The free energy principle, negative energy modes, and stability

The instability of equilibria of Hamiltonian, fluid and plasma dynamical systems is studied. Usually the dynamical equilibrium of interest is not the state of thermodynamic equilibrium, and does not correspond to a free energy minimum. The relaxation of this type of equilibrium is conventionally considered to be initiated by linear instability. However, there are many cases where linear instability is not present, but the equilibrium is nonlinearly unstable to arbitrarily small perturbations. General free energy expressions for determining the presence of linear or nonlinear instabilities is the subject of study. These expressions are simple and practical, and can be obtained for all equilibria of all ideal fluid and plasma models. By free energy, it is meant that the energy change upon perturbations of the equilibrium that respect dynamical phase space constraints. This quantity is measured by a self-adjoint quadratic form, called delta (sup 2)F. The free energy can result in instability when delta (sup 2)F is indefinite; i.e., there exist accessible perturbations that lower the free energy of the system.