Kinetic Equations and Fluctuations in Space of One-Component Dilute Plasmas

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Abstract
Kinetic equations for a spatially coarse-grained electron density in µ phase space A(p, r; t) with a length cutoff b and for its fluctuations are studied by a scaling method and a time-convolutionless approach developed by the present authors. An electron gas with a small plasma parameter ε≡1/cλD3 has three characteristic lengths; the Landau cutoff rL≡ελD, the Debye length λD≡√kBT/4πe2c and the mean free path lf≡λD/ε, e and c being electronic charge and mean electron density, respectively. It is shown that there are two characteristic regions of the length cutoff b. One is a coherent region where rL≪b≪λD. It characteristic scaling is c → 0, b → ∞, t → ∞ with b√c and t√c being kept constant. The Vlasov equation is derived in this limit. The other is a kinetic region where λD≪b≪lf. Its characteristic scaling is c → 0, b → ∞, c → ∞ with bc and tc being kept constant. The Vlasov term disappears and the Balescu-Lenard-Boltzmann-Landau equation, which is free of divergence for both close and distant collisions, is derived in this limit. It is shown that the fluctuations of A(p, r; t) obey a Markov process with scaling exponents α = 0, β = 1/2 in the coherent region near thermal equilibrium while they obey a Gaussian Markov process with α = 0, β = 1 in the kinetic region. The present theory does not need the factorization ansatz and Bogoliubov's functional ansatz.

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