Description
Landau damping is a phenomenon that occurs in collisionless plasmas governed by the Vlasov equation, which possesses time-reversal symmetry, yet it appears at first glance to be an irreversible process. On the other hand, the fluctuation theorem derived from reversible dynamics shows that the ratio of the probabilities of entropy production to entropy reduction grows exponentially with time, thereby providing a foundation for the second law of thermodynamics and nonequilibrium statistical mechanics. This study demonstrates that the fluctuation theorem is applicable to collisionless processes in kinetic plasma systems. First, the linearized Vlasov–Poisson system describing Landau damping is formulated in the form of a Schrödinger equation in order to express time-reversal symmetry and conservation laws in a concise manner. Within this formulation, it is shown that a fluctuation theorem holds for the stochastic relative entropy defined from the probability density functional of the particle velocity distribution function [H. Sugama, Phys. Plasmas 32, 080701 (2025)]. The difference between the energy fluctuation normalized by the equilibrium temperature and the entropy fluctuation constitutes a time-independent invariant of the system. This invariant takes the form of a quadratic functional of the perturbation of the velocity distribution function and corresponds to the squared amplitude of the state vector satisfying the Schrödinger equation. Eigenvectors of the Hamiltonian corresponding to the Case–van Kampen modes are derived. By constructing exact solutions from these eigenvectors, a fluctuation theorem for the Landau damping process is formulated, and its validity is verified by numerical simulations. Furthermore, the above results are extended to gyrokinetic systems. The linear electromagnetic gyrokinetic equation describing perturbations applied to a stable Maxwellian equilibrium can also be formulated in the form of a Schrödinger equation, demonstrating that the fluctuation theorem is applicable to this gyrokinetic system as well. These results contribute to a new formulation of collisionless plasma phenomena from the perspective of nonequilibrium statistical mechanics.