Bernstein–Greene–Kruskal modes (a.k.a. BGK modes) are nonlinear
electrostatic waves that propagate in an unmagnetized, collisionless
plasma. They are nonlinear solutions to the
Vlasov–Poisson equation in
,
and are named after physicists
Ira B. Bernstein
Ira Borah Bernstein (born November 8, 1924) is an American theoretical physicist specializing in plasma physics. He was the first person to formulate the theory of electrostatic waves propagating in a magnetized plasma in 1958, which are now co ...
,
John M. Greene, and
Martin D. Kruskal, who solved and published the exact solution for the one-dimensional case in 1957.
BGK modes have been studied extensively in numerical simulations for two- and three-dimensional cases,
and are believed to be produced by the
two-stream instability. They have been observed as electron phase space holes (electrostatic solitary structures) and
double layers in space plasmas, as well as in scattering experiments in the laboratory.
Small-amplitude limit: Van Kampen modes ?
It is generally claimed that in the linear limit BGK modes (e.g. in the small amplitude approximation) reduce to what is known as Van Kampen modes, named after
Nico van Kampen
Nicolaas 'Nico' Godfried van Kampen (June 22, 1921 – October 6, 2013) was a Dutch theoretical physicist, who worked mainly on statistical mechanics and non-equilibrium thermodynamics.
Van Kampen was born in Leiden, and was a nephew of Frits Zer ...
who derived the solutions in 1955.
This is wrong, however, since such a transition from a nonlinear to a linear mode does not take place even in the infinitesimal amplitude limit. A harmonic hole equilibrium of the Vlasov-Poisson system, which is correctly described as a complete solution, i.e. inclusively its phase velocity, by the
Schamel method, shows that nonlinearity persists even in the small amplitude limit. The area of trapped particles in the phase space never vanishes in this limit and there is no moment in which the distribution of trapped particles is transformed (or collapses) into a delta-function.
Another indication that this claim is unfounded is that such nonlinear modes prove to be unconditionally marginal stable in current-carrying plasmas regardless of the drift velocity between electrons and ions. Landau's theory, as a linear wave theory, is obviously not applicable in case of coherent waves such as BGK modes.
Quantum BGK (QBGK) modes
BGK modes have been generalized to
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, in which the solutions (called quantum BGK modes) solve the quantum equivalent of the Vlasov–Poisson system known as the
Wigner–Poisson system, with periodic boundary conditions. The solutions for the QBGK modes were put forth by Lange et al. in 1996, with potential applications to quantum plasmas.
References
Waves in plasmas
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