Stability of equilibrium under constraints: Role of second-order constrained derivatives

TL;DR

This paper investigates how second-order constrained derivatives can be used to accurately analyze the stability of equilibrium states under conservation constraints, providing a rigorous theoretical framework and practical examples.

Contribution

It introduces a comprehensive method for incorporating constraints into second derivative stability analysis, advancing the theoretical understanding of constrained equilibrium stability.

Findings

Constrained second derivatives effectively account for all constraint effects.

The theory is demonstrated with applications to ultrathin-film binary mixtures.

A new stability condition for constrained equilibria is rigorously derived.

Abstract

In the stability analysis of an equilibrium, given by a stationary point of a functional F[n] (free energy functional, e.g.), the second derivative of F[n] plays the essential role. If the system in equilibrium is subject to the conservation constraint of some extensive property (e.g. volume, material, or energy conservation), the Euler equation determining the stationary point corresponding to the equilibrium alters according to the method of Lagrange multipliers. Here, the question as to how the effects of constraints can be taken into account in a stability analysis based on second functional derivatives is examined. It is shown that the concept of constrained second derivatives incorporates all the effects due to constraints; therefore constrained second derivatives provide the proper tool for the stability analysis of equilibria under constraints. For a physically important type of constraints, it is demonstrated how the presented theory works. Further, the rigorous derivation of a recently obtained stability condition for a special case of equilibrium of ultrathin-film binary mixtures is given, presenting a guide for similar analyses. [For details on constrained derivatives, see also math-ph/0603027, physics/0603129, physics/0701145.]