Nonlinear Effects from Vlasov's Equation

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Abstract
Nonlinear effects from Vlasov's equation are studied for a one‐dimensional electron gas with fixed neutralizing background. The electron distribution function is expanded in Hermite polynomials and a closure condition is introduced to cut off the resulting infinite system of differential equations for the expansion coefficients. Then a nonlinear partial differential equation can be derived for the electric field. The consequences of some different closure conditions are studied. It turns out that their use implies severe restrictions on the physical situations described. In the discussed closed polynomial approximations no nonlinear instabilities are found. In particular, it results that with vanishing heat flux one has only stable linear or nonlinear oscillations showing that unstable situations are necessarily nonadiabatic. In an analogous manner, the condition that the plasma follows a state law with pressure proportional to density, like a perfect gas, eliminates unstable modes.

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