Liouville equation for granular gases Hisao Hayakawa （ YITP, Kyoto Univ. ） at 2008/10/17 & Michio Otsuki （ YITP, Kyoto Univ., Dept. of Physics, Aoyama-Gakuin Univ. ）

Aim of this talk This talk is very different from others. The purpose of this talk is what happens if local collision processes loose time-reversal symmetry.

Contents Introduction I. What is granular materials? II. Characteristics of sheared glassy or granular systems Liouville equation and MCT for sheared granular gases III. Liouville equation for sheared granular gases IV. Generalized Langevin equation V. MCT equation for sheared granular fluids Spatial correlation in sheared isothermal liquids VI. Spatial correlations in granular liquids VII. Linearized generalized fluctuating hydrodynamics VIII. Comparison between theory and simulation

I. What is granular materials? sand grains ： grain diameter is ranged in 0.01mm-1mm. Macroscopic particles Energy dissipation Repulsive systems Granular materials Many-body systems of dissipative particles

Granular shear flow Coexistence of “ solid ” region and “ fluid ” region There is creep motion in “ solid ” region. From H. M. Jeager, S. N. Nagel and R. P. Behringer, Rev. Mod. Phys. Vol. 68, 1259 (1996)

(1)Granular gases= A model of dusts (2) Uniform state is unstable. (3) It is not easy to perform experiments for gases. Granular Gases (What happens if molecules are dissipative?) I. Goldhirsch and G. Zanetti, Phys.Rev.Lett. 70, (1993).

Simulation of a freely cooling gas The restitution Area fraction 0.25 # of particles 640,000 Initial: equilibrium Time is scaled by the collision number By M. Isobe(NITECH) The correlation grows with time.

A simple model of granular gas The shear mode for the perturbation to a uniform state is always unstable because aligned motion of particles is survived. => string-like structure

Characteristics of inelastic collisions Energy is not conserved in each collision. Inelasticity is characterized by the restitution coefficient e<1. There is no time reversal symmetry in each collision. The phase volume is contracted at the instance of a collision.

Characteristics of granular hydrodynamics Theories remain in phenomenological level. Many theories are based on eigenvalue analysis of hydrodynamic equations. There is no sound wave in freely cooling case once inelasticity is introduced (HH and M.Otsuki,PRE2007) There are sound waves in sheared gases.

Contents

II. Characteristics of sheared glassy or granular systems Long time correlations: No-decay of correlations and freezing Correlated motion Dynamical heterogeneity A correlated motion of a granular system (left) and a colloidal system.

Similarity between jamming transition and glass transition Granular materials exhibit “ glass transition ” as a jamming. MCT can be used for sheared glass. Liu and Nagel, Nature (1998)

Jamming transition Jamming transition shows beautiful scalings (see right figs. by Otsuki and Hayakawa). What are the properties of dense but fluidized granular liquids?

Experimental relevancy of sheared systems Recently, there are some relevant experiments of sheared granular flows.

Simulation Shear can be added with or without gravity. For theoretical point of view, simple shear without gravity is the idealistic.

Similarity between sheared granular fluids and sheared isothermal fluids At least, the behaviors of velocity autocorrelation function, and the equal-time correlation function are common. (see M.Otsuki & HH, arXiv: )

Bagnold ’ s law for uniform sheared granular fluids The change of momentum Time scale This is the relation between the temperature and the shear rate. Shear stress

MCT for sheared granular fluids MCT equation can be derived for granular fluids starting from Liouville equation. This approach ensures formal universality in granular systems and conventional glassy systems. See HH and M. Otsuki, PTP 119, 381 (2008).

Affine transformation in sheared fluids Wave number is transferred.

Contents

III. Liouville equation for granular gases Collision operator Shear term in Liouvillian

Collision operator Here, b represents the change from a collision

Liouville equation

Properties of Liouville operator

Contents

IV. Generalized Langevin equation

Langevin equation in the steady state

Some functions in generalized Langevin equation

Remarks on steady state We should note that the steady ρ(Γ) is highly nontrivial. The steady state is determined by the balance between the external force and the inelastic collision. Thus, the eigenvalue problem cannot be solved exactly. In this sense, we adopt the formal argument. I will demonstrate how to solve linearized hydrodynamics as an eigenvalue problem, later.

Some formulae for hydrodynamic variables

Some formulae in shear flow

Generalized Langevin equation for sheared granular fluids (1) The density correlation function

Generalized Langevin equation for sheared granular fluids (2)

Equations for time-correlation

Some formulae

Contents

V. MCT equation for sheared granular fluids MCT approximation Hard-core=> all terms are balanced under Bagnold ’ s scaling

Preliminary simulation We have checked the relevancy of MCT equation for sheared dense granular liquids. MCT predicts the existence of a two- step relaxation. Parameters: 1000 LJ particles in 3D. The system contains binary particles, and has weak shear and weak dissipation.

Results of simulation for weak shear and weak dissipation The existence of the quasi- arrested state as MCT predicts.

Discussion of MCT equation Can MCT describe the jamming transition? The answer of the current MCT is NO. How can we determine S(q)? So far there is no theory to determine S(q), but it does not depend on F(q,t). No yield stress

Conclusion of MCT equation for sheared granular fluids MCT equation may be useful for very dense granular liquids. Our model starts from hard-core liquids <=The defect of this approach Nevertheless, our approach suggests that an unifying concept of sheared particles is useful.

Contents

VI. Spatial correlations in granular liquids The determination of the spatial correlations in granular liquids is important in MCT. It is known that there is a long-range velocity correlation r^{-d} (1997 Ernst, van Noije et al) for freely-cooling granular gases. It is also known that there is long-range correlation obeying a power law in sheared isothermal liquids of elastic particles. Lutsko and Dufty (1985,2002), Wada and Sasa (2003)

Spatial correlations in sheared isothermal liquids Let us explain how to determine the spatial correlations in terms of eigenvalue problems of linearized hydrodynamic equations. The result is based on M. Otsuki and HH, arXiv:

Motivation: to solve a confused situation Lutsko (2002) obtained the structure factor of sheared molecular liquids, but his result is not consistent with the long-range correlation obtained by himself. Many people believe that there is no contribution of the shear rate in the vicinity of glass transition. Is that true? The spatial correlation should be determined in MCT. Thus, we have to construct a theory to be valid for both particle scale and hydrodynamic scale.

Quantities we consider

Generalized fluctuating hydrodynamics (GFH) GFH was proposed by Kirkpatrick(1985). The basic equations consists of mass and momentum conservations. We analyze an isothermal situation obtained by the balance between the heating and inelastic collisions.

Properties of GFH The effective pressure The nonlocal viscous stress The stress has the thermal fluctuation. strain rate The direct correlation function

Characteristics of GFH GFH includes the structure of liquids. Generalized viscosities are represented by obtained by the eigenvalue problem of Enskog operator

Summary of GFH and setup We are not interested in higher order correlations. This can justify Gaussian noise We ignore the fluctuation of temperature from the technical reason. When we assume that the uniform shear flow is stable, the effect of temperature is not important. This situation can be realized in small and nearly elastic cases under Lees-Edwards boundary condition.

Contents

VII. The linearized GFH The linearized GFH is given by The linear equation can be solved analytically. The random force

Matrices

The solution of linearized equation (eigenvalue problem)

The solution of linearized GFH Steady pair correlation for unsheared system.

Contents

VIII. Comparison between theory and simulation We perform the molecular dynamics simulation for sheared granular liquids (e=0.83). We have examined cases for several densities. We also perform the simulaton for elastic cases.

Short-range density correlation The short-range density correlation can be approximated by Lutsko (2001). No fitting parameters The contribution of the shear is very small for dense case.

Long-range density correlation function However, the density correlation has a tail obeying a power law, which is the result of the shear.

Long-range momentum correlation The momentum correlation has clear a power-law tail obeying r^{-5/3}.

Discussion The effect of the temperature fluctuation is not clear. The elastic case can be analyzed within the same framework with putting e=1. The instability may destruct a power law correlation. Namely, large and strong inelastic systems encounter the violation of our theory. Quantitative calculation is still in progress.

Fugures for discussion (Left) The density correlation for e=1. (Right) The time evolution of momentum correlation. Small systems converges, but large systems do not converge. Elastic systems have the same scalings.

Conclusion We succeed to obtain the spatial correlations which covers both particle scale and hydrodynamic scale. There are long-range correlations obeying power laws. The generalized fluctuating hydrodynamics is a power tool to discuss this system.

Appendix

Parameters of our simulation

Linearized equation Random force Some additons

Matrices

The explicit forms of correlation functions

Pair-correlation by Lutsko (2001)