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We investigate the impact of weak collisions on Landau damping in theVlasov-Poisson-Fokker-Planck system on a torus, specifically focusing on itsproximity to a Maxwellian distribution. In the case where the Gevrey indexsatisfies $\frac{1}{s}<3$, we establish the global stability and enhanceddissipation of small initial data, which remain unaffected by the smalldiffusion coefficient $\nu$. For Gevrey index $\frac{1}{s}\ge3$, we prove theglobal stability and enhanced dissipation of initial data, whose size is on theorder of $O(\nu^a)$ for any $a>\frac{1-3s}{3-3s}$. Our analysis providesinsights into the effects of phase mixing, enhanced dissipation, and plasmaechoes.

Weak collision effect on nonlinear Landau damping for the Vlasov-Poisson-Fokker-Planck system