A kinetic equation for unstable plasmas in a finite space-time domain

A generalization of the Lenard–Balescu equation is derived that includes wave-particle scattering by collective instabilities in finite space-time domain plasmas. It is shown that wave-particle interactions can dominate conventional particle-particle Coulomb scattering when finite times are considered or convective instabilities are present in a medium of finite spatial extent before the instabilities grow to nonlinear or turbulent amplitudes. The modified Lenard–Balescu operator retains important physical properties including particle, momentum, and energy conservation laws and the Boltzmann $H$-theorem. To demonstrate its utility, the theory is applied to the simple example of convectively growing ion-acoustic instabilities in a finite spatial domain.