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Landau Damping, Schrödinger Equation, and Fluctuation Theorem
Hideo Sugama summarized his research as “Linear Landau damping, Schrödinger equation, and fluctuation theorem,” and his paper has been published Physics of Plasmas.
【Abstract】
Landau damping has long been studied as one of the principal physical mechanisms responsible not only for wave heating observed in high-temperature plasmas in space and in fusion devices, but also for stabilizing microinstabilities and geodesic acoustic modes (GAMs). Landau damping is a seemingly irreversible process despite occurring in collisionless plasmas governed by the Vlasov equation, which possesses time-reversal symmetry. On the other hand, the fluctuation theorem, derived from reversible dynamics, states that the probability ratio of entropy production to entropy reduction increases exponentially with time, thereby providing a microscopic foundation for the second law of thermodynamics and nonequilibrium statistical mechanics. In this study, the linear Vlasov–Poisson system is reformulated in the form of the Schrödinger equation, so that time-reversal symmetry and conservation laws can be expressed concisely. This formulation shows that the fluctuation theorem holds for the relative stochasitc entropy, defined from the probability density functional of the particle velocity distribution function. The difference between the energy perturbation normalized by the equilibrium temperature and the entropy pertubation constitutes a time-independent invariant of the system. This invariant is a quadratic form of the perturbed component of the velocity distribution function and corresponds to the squared amplitude of the state vector that satisfies the Schrödinger equation. Furthermore, the eigenvectors of the Hamiltonian corresponding to Case–Van Kampen modes are derived. By constructing exact solutions from these eigenvectors and employing them, the fluctuation theorem for the Landau damping process is formulated and its validity is verified through numerical simulations. These results provide a new formulation of collisionless plasma physical processes from the perspective of nonequilibrium statistical mechanics.
(a) Probability density function P(∆S) of the stochastic relative entropy, and
(b) ratio P(∆S)/P(−∆S) of the probabilities of entropy increase to decrease, obtained from numerical simulations.
【Publications】