Landau Damping, Schrödinger equation, and Fluctuation Theorem

Abstract: A linearized Vlasov-Poisson system of equations is transformed into a Schr\"odinger equation, which is used to demonstrate that the fluctuation theorem holds for the relative stochastic entropy, defined in terms of the probability density functional of the particle velocity distribution function in the Landau damping process. The difference between the energy perturbation, normalized by the equilibrium temperature, and the entropy perturbation constitutes a time-independent invariant of the system. This invariant takes the quadratic form of the perturbed velocity distribution function and corresponds to the squared amplitude of the state vector that satisfies the Schr\"odinger equation. Exact solutions, constructed from a discrete set of Hamiltonian eigenvectors, are employed to formulate and numerically validate the fluctuation theorem for the Landau damping process. The results offer new insights into the formulations of collisionless plasma processes within the framework of nonequilibirium statistical mechanics.