  Statistical systems with time-periodic spatially non-uniform forces are of
immense importance in several areas of physics. In this paper, we provide an
analytical expression of the time-periodic probability distribution function of
particles in such a system by perturbatively solving the 1D Vlasov equation in
the limit of high frequency and slow spatial variation of the time-periodic
force. We find that the time-averaged distribution function and density cannot
be written simply in terms of an effective potential, also known as the
fictitious ponderomotive potential. We also find that the temperature of such
systems is spatially non-uniform leading to a non-equilibrium steady state
which can further lead to a complex statistical time evolution of the system.
Finally, we outline a method by which one can use these analytical solutions of
the Vlasov equation to obtain numerical solutions of the self-consistent
Vlasov-Poisson equations for such systems.
