Vlasov equation

The Vlasov equation describes the evolution of the distribution function of particles in a plasma or self-gravitating system, where collisions between particles are negligible compared to the collective mean-field forces. Unlike the Boltzmann equation, which includes collision terms that drive the system toward equilibrium, the Vlasov equation preserves phase-space structure and admits a rich family of non-thermal steady states.

It is the continuum limit of N-particle systems where propagation of chaos holds: each particle moves in the mean field generated by all others, and correlations remain weak even as the system evolves. The equation finds applications in plasma physics, galactic dynamics, and increasingly in the analysis of large-scale multi-agent systems where pairwise interactions dominate but collisional effects are secondary.

The Vlasov-Poisson system — coupling the Vlasov equation to a self-consistent field equation — remains an active area of mathematical research, particularly regarding stability of homogeneous equilibria and the formation of coherent structures.