Phase-Transition Theory of Instabilities. I. Second-Harmonic Instability and Bifurcation Points

A free-energy minimization approach is used to address the secular & dynamical instabilities & the bifurcations along sequences of rotating, self-gravitating fluid and stellar systems. Our approach stems from the Landau-Ginzburg theory of phase transitions. We focus on the Maclaurin sequence of oblate spheroids & on the effects of second-harmonic disturbances. Second-order phase transitions appear on the Maclaurin sequence also at the points of dynamical instability & of bifurcation of the Dedekind sequence. Distinguishing characteristic of each second-order phase transition is the (non)conservation of an integral of motion (e.g. circulation) which determines the appearance of the transition. Circulation is not conserved in stellar systems because the stress-tensor gradient terms that appear in the Jeans equations of motion include viscosity-like off-diagonal terms of the same order of magnitude as the conventional pressure gradient terms. This explains why the Jacobi bifurcation is a point of dynamical instability in stellar systems but only a point of secular instability in viscous fluids. The second-order phase transitions are discussed in relation to the dynamical instability of stellar systems, the lambda-transition of liquid He-4, the second-order phase transition in superconductivity & the mechanism of spontaneous symmetry breaking.