## 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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