Perturbative solution of Vlasov equation for periodically driven systems

TL;DR

This paper derives an analytical perturbative solution to the 1D Vlasov equation for systems with high-frequency, time-periodic forces, revealing complex non-equilibrium steady states and non-uniform temperature distributions.

Contribution

It provides a novel perturbative analytical approach to solve the Vlasov equation in periodically driven systems, highlighting non-trivial steady states and temperature variations.

Findings

Time-averaged distribution cannot be simplified to an effective potential.

System temperature is spatially non-uniform, indicating non-equilibrium.

Method outlined for numerical solutions of Vlasov-Poisson equations.

Abstract

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.