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
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
This talk is very different from others.
The purpose of this talk is what
happens if local collision processes
loose time-reversal symmetry.
Contents
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Introduction
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Liouville equation and MCT for sheared granular
gases
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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
Spatial correlation in sheared isothermal liquids
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
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VI. Spatial correlations in granular liquids
VII. Linearized generalized fluctuating hydrodynamics
VIII. Comparison between theory and simulation
I. What is granular materials?
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sand grains： grain diameter is ranged
in 0.01mm-1mm.
Macroscopic particles
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Energy dissipation
Repulsive systems
Granular materials
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Many-body systems of dissipative particles
http://science.nasa.gov/headlines/y2002/06dec_dunes.htm
Granular shear flow


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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?)
(1)Granular gases= A model
of dusts
(2) 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 particles 640,000
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.

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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.
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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:
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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
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
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
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
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
p  m d,
t  1 / 
    m d 2 D |  | 
(  p / t )
This is the relation between the temperature and the shear rate.
2


  T |  |, T  
MCT for sheared granular
fluids
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

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




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


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
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








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
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.
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
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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
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

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
V. MCT equation for sheared
granular fluids
MCT approximation
Hard-core=> all terms are balanced under Bagnold’s scaling
Preliminary simulation
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
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We have checked the relevancy of
MCT equation for sheared dense
granular liquids.
MCT predicts the existence of a twostep 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 quasiarrested state
as MCT predicts.
Discussion of MCT equation

Can MCT describe the jamming
transition?

The answer of the current MCT is NO.
No yield stress

How can we determine S(q)?
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So far there is no theory to determine S(q),
but it does not depend on F(q,t).
Conclusion of MCT equation
for sheared granular fluids
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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








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
VI. Spatial correlations in
granular liquids
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

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
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
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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 direct correlation function
strain rate

The nonlocal viscous stress

The stress has the thermal fluctuation.
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.

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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








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
VII. The linearized GFH

The linearized GFH is given by
The random force
The linear equation can be solved analytically.
Matrices
The solution of linearized
equation (eigenvalue problem)
The solution of linearized GFH
Steady pair correlation for unsheared system.
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
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.
  0.093
  0.185
Long-range density correlation
function
  0.093
  0.185
However, the density correlation has a tail obeying a power law,
which is the result of the shear.
Long-range momentum
correlation
  0.093
  0.37
The momentum correlation has clear a power-law tail
obeying r^{-5/3}.
Discussion
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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
Small systems converges, but large systems do not converge.
Elastic systems have the same scalings.
(Left) The density correlation for e=1.
(Right) The time evolution of momentum correlation.
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)

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