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Kényszerek alatti egyensúlyi állapotok stabilitásának vizsgálata

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 ],.

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Kényszerek alatti egyensúlyi állapotok stabilitásának vizsgálata

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

  2. 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, 022506 (2001); J. Phys. A 35, 5899 (2002); J. Math. Chem. 42, 661 (2007); J. Phys. A 43, 425208 (2010)

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

  4. 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

  5. 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:

  6. 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,

  7. How to obtain, in practice, the constrained derivatives corresponding to a given constraint(s) ? Find a functional ρC[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: &

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

  9. 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

  10. ● in the stability analysis of droplet growth in supercooled vapors, by Uline & Corti [J. Chem. Phys. 129, 234507 (2008); also Uline et al., J. Chem. Phys. 133, 174511 (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]

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