Scaling, Multiscaling, and Nontrivial Exponents in Inelastic Collision Processes,

E. Ben-Naim and P.L. Krapivsky

We investigate velocity statistics of homogeneous inelastic gases using the Boltzmann equation. Employing an approximate uniform collision rate, we obtain analytic results valid in arbitrary dimension. In the freely evolving case, the velocity distribution is characterized by an algebraic large velocity tail, $P(v,t)\sim v^{-\sigma}$. The exponent $\sigma(d,\epsilon)$, a nontrivial root of an integral equation, varies continuously with the spatial dimension, $d$, and the dissipation coefficient, $\epsilon$. Although the velocity distribution follows a scaling form, its moments exhibit multiscaling asymptotic behavior. Furthermore, the velocity autocorrelation function decays algebraically with time, $A(t)=\langle {\bf v}(0)\cdot{\bf v}(t)\rangle\sim t^{-\alpha}$, with a non-universal dissipation-dependent exponent $\alpha=1/\epsilon$. In the forced case, the steady state Fourier transform is obtained via a cumulant expansion. Even in this case, velocity correlations develop and the velocity distribution is non-Maxwellian.

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