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

The impact of the Maxwell's theory of Saturn's ring, formulated in Aberdeen around 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 particle where the mutual gravitational attraction is replaced by the electric repulsion, the 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, the Maxwell's idea was so fertile that, later on, it was extended to the quantum-like models (f.i., Thermal Wave Model) providing the quantumlike description of coherent instability (Schroedinger{Maxwell system) and its identification with the modulational instability. The latter has been recently formulated for any nonlinear wave propagation governed by the nonlinear SchrÄodinger equation, as in the statistical approach to modulational instability (Wigner-Maxwell system). It seems that the above recent developments may provide a possible feedback to the original Maxwell's idea with the extension to quantum gravity and cosmology