Liouville equation for granular gases

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 http://science.nasa.gov/headlines/y2002/06dec_dunes.htm

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)

Granular Gases (What happens if molecules are dissipative?) • Granular gases= A model of dusts • Uniform state is unstable. • (3) It is not easy to perform experiments for gases. I. Goldhirsch and G. Zanetti, Phys.Rev.Lett. 70 , 1619-1622 (1993).

Simulation of a freely cooling gas The restitution 0.99118 Area fraction 0.25 # of particles640,000 Initial: equilibrium Time is scaled by the collision number The correlation grows with time. By M. Isobe(NITECH)

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

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:0711.1421)

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

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.

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

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

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

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 simulationfor 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.

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:0809.4799.

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. The direct correlation function strain rate

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

Liouville equation for granular gases