Eli Ben-Naim Theory Division Los Alamos National Laboratory talk, papers available: http://cnls.lanl.gov/~ebn Nonequilibrium Statistical Physics of Driven Granular Gases with Jon Machta (Massachusetts) Igor Aronson (Argonne) Jeffrey Olafsen (Kansas) August 9, 2005Pattern Formation and Transport Phenomena, João Pessoa, Paraiba, Brazil 1. E. Ben-Naim and J Machta, Phys. Rev. Lett. 95, 068001 (2005) 2. K. Kohlstedt, A. Snezhkov, M.V. Sapozhnikov, I. S. Aranson, J. S. Olafsen, and E. Ben-Naim Phys. Rev. Lett. 94 138001 (2005)
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Eli Ben-NaimTheory Division
Los Alamos National Laboratory
talk, papers available: http://cnls.lanl.gov/~ebn
Nonequilibrium Statistical Physics of Driven Granular Gases
withJon Machta (Massachusetts)
Igor Aronson (Argonne)Jeffrey Olafsen (Kansas)
August 9, 2005Pattern Formation and Transport Phenomena, João Pessoa, Paraiba, Brazil
1. E. Ben-Naim and J Machta, Phys. Rev. Lett. 95, 068001 (2005)2. K. Kohlstedt, A. Snezhkov, M.V. Sapozhnikov, I. S. Aranson, J. S. Olafsen, and E. Ben-NaimPhys. Rev. Lett. 94 138001 (2005)
1. Introduction2. kinetic theory of granular gases 3. Free cooling states4. Driven Steady states I (forcing at large scales)5. Driven steady states II (forcing at all scales)
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
Energy dissipation in granular matterEnergy dissipation in granular matter
Responsible for collective phenomena» Clustering I Goldhirsch, G Zanetti 93
» Hydrodynamic instabilities E Khain, B Meerson 04
» Pattern formation P Umbanhower, H Swinney 96
Anomalous statistical mechanicsNo energy equipartition R Wildman, D Parker 02
Nonequilibrium energy distributions
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”)?
Dissipation rate always divergentEnergy finite or infinite
The characteristic exponent σ (d=2,3)The characteristic exponent σ (d=2,3)
σ varies with spatial dimension, collision rules
Monte Carlo Simulations: Driven Steady StatesMonte Carlo Simulations: Driven Steady States
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
Further confirmation: extremal statisticsFurther confirmation: extremal statistics
Energy is injected ONLY AT LARGE VELOCITY SCALES!Energy cascades from large velocities to small velocitiesEnergy dissipated at small velocity scales
Experimental realization?
Energetic particle “shot” into static
medium
Energy balance
Extreme statisticsExtreme statistics
Scaling function
Large velocities: as in free cooling
Small velocities: non-analytic behavior
Hybrid between steady-state and time dependent state
Maxwell Model (λ=0) only unsolved case!
Time dependent solutions (1D, λ>0)Time dependent solutions (1D, λ>0)
Self-similar distribution
Cutoff velocity decays
Scaling function
Hybrid between steady-state and time dependent state
with Ben Machta (Brown)
A third family of solutions exists
Numerical confirmationNumerical confirmation
Velocity distribution Scaling function
Energy balanceEnergy balance
Energy injection rateEnergy injection scaleTypical velocity scaleBalance between energy injection and dissipation
For “lottery” injection: injection scale diverges with injection rate
Energy injection selects stationary solution
Summary: solutions of kinetic theorySummary: solutions of kinetic theory
Time dependent solution
Time independent solution
Hybrid solution
Are there other types of solutions?
Conclusions IConclusions I
New class of nonequilibrium steady 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
A Mechanically vibrated beadsF Rouyer & N Menon 00
B Electrostatically driven powdersI Aronson, J Olafsen, EB
Gaussian coreOverpopulated tail
Kurtosis Excellent agreement between theory and experiment
balance between collisional dissipation,
energy injection from walls
Conclusions IIConclusions II
Conventional nonequilibrium steady statesEnergy cascades from large to small velocitiesEnergy input at ALL scales balances dissipationStretched exponential tailsLow order moments (temperature, kurtosis) useful Excellent agreement between experiments and kinetic theory