energy cascades in granular matterebn/talks/stationary.pdf · inelastic gasinelastic gas vigorous...
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Eli Ben-NaimTheory Division
Los Alamos National Laboratory
with: Jon Machta (Massachusetts)
Energy Cascades in Granular Matter
cond-mat/0411743http://cnls.lanl.gov/~ebn
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Energy dissipation in granular mediaEnergy dissipation in granular media
Responsible for collective phenomena
» Clustering
» Hydrodynamic instabilities
» Shocks
» Pattern formation
Anomalous statistical mechanics:
No energy equipartition
Nonequilibrium distributions
)/exp()( kTEEP −≠
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Inelastic gasInelastic gas
Vigorous drivingSpatially uniform systemParticles undergo binary collisionsVelocity changes due to
1. Inelastic collisions (lose energy)2. Energy input (gain energy)
What is the typical velocity (granular “temperature”)?
What is the velocity distribution?
2vT =
)v(f
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Nonequilibrium velocity distributionsNonequilibrium velocity distributions
( ) 2/31vexp~)v( ≤≤− δδf
Mechanically vibrated beadsRouyer & Menon 2000
Electrostatically driven powdersAronson & Olafsen 2002
55.333117vv
224 ≅=
Theory: ebn & krapivsky 2002
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Inelastic CollisionsInelastic Collisions
Relative velocity reduced by 0<r<1
Momentum is conserved
Energy is dissipated
Limiting cases
2121 vv uu +=+
2v))(1( ∆−−∝∆ rE
⎩⎨⎧
=∆=∆
=)0( elastic1
max)E(inelasticcompletely0r
E
)vv 2121 ur(u −−=−
1u 2u
1v 2v
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Freely decaying statesFreely decaying states
Energy loss in a collisionCollision rateEnergy balance equation
Temperature decays, system comes to rest
2vT =∆( )λv/1~ ∆∆t
( ) 2/12v~ ~ λλ ++∆−
∆∆ −⇒ T
dtdT
tT
)v()v(~ /2 δλ →⇒− PtT
Trivial steady-stateHaff, JFM 1982
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Kinetic TheoryKinetic Theory
Collision rule (linear)
Boltzmann equation
Collision rate related to interaction potential
pr)pu,ququ(pu),u(u 21212121 −=++→
[ ])v()v()()()v(221212121 uqupuuuuPuPdudu
tP −−−−−=∂
∂∫∫ δδλ
collision rate gain loss
γλγ 121~)( −
−=− drrU⎩⎨⎧
∞==
=)( spheres Hard1
2D)2,(moleculesMaxwell0α
α
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Kinetic TheoryKinetic Theory
Collision rule (linear)
Boltzmann equation
Collision rate related to interaction potential
pr)pu,ququ(pu),u(u 21212121 −=++→
[ ])v()v()()()v(221212121 uqupuuuuPuPdudu
tP −−−−−=∂
∂∫∫ δδλ
collision rate gain loss
γλγ 121~)( −
−=− drrU⎩⎨⎧
∞==
=)( spheres Hard1
2D)2,(moleculesMaxwell0α
α
Are there nontrivial steady states?
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An exact solutionAn exact solution
One-dimensional Maxwell moleculesFourier transform obeys a closed equation
Exponential solution
Lorentzian velocity distribution
)f(ikedF(k)pkkFpkFkF vvv)()()( ∫=−=
( )0vexp)( kkF −=
( )200 v/v11
v1)v(
+=π
f
Nontrivial steady states do exist
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Properties of stationary stateProperties of stationary state
Perfect balance between collisional loss and gainPower-law high-energy tail
Infinite energy, infinite dissipation!
2v~)v( =− σσP
Is this stationary state physical?
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Cascade Dynamics (1D)Cascade Dynamics (1D)
Collision rule: arbitrary velocities
Large velocities cascade
High-energies: linearized equation
Power-law tail
)v,v(v qp→
)qu,puqu(pu),u(u 122121 ++→
0)v(v1v111 =−⎟
⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛++ f
qf
qpf
p λλ
λσσ +=− 2v~)v(f
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Cascade DynamicsCascade Dynamics
Collision process: large velocities
Stretching parameters related to impact angle
Energy decreases, velocity magnitude increases
Steady state equation
v)v,(v βα→
2/122 ]cos)1(1[cosp)-(1 θβθα p−−==
0cos)v(v1v1=⎟
⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛
++ θββαα
λλλ fff dd
1122 ≥+≤+ βαβα
θ
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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
σ-v~)v(f
( ) ⎟⎠⎞
⎜⎝⎛ +Γ⎟
⎠⎞
⎜⎝⎛Γ
⎟⎠⎞
⎜⎝⎛ +Γ⎟
⎠⎞
⎜⎝⎛ +−Γ
=−
⎟⎠⎞
⎜⎝⎛ −++−+−
−−
21
2
221
1
1,2
,2
1,2
1 221
λσ
λσλλσλ
λσ
dd
p
pddFd
2--1 ≤≤ λσ dλσ ++→ 2d
Dissipation always divergentEnergy finite or infinite
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The Characteristic ExponentThe Characteristic Exponent
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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)
N=107 N=105
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Injection, cascade, dissipationInjection, cascade, dissipation
ln f
ln vEnergy is injected at large velocity scalesEnergy cascades from large velocities to small velocitiesEnergy dissipated at small velocity scales
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ConclusionsConclusions
New class of nonequilibrium stationary statesEnergy cascades from large velocities to small velocitiesPower-law high-energy tailEnergy input at large scales balances dissipationTemperature insufficient to characterize velocities Experimental realization: requires a different driving mechanism