Temporal evolution of the distribution function of one-dimensional systems

Phase space dynamics plays an important role in systems of many particles that move towards equilibrium. In this work, we study such dynamics in some one-dimensional systems: (i) non-linear and non-interacting pendulums; and (ii) many particles interacting gravitationally. The temporal evolution of systems with many particles, in a non-collisional regime, is governed by the Vlasov equation, which we solve numerically. The original equations are separated into transport equations via Lie’s splitting method. The positive and flux conservative numerical method is used, with a monotonized central-difference (MC) slope limiter. Although we present solutions to common problems in the area, detailed discussions are not frequent. Soon this work can be explored in the teaching of physics.