Kényszerek alatti egyensúlyi állapotok stabilitásának vizsgálata Tamás Gál Department of Physics, University of Florida, Gainesville, USA

At a local minimum/maximum of a functional A[ρ], In the presence of some constraint C[ρ]=C, the above Euler equation modifies according to the method of Lagrange multipliers, Question 1: how to account for constraints apart from a local extremum ? Question 2: how to account for constraints in a stationary point analysis, based on second derivatives ? Solution: introduction of the concept of constrained functional derivatives [T. Gál, Phys. Rev. A 63, (2001); J. Phys. A 35, 5899 (2002); J. Math. Chem. 42, 661 (2007); J. Phys. A 43, (2010)

● if Idea: Under constraints, the form of a functional derivative modifies. This gives a generalization of the method of Lagrange multipliers:

Under constraints, the Taylor expansion of a functional A[ρ] becomes In the case A[ρ] has a local extremum under a constraint, while the second-order (necessary) condition for a local minimum/maximum will become

The constrained derivative formula emerges from two essential conditions: (i) The derivatives of two functionals that are equal over a given constrained domain of the functional variables should have equal derivatives over that domain: (ii) If a functional is independent of N, an N-conservation constraint does not affect the differentiation of the functional:

From condition (i), where u(x) is an arbitrary function that integrates to 1. Condition (ii) then fixes u(x) as ● This follows from the fact that for a functional for which A[λρ]=A[ρ] for any λ,

How to obtain, in practice, the constrained derivatives corresponding to a given constraint(s) ? Find a functional ρ C [ ρ] that (i) satisfies the given constraint for any ρ(x), (ii) gives an identity for any ρ ( x) that satisfies the constraint That is, and With the use of this, then, the constrained first & second derivatives: &

Why is this the proper way to obtain the constrained derivatives ? Expand ρ C [ ρ] into its Taylor series: Then, substitute this into the Taylor series expansion of A[ρ] above the constrained domain, This will give

Applications ● in the dynamical description of ultra-thin polymer binary mixtures, by Clarke [Macromolecules 38, 6775 (2005); also Thomas et al., Soft Matter 6, 3517 (2010)] – two variables describing the motion of the fluid, under the constraints of volume and material conservation: and

● in the stability analysis of droplet growth in supercooled vapors, by Uline & Corti [J. Chem. Phys. 129, (2008); also Uline et al., J. Chem. Phys. 133, (2010)] – they used fluid-dynamical DFT, with a simple particle-number conservation constraint – to determine whether the given equilibrium is stable (i.e., there is a local minimum of the free-energy functional), they applied the eigenvalues λ of which should all be positive or zero in the case of a stable stationary point of F[ρ]