Eli Ben-NaimTheory Division

Los Alamos National Laboratory

with: Jon Machta (Massachusetts)

Energy Cascades in Granular Matter

cond-mat/0411743http://cnls.lanl.gov/~ebn

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 −≠

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

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

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

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

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

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?

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

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?

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

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

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

The Characteristic ExponentThe Characteristic Exponent

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

Further confirmationFurther confirmation

Maxwell molecules (1D, 2D) Hard spheres (1D, 2D)

N=107 N=105

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

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