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Anna
Bodrova
^{ab},
Aleksei V.
Chechkin
^{acd},
Andrey G.
Cherstvy
^{a} and
Ralf
Metzler
*^{ae}
^{a}Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany. E-mail: rmetzler@uni-potsdam.de
^{b}Faculty of Physics, M.V. Lomonosov Moscow State University, Moscow 119991, Russia
^{c}Akhiezer Institute for Theoretical Physics, Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine
^{d}Max-Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
^{e}Department of Physics, Tampere University of Technology, 33101 Tampere, Finland

Received
15th May 2015
, Accepted 27th July 2015

First published on 27th July 2015

Brownian motion is ergodic in the Boltzmann–Khinchin sense that long time averages of physical observables such as the mean squared displacement provide the same information as the corresponding ensemble average, even at out-of-equilibrium conditions. This property is the fundamental prerequisite for single particle tracking and its analysis in simple liquids. We study analytically and by event-driven molecular dynamics simulations the dynamics of force-free cooling granular gases and reveal a violation of ergodicity in this Boltzmann–Khinchin sense as well as distinct ageing of the system. Such granular gases comprise materials such as dilute gases of stones, sand, various types of powders, or large molecules, and their mixtures are ubiquitous in Nature and technology, in particular in Space. We treat—depending on the physical-chemical properties of the inter-particle interaction upon their pair collisions—both a constant and a velocity-dependent (viscoelastic) restitution coefficient ε. Moreover we compare the granular gas dynamics with an effective single particle stochastic model based on an underdamped Langevin equation with time dependent diffusivity. We find that both models share the same behaviour of the ensemble mean squared displacement (MSD) and the velocity correlations in the limit of weak dissipation. Qualitatively, the reported non-ergodic behaviour is generic for granular gases with any realistic dependence of ε on the impact velocity of particles.

Ergodicity is a fundamental concept of statistical mechanics. Starting with Boltzmann, the ergodic hypothesis states that long time averages of a physical observable are identical to their ensemble averages .^{11,12} In this sense, Brownian motion is ergodic even at out-of-equilibrium conditions, while a range of anomalous diffusion processes exhibit a distinct disparity : for instance, for sufficiently long observation times the time averaged mean squared displacement (MSD) of a Brownian particle converges to the corresponding ensemble average 〈R^{2}(t)〉^{13,14} calling for generalisation of the classical ergodic theories.^{12} In fact, similar concepts were already discussed in the context of glassy systems.^{15} In the wake of modern microscopic techniques, such as single particle tracking,^{16} in which individual trajectories of single molecules or submicron tracers are routinely measured, knowledge of the ergodic properties of the system is again pressing. While the time averages are measured in single particle assays or massive computer simulations, generally ensemble averages are more accessible theoretically. How measured time averages can be interpreted in terms of ensemble approaches and diffusion models is thus an imminent topic.^{13,14}

Here we quantify in detail from analytical derivations and extensive simulations how exactly the ergodicity is violated in simple mechanical systems such as force-free granular gases. Our results for generic granular gases are relevant both from a fundamental statistical mechanical point of view and for the practical analysis of time series of granular gas particles from observations and computer simulations. Specifically, (i) we here derive the time and ensemble averaged MSDs and show that for both constant and viscoelastic restitution coefficients the time averaged MSD is fundamentally different from the corresponding ensemble MSD. (ii) Moreover, the amplitude of the time averaged MSD is shown to be a decaying function of the length of the measured trajectory (ageing). (iii) We study an effective single particle mean field approach to the granular gas dynamics. This underdamped scaled Brownian motion (SBM) demonstrates how non-ergodicity and ageing emerge from the non-stationarity invoked by the time dependence of the granular temperature, which translates into the power-law time dependence of the diffusion coefficient of SBM. We note that systems with time dependent diffusion coefficients are in fact common in nature, ranging from mobility of proteins in cell membranes,^{17} motion of molecules in porous environments,^{18} water diffusion in brain as measured by magnetic resonance imaging,^{19} to snow-melt dynamics.^{20,21}

The energy dissipation in a pair-wise collision event of granular particles is quantified by the restitution coefficient

(1) |

(2) |

T(t) = m〈v^{2}〉/2 | (3) |

T(t) = T_{0}/(1 + t/τ_{0})^{2}. | (4) |

D(t) = T(t)τ_{v}(t)/m = D_{0}/(1 + t/τ_{0}), | (5) |

Most studies of granular gases assume that ε is constant. Different approaches consider the dependence of ε on the relative collision speed of the form^{30,31}

ε(v_{12}) ≃ 1 − C_{1}Aκ^{2/5}(v_{12}·e)^{1/5} + C_{2}A^{2}κ^{4/5}(v_{12}·e)^{2/5}. | (6) |

D(t) ∼ t^{−5/6}, | (7) |

Fig. 2 Ensemble (〈R^{2}(t)〉) and time averaged MSDs versus (lag) time (upper graph) and versus length t of the time series (lower graph), from event-driven MD simulations of a granular gas with two different values of the restitution coefficient, ε = 0.3 and 0.8. While the ensemble MSD crosses over from ballistic motion 〈R^{2}(t)〉 ∼ t^{2} for t ≪ τ_{0} to the logarithmic law 〈R^{2}(t)〉 ∼ log(t) for t ≫ τ_{0}, the time averaged MSD starts ballistically and crosses over to the scaling given by eqn (A10). |

Fig. 3 MSDs 〈R^{2}(t)〉 and as function of (lag) time (top) and versus the measurement time t (bottom) from MD simulations (symbols) of a granular gas with viscoelastic ε(v_{12}). We observe the scaling 〈R^{2}(t)〉 ∼ t^{1/6} in the limit t ≫ τ_{0}. The scaling of the time averaged MSD slowly changes between the indicated slopes (dashed lines). The continuous change of slope of as function of the length t of time traces from slope −5/6 to −1 is seen in the inset of the bottom graph. The results for the time averaged MSD with the restitution coefficient computed according to the Padé approximation^{32} (see text for details) are shown as the red filled squares in the top panel. |

We evaluate the gas dynamics in terms of the standard ensemble MSD 〈R^{2}(t)〉, obtained from averaging over all gas particles at time t, as well as the time averaged MSD

(8) |

(9) |

Fig. 2 shows the results of our computer simulations of a granular gas with constant ε = 0.8 and 0.3. The ensemble MSD shows initial ballistic particle motion, 〈R^{2}(t)〉 ∼ t^{2}. Eventually, the particles start to collide and gradually lose kinetic energy. The ensemble MSD of the gas in this regime follows the logarithmic law 〈R^{2}(t)〉 ∼ log(t) (the red line in Fig. 2, top panel).^{2} The time averaged MSD at short lag times Δ preserves the ballistic law . At longer lag times, we observe the linear growth (black symbols in Fig. 2, top). In addition to this non-ergodic behaviour, the time averaged MSD decreases with increasing length t of the recorded trajectory, . This highly non-stationary behaviour is also referred to as ageing, the dependence of the system dynamics on its time of evolution.^{41} The dependence on the trace length we observe in the bottom panel of Fig. 2 implies that the system is becoming progressively slower. We observe the convergence .

Fig. 3 depicts the results of MD simulations for a granular gas with viscoelastic restitution coefficient (6) with Aκ^{2/5} = 0.2. In this case the ensemble MSD scales as

〈R^{2}(t)〉 ∼ t^{1/6} | (10) |

(11) |

τ = τ_{0}log(1 + t/τ_{0}). | (12) |

〈c(τ_{1})c(τ_{2})〉 = (3/2)exp(−|τ_{2} − τ_{1}|/τ_{v}(0)). | (13) |

(14) |

β = τ_{0}/τ_{v}(0). |

(15) |

From the autocorrelation function (14) we obtain the time averaged MSD (see Appendix A)

(16) |

〈R^{2}(t)〉 ∼ 36D_{0}τ_{0}^{5/6}t^{1/6}, |

and

compare the details in Appendix A. These bounds are given by the dashed lines in the top panel of Fig. 3. Concurrent to this change of slopes as a function of the lag time, the bottom panel of Fig. 3 shows the change of slope of as function of the trajectory length t from the slope −5/6 to −1 at a fixed lag time Δ.

We note that a more explicit expression for the viscoelastic restitution coefficient can be obtained in terms of the Padé approximant [3/6]_{ε}, as derived in ref. 32. In Fig. 3 we demonstrate, however, that for the range of parameters used in our simulations—corresponding to relatively slow collision velocities of granular particles (scaled thermal velocity v* < 0.3)—we obtain nearly the same results for the time averaged MSD as our previous simulations with the restitution coefficient (6), see the red filled squares in Fig. 3.

D(t) ∼ t^{α−1} | (17) |

To study whether SBM provides an effective single particle description of diffusion in dissipative granular gases we extend SBM to the underdamped case. We thus take the inertial term explicitly into account when considering the dynamics,

(18) |

For α = 0 the velocity correlation may be derived from the Langevin eqn (18), namely

(19) |

Here, we demonstrated how non-ergodicity arises in a simple mechanistic systems such as force-free granular gases. Physically, it stems from a strong non-stationary character of this process brought about by the continuous decay of the gas temperature. Therefore, the ergodicity breaking is expected independent of the particular model of the restitution coefficient ε, while the precise behaviour of the MSD and time averaged MSD clearly depends on the specific law for ε.

For a constant restitution coefficient, the MSD of gas particles 〈R^{2}(t)〉 grows logarithmically, while the time averaged MSD is linear in the lag time and decays inverse proportionally with the trace length (ageing). We derived the observed non-ergodicity and the ageing behaviour of granular gases from the velocity autocorrelation functions. We note that ageing in the homogeneous cooling state of granular gases was reported previously,^{48} however, it was not put in context with the diffusive dynamics of gas particles.

The decaying temperature of the dissipative force-free granular gas corresponds to an increase of the time span between successive collisions of gas particles, a feature directly built into the SBM model.^{44} As we showed here, SBM and its ultraslow extension with the logarithmic growth of the MSD indeed captures certain features of the observed motion and may serve as an effective single particle model for the granular gas. It is particularly useful when more complex situations are considered, such as the presence of external force fields. Our results shed new light on the physics of granular gases with respect to their violation of ergodicity in the Boltzmann sense. They are important for a better understanding of dissipation in free gases as well as the analysis of experimental observations and MD studies of granular gases.

It will be interesting to compare the results obtained herein—based on the two standard assumptions for the restitution coefficient—with experimental observations of granular gas systems. Similarly, it might be of interest to see to what extent the present scenario pertains to dilute gases of complex molecules with a large number of internal degrees of freedom ready to absorb a part of the collision energies.^{2–4}

The time averaged MSD for the granular gas with constant restitution coefficient, eqn (8) in the main text, may be written as

(A1) |

(A2) |

(A3) |

(A4) |

(A5) |

where β = τ

(A6) |

(A7) |

(A8) |

(A9) |

(A10) |

(B1) |

(B2) |

(B3) |

1 ≪ k_{1} ≪ τ_{0}Δ^{1/5}/τ_{v}^{6/5}(0) |

τ
_{0}Δ^{1/5}/τ_{v}^{6/5}(0) ≪ k_{2} ≪ t/Δ. |

(B4) |

(B5) |

(B6) |

(B7) |

(B8) |

Fig. 4 Time averaged MSD divided by lag time Δ as function of Δ from MD simulations (symbols) of a granular gas with velocity-dependent restitution coefficient. The lines connecting the symbols guide the eye. Red line corresponds to numerical calculation of in eqn (A3) for τ_{0} = 25, τ_{v} = 2, D_{0} = 2. These values ensure the closest agreement and are consistent with the parameters of the granular gas as used in MD simulations apart from very short lag times. Dashed line shows the asymptotic behaviour according to eqn (B8). |

- H. M. Jaeger, S. R. Nagel and R. P. Behringer, Rev. Mod. Phys., 1996, 68, 1259 CrossRef; Physics of Dry Granular Media, ed. H. J. Herrmann, J.-P. Hovi and S. Luding, NATO ASI Series, Kluwer, Dordrecht, 1998 Search PubMed.
- N. V. Brilliantov and T. Pöschel, Kinetic theory of Granular Gases, Oxford University Press, 2004 Search PubMed.
- Mathematics and Mechanics of Granular Materials, ed. J. M. Hill and A. P. S. Selvadurai, Springer, 2005 Search PubMed.
- Unifying Concepts in Granular Media and Glasses, ed. A. A. Coniglio, H. J. Herrmann and M. Nicodemi, Elsevier, Amsterdam, 2004 Search PubMed.
- A. Mehta, in Granular Physics, Cambridge University Press, 2011 Search PubMed; Granular Matter, ed. A. Mehta, Springer, Berlin, 2011 Search PubMed.
- R. D. Wildman and D. J. Parker, Phys. Rev. Lett., 2002, 88, 064301 CrossRef CAS; A. Prevost, D. A. Egolf and J. S. Urbach, Phys. Rev. Lett., 2002, 89, 084301 CrossRef.
- O. Zik, D. Levine, S. Lipson, S. Shtrikman and J. Stavans, Phys. Rev. Lett., 1994, 73, 644 CrossRef.
- I. S. Aranson and J. S. Olafsen, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2002, 66, 061302 CrossRef CAS.
- A. Snezhko, I. S. Aranson and W.-K. Kwok, Phys. Rev. Lett., 2005, 94, 108002 CrossRef CAS; C. C. Maaß, N. Isert, G. Maret and C. M. Aegerter, Phys. Rev. Lett., 2008, 100, 248001 CrossRef.
- R. Greenberg and A. Brahic, Planetary Rings, University of Arizona Press, Tucson, AZ, 1984 CrossRef CAS PubMed; A. Bodrova, J. Schmidt, F. Spahn and N. V. Brilliantov, Icarus, 2012, 218, 60 CrossRef CAS PubMed; F. Spahn, U. Schwarz and J. Kurths, Phys. Rev. Lett., 1997, 78, 1596 CrossRef.
- A. I. Khinchin, Mathematical foundations of statistical mechanics, Dover, New York, NY, 1949 Search PubMed; M. Toda, R. Kubo and N. Saitô, Statistical Physics I, Springer, Berlin, 1992 Search PubMed.
- S. Burov, R. Metzler and E. Barkai, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 13228 CrossRef CAS PubMed.
- E. Barkai, Y. Garini and R. Metzler, Phys. Today, 2012, 65(8), 29 CrossRef CAS PubMed; I. M. Sokolov, Soft Matter, 2012, 8, 9043 RSC; S. Burov, J.-H. Jeon, R. Metzler and E. Barkai, Phys. Chem. Chem. Phys., 2011, 13, 1800 RSC.
- R. Metzler, J.-H. Jeon, A. G. Cherstvy and E. Barkai, Phys. Chem. Chem. Phys., 2014, 16, 24128 RSC.
- J.-P. Bouchaud, J. Phys. I, 1992, 2, 1705 CrossRef.
- C. Bräuchle, D. C. Lamb and J. Michaelis, Single Particle Tracking and Single Molecule Energy Transfer, Wiley-VCH, Weinheim, Germany, 2012 CrossRef CAS PubMed; X. S. Xie, P. J. Choi, G.-W. Li, N. K. Lee and G. Lia, Annu. Rev. Biophys., 2008, 37, 417 CrossRef CAS PubMed.
- T. J. Feder, I. Brust-Mascher, J. P. Slattery, B. Baird and W. W. Webb, Biophys. J., 1996, 70, 2767 CrossRef CAS.
- P. N. Sen, Concepts Magn. Reson., Part A, 2004, 23, 1 CrossRef PubMed.
- D. S. Novikov, J. H. Jensen, J. A. Helpern and E. Fieremans, Proc. Natl. Acad. Sci. U. S. A., 2014, 111, 5088 CrossRef CAS PubMed.
- A. Molini, P. Talkner, G. G. Katul and A. Porporato, Physica A, 2011, 390, 1841 CrossRef CAS PubMed.
- D. De Walle and A. Rango, Principles of Snow Hydrology, Cambridge University Press, 2008 Search PubMed.
- Y. Grasselli, G. Bossis and G. Goutallier, Europhys. Lett., 2009, 86, 60007 CrossRef.
- M. J. Ruiz-Montero and J. J. Brey, Eur. Phys. J.: Spec. Top., 2009, 179, 249 CrossRef PubMed.
- K. Saitoh, A. Bodrova, H. Hayakawa and N. V. Brilliantov, Phys. Rev. Lett., 2010, 105, 238001 CrossRef.
- P. K. Haff, J. Fluid Mech., 1983, 134, 401 CrossRef.
- N. V. Brilliantov and T. Pöschel, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2000, 61, 1716 CrossRef CAS.
- J. J. Brey, M. J. Ruiz-Montero, D. Cubero and R. Garcia-Rojo, Phys. Fluids, 2000, 12, 876 CrossRef CAS PubMed.
- J. W. Dufty, J. J. Brey and J. Lutsko, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2002, 65, 051303 CrossRef; J. Lutsko, J. J. Brey and J. W. Dufty, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2002, 65, 051304 CrossRef.
- A. Bodrova and N. Brilliantov, Granular Matter, 2012, 14, 85 CrossRef CAS.
- R. Ramirez, T. Pöschel, N. V. Brilliantov and T. Schwager, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1999, 60, 4465 CrossRef CAS; D. L. Blair and A. Kudrolli, Phys. Rev. E, 2003, 67, 041301 CrossRef.
- T. Schwager and T. Pöschel, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 78, 051304 CrossRef.
- P. Müller and T. Pöschel, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2011, 84, 021302 CrossRef.
- T. Schwager and T. Pöschel, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1998, 57, 650 CrossRef CAS.
- N. V. Brilliantov and T. Pöschel, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2000, 61, 5573 CrossRef CAS.
- A. Bodrova, A. K. Dubey, S. Puri and N. Brilliantov, Phys. Rev. Lett., 2012, 109, 178001 CrossRef.
- T. Pöschel and T. Schwager, Computational Granular Dynamics, Springer, Berlin, 2005, see also http://www.mss.cbi.uni-erlangen.de/cgd/ Search PubMed.
- I. Golding and E. C. Cox, Phys. Rev. Lett., 2006, 96, 092102 CrossRef CAS PubMed; Y. M. Wang, R. H. Austin and E. C. Cox, Phys. Rev. Lett., 2006, 97, 048302 CrossRef.
- A. V. Weigel, B. Simon, M. M. Tamkun and D. Krapf, Proc. Natl. Acad. Sci. U. S. A., 2011, 108, 6438 CrossRef CAS PubMed; S. M. A. Tabei, S. Burov, H. Y. Kim, A. Kuznetsov, T. Huynh, J. Jureller, L. H. Philipson, A. R. Dinner and N. F. Scherer, Proc. Natl. Acad. Sci. U. S. A., 2013, 110, 4911 CrossRef PubMed; J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Oddershede and R. Metzler, Phys. Rev. Lett., 2011, 106, 048103 CrossRef.
- Y. He, S. Burov, R. Metzler and E. Barkai, Phys. Rev. Lett., 2008, 101, 058101 CrossRef CAS; A. Lubelski, I. M. Sokolov and J. Klafter, Phys. Rev. Lett., 2008, 100, 250602 CrossRef.
- A. Godec, A. V. Chechkin, H. Kantz, E. Barkai and R. Metzler, J. Phys. A: Math. Theor., 2014, 47, 492002 CrossRef.
- J. H. P. Schulz, E. Barkai and R. Metzler, Phys. Rev. Lett., 2013, 110, 020602 CrossRef; J. H. P. Schulz, E. Barkai and R. Metzler, Phys. Rev. X, 2014, 4, 011028 Search PubMed.
- T. P. C. van Noije, M. H. Ernst, R. Brito and J. A. G. Orza, Phys. Rev. Lett., 1997, 79, 411 CrossRef CAS; J. F. Lutsko, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2001, 63, 061211 CrossRef.
- S. C. Lim and S. V. Muniandy, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2002, 66, 021114 CrossRef CAS; M. J. Saxton, Biophys. J., 2001, 81, 2226 CrossRef; P. P. Mitra, P. N. Sen, L. M. Schwartz and P. Le Doussal, Phys. Rev. Lett., 1992, 68, 3555 CrossRef; J. F. Lutsko and J. P. Boon, Phys. Rev. Lett., 2013, 88, 022108 Search PubMed.
- A. Fuliński, J. Chem. Phys., 2013, 138, 021101 CrossRef CAS PubMed; A. Fuliński, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2011, 83, 061140 CrossRef; F. Thiel and I. M. Sokolov, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2014, 89, 012115 CrossRef; J.-H. Jeon, A. V. Chechkin and R. Metzler, Phys. Chem. Chem. Phys., 2014, 16, 15811 RSC.
- H. Safdari, A. G. Cherstvy, A. V. Chechkin, F. Thiel, I. M. Sokolov and R. Metzler, J. Phys. A, E-print arXiv:1507.02450 Search PubMed.
- H. Safdari, A. V. Chechkin, G. R. Jafari and R. Metzler, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2015, 91, 042107 CrossRef.
- A. Bodrova, A. V. Chechkin, A. G. Cherstvy and R. Metzler, New J. Phys., 2015, 17, 063038 CrossRef.
- J. J. Brey, A. Prados, M. I. Garca de Soria and P. Maynar, J. Phys. A: Math. Theor., 2007, 40, 14331 CrossRef CAS.

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