Time-asymptotic behavior of the Vlasov–Poisson–Boltzmann system near vacuum

We present a collision potential for the Vlasov–Poisson–Boltzmann system near vacuum in plasma physics case. This potential measures the future possible collisions between charged particles with different velocities and satisfies a time-decay estimate. We use this time-decay property of the functional to show that the dynamics of the Vlasov–Poisson–Boltzmann is time-asymptotically equivalent to that of the corresponding linear Vlasov equation, when initial datum is small and decays fast enough in phase space as in [Y. Guo, The Vlasov–Poisson–Boltzmann system near vacuum, Comm. Math. Phys. 218 (2001) 293–313].