Bounds on Fluctuations in a Homogeneous Plasma

A homogeneous collisionless plasma is considered whose velocity distribution is initially unstable. Making use of the constraints inherent in the Maxwell‐Vlasov equations—along the lines of C. S. Gardner and of T. K. Fowler—it is shown that the nonlinear fluctuations grown out of the instability are subject to the following bounds: $$\frac{\overline{{\mathrm{(\Delta \rho )}}^2}}{{\mathrm{(\rho ̄)}}^2}\text{⪅0.4}\frac{{\epsilon }_{\mathrm{G}}}{{\epsilon }_{\mathrm{total}}{\mathrm{-\epsilon }}_{\mathrm{G}}}\mathrm{,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\displaystyle \sum _{{\mathrm{k\ge k}}_0}}\frac{E^2(\mathbf{k})}{\text{8π}}⪅\frac{{\epsilon }_{\mathrm{G}}}{{\text{1+1.2(hk}}_0{)}^2},$$ where ρ and Δρ are density and its fluctuation, ε_G the available energy in the plasma in the sense of Gardner's, E(k) the Fourier component of the electrostatic oscillations, and h the Debye length. The variation of the density function f(r, v) is considered from an originally homogeneous and Maxwellian density f_o(v) under the constraint of Liouville invariance. The total kinetic energy is then minimized under the further constraint that a given level of density perturbation Δρ is maintained. Solutions to this variational problem yield minimum energies associated with density fluctuations and lead to results which, in their approximate forms, are given in the above inequalities.