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Velocity Tails for Inelastic Maxwell Models

We study the velocity distribution function for inelastic Maxwell models, characterized by a Boltzmann equation with constant collision rate, independent of the energy of the colliding particles. By means of a nonlinear analysis of the Boltzmann equation, we find that the velocity distribution function decays algebraically for large velocities, with exponents that are analytically calculated.


I. INTRODUCTION
Velocity distributions have over-populated high energy tails in many particle systems with in-elastic interactions, as has been discovered theoretically, and later observed in laboratory experiments with granular materials [1]. Instead of Gaussian tail distributions, many kinetic equations typically predict stretched exponentials, like exp[−A|v| 3/2 ] in driven inelastic hard sphere systems (IHS) [2], or even tails with a higher overpopulation, like exp[−A|v|] in the freely evolving IHS fluids [2,3]. These theoretical predictions have been extensively verified in Direct Monte Carlo Simulations (DSMC) of the Boltzmann equation [4,5], and the stretched exponentials have been observed in laboratory experiments with granular matter on vibrating plates [1]. However, the presence and also the absence of over-populated tails depends very strongly on the energy input or on how the system is thermo-statted [5,6].
Recently classes of simplified kinetic models have been studied, so-called Maxwell models, which are characterized by a Boltzmann equation with a collision rate that is independent of the relative kinetic energy of the colliding particles. Maxwell molecules are for kinetic theory, what harmonic oscillators are for quantum mechanics, and dumb-bells for polymer physics.
Ben-Naim and Krapivsky [7] have presented arguments about the non-existence of scaling solutions of the Boltzmann equation for the one-dimensional Maxwell model, and argue in favor of multi-time scales [8]. However, Puglisi et al. [9,10] have found an exact scaling solution f (v, t) ∼ (1/v d 0 (t))f (v/v 0 (t)) for that equation, i.e.f (c) = (2/π)(1 + c 2 ) 2 . It does have a power law tail (1/c 4 ). Puglisi et al. also solve the spatially homogeneous Maxwell-Boltzmann equation using MC simulations in one-and two-dimensions, and, more importantly, they show that an arbitrary initial distribution approaches this scaling solution, with power law tails at high energies. In two-dimensions the exponent a of the power law tails depends on the degree of in-elasticity, i.e. on the coefficient of restitution α. The goal of this article is to derive these power laws from the dominant small-k singularity in the Fourier transform of the velocity distribution function.

II. DOMINANT SMALL k SINGULARITY
In order to analyze the large v-behavior of the distribution function, it is convenient to use the Fourier transform φ(k, t) of f (v, t). It is the generating function of the moments v n t . If f (v, t) has a tail ∼ 1/|v| a+d , then the moments with n > a are divergent, and so is the n-th derivative of the generating function at k = 0, i.e. φ(k, t) is singular at k = 0. Suppose the dominant small-k singularity of φ(k, t) is ∼ |k| a , where a is different from an even integer (even powers of k represent contributions that are regular at small k), then the inverse Fourier transform scales as 1/|v| a+d at large v.
By applying Bobylev's Fourier transform method [11,12], we obtain a nonlinear equation for φ(k, t), and we determine its dominant small-k singularity. In doing so we have derived a transcendental equation for the exponent a in the power law tail ∼ 1/|c| a+d of the scaling solutionsf (v/v 0 ) of the Boltzmann equation for Maxwell molecules with in-elastic hard sphere interactions in arbitrary dimensions.
The Boltzmann equation for the d-dimensional in-elastic Maxwell model reads, Here n (· · ·) = (1/Ω d ) dn(· · ·) is an average over a d-dimensional solid angle where Ω d = 2π d/2 /Γ( 1 2 d). The velocities v * * i with i, j = {1, 2|} denote the d-dimensional restituting velocities, and v * i the corresponding post-collision velocities. They are defined as, with v ij = v i − v j , and n is a unit vector along the line of centers of the interacting particles. In one-dimension, the tensorial product nn can be replaced by 1. From the normalization of f it follows that the loss term reduces to −f (v 1 , t), i.e. the collision frequency is unity, and the dimension-less time t counts the average number of collisions per particle. We first illustrate the method for the one-dimensional case. Fourier transformation of the Boltzmann equation yield then, where we have used that φ(0, t) = 1 and p = 1 2 (1 + α). The equation for the scaling solution, φ(k, t) = Φ(v 0 (t)k) simplifies to, where the exponent γ in v 0 (t) = v 0 (0) exp[−γt] is still to be determined. The requirement that the total energy be finite, imposes the the lower bound a > 2 on the exponent. We therefore make the ansatz that the dominant small-k singularity has the form, Inserting this in (4), and equating the coefficient of equal powers of k yields the equation, The smallest root of this equation, satisfying a > 2, is a = 3, and A is left undetermined. Consequently the scaling solution has a power law tail,f (c) ∼ 1/c 4 . The same method can be applied to the d-dimensional case. Application of Bobylev's Fourier transform method to the Boltzmann equation for this case [13,11] yields the transformed equation, where the n-average is defined below (1), and where q = p(2 − p) is a positive number. We proceed in the same way as in the one-dimensional case, and obtain the equation for the scaling solution, Inserting the ansatz (5) into (8), and equating the coefficients of equal powers of k s yields, for s = 2, a. In order to carry out the angularĉ-average in |q · c| a with q = {k, k + , k − } we choose q as polar axis, and denote q · c = q cq ·ĉ = q c cos θ, then |q · c| a = |q| a |c| a K (d) is the average of | cos θ| a over a d-dimensional solid angle, which equals where q = p(2 − p). Finally we carry out the angular n-averages using (8), and obtain, Insertion of these results in (10) for a=2, yields For the exponent a, featuring in the power law tail of the scaling functionf (c) ∼ 1/c a+d , we obtain the transcendental equation, The two most interesting cases are d = 2, 3, where and one can verify that a = 2 is also a solution of (14). We look for the smallest solution a(α) of this transcendental equation with a > 2. The numerical solutions for d = 2, 3 are shown in Figure 1 as a function of α. If p = 1 2 (1 + α) ↑ 1 the root a(α) moves to ∞, as it should, which is consistent with a Maxwellian tail distribution for the elastic case. The solution diverges as α → 1 (elastic limit), becausef becomes Maxwellian in this limit. We note that a = 2 always satisfies Eq.(14), and the solution shown here is the one different from a = 2.
The simulations in Ref. [9] of the two-dimensional Maxwell -Boltzmann equation show for the exponent a(α = 0) + 2 ≃ 5,where our analytical method predicts a(α = 0) + 2 ≃ 6.2. In fact, closer inspection of their two-dimensional scaling plot shows that the slope of their log-log plot off (c) versus c increases at larger velocities, approaching the exact prediction of the Boltzmann equation. However, at these large velocities the statistical errors in their simulations are too large to make a quantitative comparison for larger α-values.