Asymptotic behavior of the two-species Vlasov-Poisson-Boltzmann system

This paper provides an estimation of the convergence rate to the steady state for the two-species Vlasov-Poisson-Boltzmann system. By leveraging methods used by Desvillettes and Villani for the Boltzmann equation and applying a quantitative H-theorem to the two-species solutions, we establish a series of second-order differential inequalities to estimate the duration of time intervals over which the relative entropy decreases to a constant multiple of itself, thereby achieve an almost exponential convergence rate of the solutions. This result is applicable to large initial data since the proof does not employ linearization around the stationary state. The main challenge of this work lies in handling interactions between particles and elucidating the relationship between the macroscopic quantities of different particles.