A kinetic description for the electromagnetic response of the charged particles to Chern–Simons gauge fields

In this paper, we introduce the Vlasov–Chern–Simons (VCS) equation, a Vlasov-type equation that describes the two-dimensional dynamics of charged particles affected by the Chern–Simons electromagnetic potentials. First, we derive the VCS equation from the Chern–Simons–Schrödinger equations, a quantum mechanical model for the particle affected by Chern–Simons gauge fields, via the Wigner transform. Subsequently, we study the local-in-time well-posedness for the strong solution and the global-in-time existence for weak solutions to the VCS equation, respectively. Additionally, we propose a simple semi-Lagrangian numerical scheme for solving the VCS equation and validate the conservation of total moments and $L^p$-norms through numerical tests.