theory division los alamos national laboratorycnls.lanl.gov/~ebn/talks/kitp.pdfeli ben-naim theory...
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Eli Ben-NaimTheory Division
Los Alamos National Laboratory
talk, papers available: http://cnls.lanl.gov/˜ebn
Kinetic Theory of Granular Gases
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PlanPlan
1. Basics: collision rules, collision rates2. The Boltzmann equation3. Extreme statistics and the linearized
Boltzmann equation4. Forced steady states5. Freely cooling states6. Stationary states and energy cascades7. Hybrid solutions
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ExperimentsExperiments
Friction D Blair, A Kudrolli 01
RotationK Feitosa, N Menon 04
Driving strengthW Losert, J Gollub 98
DimensionalityJ Urbach & Olafsen 98
BoundaryJ van Zon, H Swinney 04
Fluid dragK Kohlstedt, I Aronson, EB 05
Long range interactionsD Blair, A Kudrolli 01; W Losert 02 K Kohlstedt, J Olafsen, EB 05
SubstrateG Baxter, J Olafsen 04
Deviations from equilibrium distribution
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Driven Granular gasDriven Granular gas
Vigorous drivingSpatially uniform systemParticles undergo binary collisionsVelocities change due to
1. Collisions: lose energy2. Forcing: gain energy
What is the typical velocity (granular “temperature”)?
What is the velocity distribution?
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Inelastic Collisions (1D)Inelastic Collisions (1D)
Relative velocity reduced by
Momentum is conserved
Energy is dissipated
Limiting cases
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Inelastic Collisions (any D)Inelastic Collisions (any D)
Normal relative velocity reduced by
Momentum conservation
Energy loss
Limiting cases
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Non-Maxwellian velocity distributionsNon-Maxwellian velocity distributions
1. Velocity distribution is isotropic
2. No correlations between velocity components
Only possibility is Maxwellian
Granular gases: collisions create correlations
JC Maxwell, Phil Trans Roy. Soc. 157 49 (1867)
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The Boltzmann equation (1D)The Boltzmann equation (1D)
Collision rule (linear)
Boltzmann equation (nonlinear and nonlocal)
Collision rate related to interaction potentialcollision rate gain loss
Theory: non-linear, non-local, dissipative
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The collision rateThe collision rate
Collision rate
Collision rate related to interaction potential
Balance kinetic and potential energy
Collisional cross-section
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Extreme Statistics ( 1D)Extreme Statistics ( 1D)
Collision rule: arbitrary velocities
Large velocities: linear but nonlocal process
High-energies: linear equation
gain lossgain
Linear, nonlocal evolution equation
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Extreme Statistics (any D)Extreme Statistics (any D)
Collision process: large velocities
Stretching parameters related to impact angle
Energy decreases, velocity magnitude increases
Linear equation
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Forced steady states: overpopulated tailsForced steady states: overpopulated tails
Energy injection: thermal forcing (at all scales)
Energy dissipation: inelastic collision
Steady state equation
Stretched exponentials
T van Noije, M Ernst 97
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Nonequilibrium velocity distributionsNonequilibrium velocity distributions
A Mechanically vibrated beadsF Rouyer & N Menon 00
B Electrostatically driven powdersI Aronson & J Olafsen 05
Gaussian coreOverpopulated tail
Kurtosis Excellent agreement between theory and experiment
balance between collisional dissipation,
energy injection from walls
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Comparing kinetic theoriesComparing kinetic theories
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Freely cooling states: temperature decayFreely cooling states: temperature decay
Energy lossCollision rateEnergy balance equation
Temperature decays, system comes to rest
System comes to rest
Haff, JFM 1982
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Temperature decay: dimensional analysisTemperature decay: dimensional analysis
Collision rate
Collision rate inversely proportional to time
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Freely cooling states: similarity solutionsFreely cooling states: similarity solutions
Linearized equation
Similarity solution
Steady state equation
Stretched exponentials (overpopulation)
Esipov, Poeschel 97
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An exact solutionAn exact solution
One-dimensional Maxwell moleculesFourier transform obeys a closed equation
Exponential solution
Lorentzian velocity distribution
Nontrivial stationary states do exist!
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Are there nontrivial steady states?Are there nontrivial steady states?
Stationary Boltzmann equation
Naive answer: NO!According to the energy balance equation
Dissipation rate is positive
collision rate gain loss
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Cascade Dynamics (1D)Cascade Dynamics (1D)
Collision rule: arbitrary velocities
Large velocities: linear but nonlocal process
High-energies: linear equation
Power-law tailgainloss gain
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Cascade Dynamics (any D)Cascade Dynamics (any D)
Collision process: large velocities
Stretching parameters related to impact angle
Energy decreases, velocity magnitude increases
Steady state equation
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Power-laws are genericPower-laws are generic
Velocity distributions always has power-law tail
Exponent varies with parameters
Tight boundsElastic limit is singular
Dissipation rate always divergentEnergy finite or infinite
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The characteristic exponent σThe characteristic exponent σ
σ varies with spatial dimension, collision rules
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Monte Carlo SimulationsMonte Carlo Simulations
Compact initial distributionInject energy at very large velocity scales onlyMaintain constant total energy“Lottery” implementation: – Keep track of total energy
dissipated, ET
– With small rate, boost a particle by ET
Excellent agreement between theory and simulation
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Further confirmationFurther confirmation
Maxwell molecules (1D, 2D) Hard spheres (1D, 2D)
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Injection, cascade, dissipationInjection, cascade, dissipation
Energy is injected at large velocity scalesEnergy cascades from large velocities to small velocitiesEnergy dissipated at small velocity scales
Experimental realization?
Energetic particle “shot” into static
medium
Energy balance
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Energy balanceEnergy balance
Energy injection rateEnergy injection scaleTypical velocity scaleBalance between energy injection and dissipation
For “lottery” injection: injection scale diverges with injection rate
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Self-similar collapseSelf-similar collapse
Self-similar distribution
Cutoff velocity decays
Scaling function
Hybrid between steady-state and time dependent state
with Ben Machta (Brown)
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A third family of solutions exists
Numerical confirmationNumerical confirmation
Velocity distribution Scaling function
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ConclusionsConclusions
New class of nonequilibrium stationary statesEnergy cascades from large to small velocitiesPower-law high-energy tailEnergy input at large scales balances dissipationAssociated similarity solutions exist as wellTemperature insufficient to characterize velocities Experimental realization: requires a different driving mechanism
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OutlookOutlook
Spatially extended systemsSpatial structuresPolydisperses granular mediaExperimental realization
E. Ben-Naim and J. Machta, Phys. Rev. Lett. 94, 138001 (2005)
E. Ben-Naim, B. Machta, and J. Machta, cond-mat/0504187
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Deviation from Maxwell-BoltzmannDeviation from Maxwell-Boltzmann
Kurtosis
Restitution coefficient
Exact solution of Maxwell’s kinetic theory: thermal forcing balances dissipationEB, Krapivsky 02
1. Velocity distribution independent of driving strength2. Stronger dissipation yields stronger deviation