Instability power laws and planes from the variational principle

Stable thermodynamic equilibrium of a physical system exists for a finite range of its physical parameters. Beyond this range, the system may transition to a dynamical one or undergo a phase change; determining the stability limit is often challenging for complex and nonlinear systems. We show that, for a broad range of physical systems, the stability limit may be formulated in terms of physical parameters that are independent with respect to an arbitrary variation of the system. Consequently, the stability limit is simply a power law if the thermodynamic potential describing the system comprises only two energy quantities; if comprised of more than two quantities, the stability limit is generally represented as a plane. Our result is shown to be valid for several solid, fluid and gas systems, and is especially useful for determining the stability limit of systems for which no analytical solution exists. As an example of the latter, we experimentally and theoretically investigate the stability limit of a liquid droplet exposed to an electric and gravitational field. The connection with statistical physics is also discussed.