Stationary states and energy cascades in inelastic gases

Transcription

1 Los Alamos National Laboratory From the SelectedWorks of Eli Ben-Naim May, 005 Stationary states and energy cascades in inelastic gases E. Ben-Naim, Los Alamos National Laboratory J. Machta, Uniersity of Massachusetts Aailable at:

2 Stationary states and energy cascades in inelastic gases E. Ben-Naim 1, and J. Machta, 1 Theoretical Diision and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico Department of Physics, Uniersity of Massachusetts, Amherst, Massachusetts We find a general class of nontriial stationary states in inelastic gases where, due to dissipation, energy is transfered from large elocity scales to small elocity scales. These steady-states exist for arbitrary collision rules and arbitrary dimension. Their signature is a stationary elocity distribution f() with an algebraic high-energy tail, f() σ. The exponent σ is obtained analytically and it aries continuously with the spatial dimension, the homogeneity index characterizing the collision rate, and the restitution coefficient. We obsere these stationary states in numerical s in which energy is injected into the system by infrequently boosting particles to high elocities. We propose that these states may be realized experimentally in drien granular systems. PACS numbers: Mg, Nd, a, Rm Energy dissipation has profound consequences in granular media. It is responsible for collectie phenomena such as hydrodynamic instabilities [1, ], shocks [3 5], and clustering [6 8]. The inelastic nature of the collisions remains crucially important in dilute settings and under igorous forcing where, in contrast with molecular gases, there is no energy equipartition [9, 10] and the elocity distributions are typically non-maxwellian [11 16]. In this letter, we show that energy dissipation also results in nonequilibrium stationary states where energy cascades from large elocity scales to small elocity scales, and we propose that these states may be realized experimentally in drien granular gases. Hydrodynamic of granular flows is formulated using inelastic gases as a starting point [17 1]. Without forcing, dissipation is quantified ia the energy balance equation, dt/dt = Γ, where T = is the granular temperature and Γ the dissipation rate. Collisions inoling a pair of particles with relatie elocity occur with rate ( ) λ and result in energy loss ( ). Assuming that a single elocity scale characterizes the system, Γ ( ) +λ T 1+λ/, leads to Haff s cooling law, T t /λ (exponential decay occurs when λ = 0). Hence, the temperature decays indefinitely and the elocity distribution f() approaches the triial steady-state f() δ() where all particles are at rest, a stationary state that can be considered as a dynamical fixed point. Howeer, the energy balance equation assumes a finite dissipation rate. This need not be the case! Our main result is that there is a general class of nontriial stationary states where the elocity distribution has an algebraic high-energy tail f() σ (1) as. First, we obtain this result in one-dimension and then, generalize it to arbitrary dimension. Our system is a spatially homogeneous ensemble of identical particles that undergo inelastic collisions 1, = pu 1, + qu,1 () with u 1, ( 1, ) the pre-collision (post-collision) elocities. The collision parameters p and q obey p + q = 1 so momentum is consered. In each collision, the relatie elocity is reduced by the restitution coefficient r = 1 p and the energy loss equals pq(u 1 u ). The energy dissipation is maximal for completely inelastic collisions (r = 0, p = 1/), and it anishes for elastic collisions (r = 1, p = 0). We consider general collision rates u 1 u λ with 0 λ 1 the homogeneity index. Hard-spheres, λ = 1, model ordinary granular media. Maxwell molecules, where the collision rate is elocityindependent [], λ = 0, model granular particles with dipole interactions, such as magnetic particles [3]. We seek stationary elocity distributions f() that obey the Boltzmann equation, with the space and time deriaties set to zero, du 1 du u 1 u λ f(u 1 )f(u ) (3) [δ( pu 1 qu ) δ( u 1 )] = 0 reflecting balance of gain and loss due to collisions. The case λ = 0 is instructie as the elocity distribution can be obtained explicitly. The Boltzmann equation is a simple conolution because the collision rate is constant, and it is coneniently studied using the Fourier transform F (k) = d f() e ik that satisfies the nonlinear, nonlocal equation F (k) = F (pk)f (qk). Normalization implies F (0) = 1. For elastic collisions (p = 0) any distribution is stationary, but this is just a one-dimensional pathology. Generally, for any alue of the restitution coefficient, there is a family of solutions F (k) = exp ( 0 k ) with arbitrary typical elocity 0. As a result, the elocity distribution is Lorentzian f() = 1 π (/ 0 ). (4) We note that this distribution is independent of the restitution coefficient. The typical elocity 0 is finite, but the aerage energy is infinite.

3 This distribution does not eole under the inelastic collision dynamics. From the energy balance equation, one might hae expected that the elocity distribution constantly narrows because particles dissipate energy. Surprisingly, there is perfect balance between collisional loss and gain at eery elocity so that a steady elocity distribution is maintained. We conclude that in addition to the triial state where all particles are at rest, there is another fixed point, a nonequilibrium stationary state. For general collision rules, the characteristic exponent σ can be found analytically. For large, the conolution in Eq. (3) is goerned by the product f(u 1 )f(u ) with one of the pre-collision elocities large and the other small because the distribution decays sharply at large elocities. The Boltzmann equation includes a loss term and a gain term. For the loss term, u 1 = is large and u is small. For the gain term, there are two separate contributions. Either = pu 1 and then u is small or = qu and then u 1 is small. In either case, the double integral separates. The integral oer the smaller elocity equals one. With the remaining integral oer the larger elocity, denoted by u, the nonlinear Boltzmann equation becomes linear for large du u λ f(u) [δ( pu)+δ( qu) δ( u)] = 0. (5) Consequently, the tail of the distribution satisfies the functional equation ( ) 1 p 1+λ f + 1 ( ) p q 1+λ f f() = 0 (6) q describing cascade of energy from large elocities to small ones, (p, q). The power-law decay (1) satisfies this equation when p σ 1 λ + q σ 1 λ = 1, and since p + q = 1, the characteristic exponent is σ = + λ. (7) In one-dimension, σ is independent of the restitution coefficient. We comment that the full nonlinear Boltzmann equation assumes molecular chaos as the two-particle elocity distribution is a product of one-particle distributions. Howeer, the equation goerning the tail is linear and it is alid under less stringent conditions. The only requirement is that energetic particles are weakly correlated with slower ones. In arbitrary dimension d, the collision law is 1 = u 1 (1 p)(u 1 u ) ˆn ˆn. (8) Momentum transfer occurs only along the normal direction with ˆn a unit ector parallel to the impact direction connecting the particles. The energy dissipation equals p(1 p) (u 1 u ) ˆn and the collision rate is (u 1 u ) ˆn λ. We derie the linear equation goerning the tail of the distribution directly from the collision rule. With the large elocity u, its magnitude u u and µ = (û ˆn), the collision rate is u λ µ λ/. The cascade process is (α, β) with the stretching parameters α = (1 p)µ 1/ and β = [1 (1 p )µ] 1/ for = (1 p)u ˆn ˆn and = u (1 p)u ˆn ˆn, respectiely [4]. Integrating oer the impact angle ˆn and the elocity magnitude u, Eq. (5) generalizes as follows du u λ+d 1 f(u)[δ( αu)+δ( βu) f()]µ =0. λ/ Here, the brackets are used as shorthand for the angular integration, g = dˆn g(ˆn). Integrating oer u yields [ 1 ( ) α d+λ f + 1 ( ) α β d+λ f f() ]µ λ/ = 0. (9) β The power-law decay (1) satisfies this equation proided that the exponent σ is root of the equation (α σ d λ + β σ d λ 1 ) µ λ/ = 0. (10) The angular integration is performed using spherical coordinates. Gien µ = cos θ, then dˆn (sin θ) d dθ, and g 1 0 dµ µ 1 (1 µ) d 3 g(µ). The equality for the exponent σ can be written in terms of the gamma function and the hypergeometric function [5] ( 1 F d+λ σ 1, λ+1, d+λ, 1 p) σ d+1 Γ( )Γ( d+λ (1 p) σ d λ = ) Γ( σ λ+1 )Γ( ). (11) In the cascade process (α, β), the total elocity magnitude increases (α + β 1) een though the total energy decreases (α + β 1). From these inequalities, the left-hand-side of (10) is positie when σ d λ = 1 but negatie when σ d λ =. This gies the bounds 1 σ d λ. The lower bound (7) is realized in one-dimension where collisions are head-on (µ = 1). The upper bound is approached, σ d + + λ, in the quasielastic limit, r 1. Clearly, dissipation is responsible for these steady states. For elastic collisions, the elocity distributions are Maxwellian, so the no-dissipation limit is singular. Interestingly, the energy may be either finite or infinite, depending on whether σ is larger or smaller than d + [6]. In either case, the integral underlying the dissipation rate is diergent, a general characteristic of the stationary distributions. Generally, the exponent σ increases monotonically with the spatial dimension d, the homogeneity index λ, and the restitution coefficient r. For completely inelastic (r = 0) hard spheres, we find σ = and σ = for d =, 3, respectiely. The exponents for completely inelastic Maxwell molecules are σ = and σ = for d =, 3. These alues proide lower bounds on σ with respect to r, as shown in Fig. 1.

4 3 σ λ=0 λ= r FIG. 1: The exponent σ as a function of the restitution coefficient r for λ = 0 (circles) and λ = 1 (squares). The lower cures correspond to d = and the upper to d = 3. f() FIG. : The elocity distribution f() ersus for Maxwell molecules in one-dimension. The ratio between the typical injection energy V and typical elocity 0 is V/ A remarkable feature is that the characteristic exponent changes continuously with the parameters d, λ, and r. Hence, the tails are not uniersal. The power-law decay stands in contrast with the stretched exponentials, f() exp( ν ), with ν = λ and ν = 1 + λ/, respectiely, for unforced and thermally forced inelastic gases [1 15]. Preiously, algebraic tails with different exponents were found but for non-stationary distributions describing freely cooling Maxwell molecules [7 9]. These steady-state distributions can be realized, up to some cutoff, in finite systems of drien inelastic particles. The key is to inject energy at a ery large elocity scale. We used a Discrete Simulation Monte Carlo method in which pairs of particles are chosen at random with rate proportional to the collision rate and collided according to the collision rule (8). Initially, we start with an in- f() f() (a) (b) FIG. 3: The elocity distribution f() ersus for Maxwell molecules (a) and hard spheres (b) in one-dimension (upper lines) and two-dimension (lower lines). The is shown for reference and V/ , 10 for (a), (b) respectiely. nocuous elocity distribution, uniform with support in [ 1 : 1]. An energy loss counter keeps track of the aggregate energy loss. With rate γ, small compared with the collision rate, a particle is selected at random and it is energized by an amount equal to the aggregate energy loss. Subsequently, the energy loss counter is reset to zero. In this, the total energy remains practically constant and more importantly, energy is injected only at the tail of the distribution. In effect, this procedure does not alter the purely dissipatie dynamics, except for setting a scale for the most energetic particles. We simulated completely inelastic hard-spheres and Maxwell molecules in one- and two-dimensions. We used N = 10 7 (N = 10 5 ) particles and injection rate γ = 10 4 (γ = 10 ) for λ = 0 (λ = 1). We erified that: (i) after a transient, the elocity distribution becomes stationary, (ii) the elocity distribution is Lorentzian for one-dimensional Maxwell molecules (Fig. ), (iii) the elocity distribution has an algebraic tail, and (i) σ is consistent with (Fig. 3). The ery same stationary distributions can be reached

5 4 using other methods. For example, particles may be re-energized such that their elocity is drawn from a Maxwellian distribution with a typical energy proportional to the system size (the data presented for onedimensional hard-spheres are from such a ). We conclude that the s confirm the existence of the nontriial stationary states with power-law tails. These steady-states are stable fixed points as the system is drien into them een when starting from compact distributions. Moreoer, stability analysis of the time dependent ersion of Eq. (9) shows that the stationary distribution (1) is stable with respect to perturbations consisting of steeper algebraic tails. Clearly, if f() is a steady-state so is 0 d f(/ 0) for an arbitrary typical elocity 0. The injection protocol selects 0. Suppose that particles are boosted at rate γ per particle to elocity V, a scale that sets an upper cutoff on the elocity distribution [30]. The energy injection rate is γv, and the energy dissipation rate is Γ +λ with g V d d 1 f()g(). Energy balance, Γ γv, relates the injection rate, the injection scale, and the typical elocity scale, γ V λ (V/ 0 ) d σ. (1) In our s, energy is maintained constant, 1. When σ < d +, the constant energy constraint implies V d+ σ 0 d σ, that combined with energy balance (1) reeals how the maximal elocity, d+ σ (σ d)( λ), V γ 1/( λ), and the typical elocity, 0 γ scale with the injection rate. When σ > d +, the initial conditions set the typical elocity: 0 implies 0 1, and energy balance (1) yields V γ 1/(σ d λ). The s are consistent with these scaling laws. For example, V 10 and 0 10 for one-dimensional Maxwell-molecule s with γ = This, combined with the s, demonstrates that finite energy and infinite energy cases differ quantitatiely but not qualitatiely. In conclusion, we find that the Boltzmann equation for homogeneous inelastic gases has nontriial stationary solutions where the elocity distribution has an algebraic high-energy tail corresponding to an energy cascade from high to low elocities. These stationary states exist for arbitrary collision rates, arbitrary collision rules, and arbitrary spatial dimensions. Nonequilibrium steady states with power law tails oer a finite range can be maintained by injecting energy at a large elocity scale. We hae obsered these steady states in numerical s and propose that they may be realized in granular gas experiments in which energy is added at large elocity scales ia rare but powerful energy injections. We thank A. Baldassarri, N. Menon, and H. A. Rose for useful discussions. We acknowledge DOE W-7405-ENG- 36 and NSF DMR-0440 for support of this work. Electronic address: ebn@lanl.go Electronic address: machta@physics.umass.edu [1] I. Goldhirsch, and G. Zanetti, Phys. Re. 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