liouville equation for granular gases
Post on 05-Jan-2016
35 Views
Preview:
DESCRIPTION
TRANSCRIPT
Liouville equation for granular gasLiouville equation for granular gaseses
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
particleshttp://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)
(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 , 1619-1622 (1993).
Simulation of a freely cooling gas
The restitution 0.99118Area fraction 0.25# of particles 640,000Initial: equilibriumTime 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 an
alysis 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?I. What is granular materials? II. Characteristics of sheared glassy or granul
ar systems III. Liouville equation for sheared granular gasIII. Liouville equation for sheared granular gas
eses IV. Generalized Langevin equationIV. Generalized Langevin equation V. MCT equation for sheared granular fluidsV. MCT equation for sheared granular fluids VI. Spatial correlations in granular liquidsVI. Spatial correlations in granular liquids VII. Linearized generalized fluctuating hydrodyVII. Linearized generalized fluctuating hydrody
namicsnamics VIII. Comparison between theory and simulatiVIII. Comparison between theory and simulati
onon
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
)/(
||
/1,
2
tp
md
tmdp
D
The change of momentum
Time scale
This is the relation between the temperature and the shear rate.
Shear stress
2|,| TT
MCT for sheared granular fluids MCT equation can be derived for granula
r fluids starting from Liouville equation. This approach ensures formal universalit
y 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 I. What is granular materials?I. What is granular materials? II. Characteristics of sheared glassy or granulII. Characteristics of sheared glassy or granul
ar systemsar systems III. Liouville equation for sheared granular gasIII. Liouville equation for sheared granular gas
eses IV. Generalized Langevin equationIV. Generalized Langevin equation V. MCT equation for sheared granular fluidsV. MCT equation for sheared granular fluids VI. Spatial correlations in granular liquidsVI. Spatial correlations in granular liquids VII. Linearized generalized fluctuating hydrodyVII. Linearized generalized fluctuating hydrody
namicsnamics VIII. Comparison between theory and simulatiVIII. Comparison between theory and simulati
onon
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?I. What is granular materials? II. Characteristics of sheared glassy or granulII. Characteristics of sheared glassy or granul
ar systemsar systems III. Liouville equation for sheared granular gasIII. Liouville equation for sheared granular gas
eses IV. Generalized Langevin equationIV. Generalized Langevin equation V. MCT equation for sheared granular fluidsV. MCT equation for sheared granular fluids VI. Spatial correlations in granular liquidsVI. Spatial correlations in granular liquids VII. Linearized generalized fluctuating hydrodyVII. Linearized generalized fluctuating hydrody
namicsnamics VIII. Comparison between theory and simulatiVIII. Comparison between theory and simulati
onon
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 betw
een 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 I. What is granular materials?I. What is granular materials? II. Characteristics of sheared glassy or granulII. Characteristics of sheared glassy or granul
ar systemsar systems III. Liouville equation for sheared granular gasIII. Liouville equation for sheared granular gas
eses IV. Generalized Langevin equationIV. Generalized Langevin equation V. MCT equation for sheared granular fluidsV. MCT equation for sheared granular fluids VI. Spatial correlations in granular liquidsVI. Spatial correlations in granular liquids VII. Linearized generalized fluctuating hydrodyVII. Linearized generalized fluctuating hydrody
namicsnamics VIII. Comparison between theory and simulatiVIII. Comparison between theory and simulati
onon
V. MCT equation for sheared granular fluids
MCT approximationHard-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 transitio
n? 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 I. What is granular materials?I. What is granular materials? II. Characteristics of sheared glassy or granulII. Characteristics of sheared glassy or granul
ar systemsar systems III. Liouville equation for sheared granular gasIII. Liouville equation for sheared granular gas
eses IV. Generalized Langevin equationIV. Generalized Langevin equation V. MCT equation for sheared granular fluidsV. MCT equation for sheared granular fluids VI. Spatial correlations in granular liquidsVI. Spatial correlations in granular liquids VII. Linearized generalized fluctuating hydrodyVII. Linearized generalized fluctuating hydrody
namicsnamics VIII. Comparison between theory and simulatiVIII. Comparison between theory and simulati
onon
VI. Spatial correlations in granular liquids The determination of the spatial correlations i
n granular liquids is important in MCT. It is known that there is a long-range velocity c
orrelation 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 (2
003)
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 s
heared 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 considerQuantities we consider
Generalized fluctuating Generalized fluctuating hydrodynamics (GFH)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 rateThe direct correlation function
Characteristics of GFHCharacteristics 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 I. What is granular materials?I. What is granular materials? II. Characteristics of sheared glassy or granulII. Characteristics of sheared glassy or granul
ar systemsar systems III. Liouville equation for sheared granular gasIII. Liouville equation for sheared granular gas
eses IV. Generalized Langevin equationIV. Generalized Langevin equation V. MCT equation for sheared granular fluidsV. MCT equation for sheared granular fluids VI. Spatial correlations in granular liquidsVI. Spatial correlations in granular liquids VII. Linearized generalized fluctuating hydrodyVII. Linearized generalized fluctuating hydrody
namicsnamics VIII. Comparison between theory and simulatiVIII. Comparison between theory and simulati
onon
VII. The linearized GFHVII. 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 GFHThe solution of linearized GFH
Steady pair correlation for unsheared system.
Contents I. What is granular materials?I. What is granular materials? II. Characteristics of sheared glassy or granulII. Characteristics of sheared glassy or granul
ar systemsar systems III. Liouville equation for sheared granular gasIII. Liouville equation for sheared granular gas
eses IV. Generalized Langevin equationIV. Generalized Langevin equation V. MCT equation for sheared granular fluidsV. MCT equation for sheared granular fluids VI. Spatial correlations in granular liquidsVI. Spatial correlations in granular liquids VII. Linearized generalized fluctuating hydrodyVII. Linearized generalized fluctuating hydrody
namicsnamics VIII. Comparison between theory and simulatiVIII. Comparison between theory and simulati
onon
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 Short-range density correlationcorrelation
The short-range density correlation can be approximated by Lutsko (2001). No fitting paramet
ers The contribution of th
e shear is very small for dense case.
185.0
093.0
Long-range density correlation function
However, the density correlation has a tail obeying a power law,
which is the result of the shear.
185.0093.0
Long-range momentum correlation
The momentum correlation has clear a power-law tail
obeying r^{-5/3}.
37.0093.0
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)
top related