Conductivity tensors of all orders in a collisionless plasma

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Abstract
Expressions for the conductivity tensors of all orders are obtained, for both relativistic and unrelativistic magnetized Vlasov–Maxwell plasmas, closely related to the well-known formula for the linear conductivity tensor. Thus the unperturbed orbit integrations and the ø-integration in velocity space have been performed and each conductivity tensor is expressed as an infinite series with terms involving ordinary Bessel functions. Jα(K⊥V⊥/ωc) and denominators of type ω–KzVz–αωc for integers.α a. The symmetries related to the Manley-Rowe relations are explicitly seen in the formulae presented. These conductivity tensor formulae provide a much better starting point than the basic equations for many nonlinear investigations.