From Maxwell's theory of Saturn's rings to the negative mass instability

Abstract

The impact of Maxwell's theory of Saturn's rings, formulated in Aberdeen ca 1856, is discussed. One century later, Nielsen, Sessler and Symon formulated a similar theory to describe the coherent instabilities (in particular, the negative mass instability) exhibited by a charged particle beam in a high-energy accelerating machine. Extended to systems of particles where the mutual gravitational attraction is replaced by the electric repulsion, Maxwell's approach was the conceptual basis to formulate the kinetic theory of coherent instability (Vlasov–Maxwell system), which, in particular, predicts the stabilizing role of the Landau damping. However, Maxwell's idea was so fertile that, later on, it was extended to quantum-like models (e.g. thermal wave model), providing the quantum-like description of coherent instability (Schrödinger–Maxwell system) and its identification with the modulational instability (MI). The latter has recently been formulated for any nonlinear wave propagation governed by the nonlinear Schrödinger equation, as in the statistical approach to MI (Wigner–Maxwell system). It seems that the above recent developments may provide a possible feedback to Maxwell's original idea with the extension to quantum gravity and cosmology.