})\cdots \delta (t-t')\delta (t'-t'')\delta (t^{(n-1)}-t^{(n)}). \end{aligned}$$
(92)

Thus, in the limit as \(L \rightarrow \infty \),

$$\begin{aligned} \langle \tilde{\xi }(x,t)\tilde{\xi }(x',t')\cdots \tilde{\xi }(x^{(n)},t^{(n)})\rangle =0, \quad \text {for all}\; n>2, \end{aligned}$$
(93)

and the vanishing of all the cumulants for \(n>2\) means that the momentum current noise is Gaussian in the infinite size limit.

The same procedure can be repeated for the energy current noise, by defining \(\xi ^{(J)}(x,t)=L^{-1/2}\tilde{\eta }(x,t)\)), with the result

$$\begin{aligned}&\langle \langle \tilde{\eta }(x,t)\tilde{\eta }(x',t')\cdots \tilde{\eta }(x^{(n)},t^{(n)})\rangle \rangle \sim {L^{3\left( 1-\frac{n}{2} \right) }}{\langle D(x,t)\rangle }\times \nonumber \\&\qquad \delta (x-x')\delta (x'-x'')\delta (x^{(n-1)}-x^{(n)})\cdots \delta (t-t')\delta (t'-t'')\delta (t^{(n-1)}-t^{(n)}). \end{aligned}$$
(94)

In the equation above, \(\langle D(x,t)\rangle \) is a certain average, different from \(\langle C(x,t)\rangle \), but also finite in the large system size limit. Thus, we have that

$$\begin{aligned} \langle \tilde{\eta }(x,t)\tilde{\eta }(x',t')\cdots \tilde{\eta }(x^{n},t^{n})\rangle =0, \quad \text {for all}\; n>2, \end{aligned}$$
(95)

and the energy current noise also becomes Gaussian in the continuum limit.

Note that the Gaussianity of the noises is independent of the validity of the local equilibrium approximation, which is only needed to write \(\langle C(x,t)\rangle \) and \(\langle D(x,t)\rangle \) in terms of the hydrodynamic fields u(xt) and T(xt). Besides, a similar procedure for the dissipation noise gives that the corresponding scaled noise vanishes in the continuum limit, since the power of L in the dominant contribution to the nth order cumulant is \(3-5n/2\) instead of \(3-3n/2\). This means that the dissipation noise is subdominant as compared to the currents noises in the continuum limit, and can be neglected.

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Manacorda, A., Plata, C.A., Lasanta, A. et al. Lattice Models for Granular-Like Velocity Fields: Hydrodynamic Description. J Stat Phys 164, 810–841 (2016). https://doi.org/10.1007/s10955-016-1575-z

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Keywords

  • Granular fluids
  • Hydrodynamics
  • Momentum conservation