PHASE-TRANSITION THEORY OF INSTABILITIES .1. 2ND-HARMONIC INSTABILITYAND BIFURCATION POINTS

We use a free-energy minimization approach to describe in simple and c lear physical terms the secular and dynamical instabilities as well as the bifurcations along well-known sequences of rotating, self-gravita ting fluid and stellar systems such as the Maclaurin spheroids, the Ja cobi, Dedekind, and Riemann ellipsoids, and the fluid/stellar disks. O ur approach stems from the Ginzburg-Landau theory of phase transitions . In this paper, we focus on the Maclaurin sequence of oblate spheroid al equilibria and on the effects of nonaxisymmetric, second-harmonic d isturbances. The free-energy approach has been pioneered in astrophysi cs by Bertin and Radicati (1976) who showed that the secular instabili ty beyond the Maclaurin-Jacobi bifurcation can be interpreted as a sec ond-order phase transition. We show that second-order phase transition s appear on the Maclaurin sequence also at the points of dynamical ins tability (bifurcation of the x = +1 self-adjoint Riemann sequence) and of bifurcation of the Dedekind sequence. The distinguishing character istic of each second-order phase transition is the conservation/noncon servation of an integral of motion (a ''conserved/nonconserved current '') which, in effect, determines uniquely whether the transition appea rs or not. The secular instability beyond the Jacobi bifurcation appea rs only if circulation is not conserved. The secular instability at th e Dedekind bifurcation appears only if angular momentum is not conserv ed. We show by an explicit calculation that, in the presence of dissip ation agents that violate one or the other conservation law, the globa l minimum of the free-energy function beyond the onset of secular inst ability belongs to the Jacobi and to the Dedekind sequence, respective ly. In the case of a ''perfect'' fluid which conserves both circulatio n and angular momentum, the ''secular'' phase transitions are no longe r realized and the Jacobi/Dedekind bifurcation point becomes irrelevan t. The Maclaurin spheroid remains at the global minimum of the free-en ergy function up to the bifurcation point of the x = +1 Riemann sequen ce. The x = +1 equilibria have lower free energy than the correspondin g Maclaurin spheroids for the same values of angular momentum and circ ulation. Thus, a ''dynamical'' second-order phase transition is allowe d to take place beyond this bifurcation point. This phase transition b rings the spheroid, now sitting at a saddle point of the free-energy f unction, to the new global minimum on the x = +1 Riemann sequence. Cir culation is not conserved in stellar systems because the stress-tenser 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. For this reason, global ly unstable axisymmetric stellar systems evolve toward the ''stellar'' Jacobi sequence on dynamical timescales. This explains why the Jacobi bifurcation is a point of dynamical instability in stellar systems bu t only a point of secular instability in viscous fluids. The second-or der phase transitions on the Maclaurin sequence are discussed in relat ion to the dynamical instability of stellar systems, the lambda-transi tion of liquid He-4, the second-order phase transition in superconduct ivity, and the mechanism of spontaneous symmetry breaking.