## The Inelastic Maxwell Model

### E. Ben-Naim and P.L. Krapivsky

Dynamics of inelastic gases are studied within the framework of
random collision processes. The corresponding Boltzmann equation
with uniform collision rates is solved analytically for gases,
impurities, and mixtures. Generally, the energy dissipation leads
to a significant departure from the elastic case. Specifically,
the velocity distributions have overpopulated high energy tails
and different velocity components are correlated. In the freely
cooling case, the velocity distribution develops an algebraic
high-energy tail, with an exponent that depends sensitively on
the dimension and the degree of dissipation. Moments of the
velocity distribution exhibit multiscaling asymptotic behavior,
and the autocorrelation function decays algebraically with time.
In the forced case, the steady state velocity distribution decays
exponentially at large velocities. An impurity immersed in a
uniform inelastic gas may or may not mimic the behavior of the
background, and the departure from the background behavior is
characterized by a series of phase transitions.

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