Fluctuation Relations of Dissipative Systems

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Fluctuation Relations of Dissipative Systems
Hisao Hayakawa (YITP, Kyoto University)
collaboration with Song-Ho Chong (IMS), Michio
Otsuki (Aoyama Gakuin Univ.)
YKIS 2009(2009/07/31)
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This is the last talk .

• I am sorry that I planned to talk on quantum
  systems, but I could not find time to fix it.
• My talk discusses classical systems but it may be
  useful to think of properties of nonequilibrium
  steady state, generalized Green-Kubo formula,
  Markovian approximation, long tails, and memory
  effects etc.

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Connection of each subject
Glass Transition
Granular Physics
Nonequilibrium Physics
Quantum Nonequilibrium Physics
Computational Physics
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Contents

• Introduction
• Part I Sheared granular systems
• Part II Fluctuation theorem and generalized
  Green-Kubo formula
• Part III On initial condition
• Part IV Liquid theory, Nonequilibrium MCT
• Summary

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Introduction
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What is granular material?

• Each grain is large and the heat can be absorbed
  in it.
• The grain is coupled with the heat bath at T0.
• No detailed balance

T0
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The characteristics of granular materials

• Granular materials are collection of dissipative
  macroscopic particles.
• To discuss NESS (nonequilibrium steady state), we
  need input of the energy and dissipation.
• We focus on statistical mechanics of sheared
  systems

H.Kuninaka and HH, PRE 2009
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Can we describe granular or jammed materials by
the theory?

• Yes, we can.
• Liquid theory can describe
• Generalized Green-Kubo formula
• Integral Fluctuation Theorem
• Markovian approximation
• Long-time tail and long-range correlation
• Nonequilibrium MCT
• Jamming transition for dense granular materials

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Part I

• Fluctuation relations and generalized Green-Kubo
  in sheared granular liquids

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Shear induced NESS

• A balance between viscous heating and dissipation
  produces NESS.

T. Hatano (2008)
No detailed balance and no equilibrium state
Rayleighs dissipation function
stress tensor
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Basic Equations (1)

• SLLOD equations

Heaviside function represents range of
interaction
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Basic Equations (2)

• Liouville equation
• Phase volume contraction
• Rayleighs dissipation function

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Transient time correlation function formalism
(Evans and Morriss, Statistical mechanics of
nonequilibrium liquids)
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Liouville operator
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Identities

• Identities
• Initial conditions

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Kawasaki representation Jarzynski equality

• Kawasaki representation
• where
• Thus

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Integral FT and 2nd law

• We can obtain integral FT

This represents a generalized 2nd law of
thermodynamics.
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Generalized Green-Kubo formula
t-gt8
If
where we use ltOgt0
If there are no energy dissipation, it reduces to
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We can apply this formulation to driven inelastic
Lorentz gas

• Balance between driven force and dissipation can
  lead to a steady state.

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Basic equations for Lorentz gas
Later OB can be replaced by O. Then, all
discussion of sheared granular systems can be
used.
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Part III

• On initial conditions

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Comments on the initial condition

• The expression seems to depend on the initial
  temperature and initial distribution.
• However, when we assume mixing properties, we can
  prove
• The result is independent of the initial
  temperature.
• The result is independent of the choice of the
  distribution.

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Independence of initial temperature for canonical
case
mixing
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It is unnecessary that the initial condition
satisfies canonical distribution.
Two-body spatial correlation function
Time evolution of temperature
g(r)
We can prove this statement mathematically.
r
Red From the canonical distribution at
T4 Green From a freely cooling granular state
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Irrelevancy of canonical initial condition
canonical distribution
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Part IV

• Correlation and Nonequilibrium MCT

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Formulation of liquid theory of granular materials

• Is the glass transition equivalent to jamming
  transition?
• Cage effect is not important for granular
  liquids.
• The plateau cannot be observed for granular
  liquids.

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Zwanzig-Mori equation

• Current-current correlation and current-density
  correlation are important as density-density
  correlation function in granular liquids, because
  the viscous force depends on the relative contact
  speed for granular materials.
• Identities can be derived based on projection
  operator formalism.
• Generalized Langevin equation

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Equation of continuity
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Projection operator formalism

• Projection onto the density and the current

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Contribution from viscous force (1)
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Contribution from viscous force (2)
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Memory kernels
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Generalized noise
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Time evolution of current
noise
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Nearly elastic and weak shear case
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Markovian approximation

• The set of equations for the density and the
  current is fluctuating hydrodynamics.
• Noise reduces to the random part of the stress
  tensor satisfying

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Long-range momentum correlation (PRE 79, 021502)
The momentum correlation has clear a power-law
tail obeying r-5/3.
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Long-time tails (J. STAT. MECH. in press)
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Brief comments on correlations

• There are long-range correlations and long-time
  tails for sheared granular liquids.
• These are essentially same as those for sheared
  isothermal systems.
• See e.g. M. Otsuki and HH, PRE 79, 021502 (2009)
  and J.STAT. MECH. (in press).

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Beyond Markovian approximation

• Memory kernels play important roles at least for
  glass transition.
• Thus, we need a simple closure form of
  correlation functions.
• Simplest one might be MCT which may correspond to
  RPA or Hatree-Fock approximation.
• Map to slow variables.

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Mode-coupling approximation
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Mode-coupling equations revisted

• It is possible to obtain a closed set of
  equations of correlation function under MCT
  approximation. I skip this because of its
  complicated form.
• In equilibrium MCT is a closed equation for
  density correlation which describes the ideal
  glass transition, but in sheared granular systems
  correlations including current are also
  important.
• Thus, plateau might disappear.

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Discussion

• To calculate Green-Kubo relation we should adopt
  some approximation such as MCT.
• However, Markovian approximation recovers
  long-range correlation and long time tail.
• Non-Markovian may describe jamming??
• Not yet confirmed.

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Summary

• We have formulated a general framework to
  describe sheared granular liquids and driven
  inelastic Lorentz gas.
• We obtain generalized Green-Kubo formula.
• We also obtained the integral Fluctuation Theorem.

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Thank you for your attention.
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Closing address
Hisao Hayakawa (YITP, Kyoto University)
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Schedule of Workshop

• Granular Physics (July 21-24)
• Physics of Glassy Materials (July 27-30)
• YKIS Symposium (July31-Aug.1)
• Nonequilibrium Statistical Mechanics (Aug.3-7)
• Computational Physics (Aug.10-14)
• Quantum Nonequilibrium Systems (Aug. 17-21)

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Achievement of this long-term workshop

• Intensive discussion among participants leads to
  mutual understanding.
• I hope that collaborative works are initiated
  from this workshop.
• Please acknowledge this workshop when the
  participants will write papers if this workshop
  helps the research.

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Acknowledgment etc

• Please return your keys.
• Please do not forget to submit the paper (invited
  speakers).
• I would like to thank Tomio to organize this week
  symposium.
• I want to express my special thanks to Prof.
  Sudarshan.
• Let us thank Ms. Yagi, Dr. Wada and students to
  support this workshop.

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See you someday!