Abstract
In this paper, we consider the problem of characterizing the minimum energy configurations of a finite system of particles interacting between them due to attractive or repulsive forces given by a certain intermolecular potential. We limit ourselves to the cases of three particles arranged in a triangular array and that of four particles in a tetrahedral array. The minimization is constrained to a fixed area in the case of the triangular array, and to a fixed volume in the tetrahedral case. For a general class of intermolecular potentials we give conditions for the homogeneous configuration (either an equilateral triangle or a regular tetrahedron) of the array to be stable that is, a minimizer of the potential energy of the system. To determine whether or not there exist other stable states, the system of first-order necessary conditions for a minimum is treated as a bifurcation problem with the area or volume variable as the bifurcation parameter. Because of the symmetries present in our problem, we can apply the techniques of equivariant bifurcation theory to show that there exist branches of non-homogeneous solutions bifurcating from the trivial branch of homogeneous solutions at precisely the values of the parameter of area or volume for which the homogeneous configuration changes stability. For the triangular array, we construct numerically the bifurcation diagrams for both a Lennard–Jones and Buckingham potentials. The numerics show that there exist non-homogeneous stable states, multiple stable states for intervals of values of the area parameter, and secondary bifurcations as well.
Highlights
Consider a system of N molecules, modeled as identical spherical particles, enclosed in a bounded region B of R3
We do not commit to any particular intermolecular potential φ so that our results are applicable to any such smooth potential
We find that the necessary condition for bifurcation from the trivial branch for this system occurs exactly at the boundary points of the set A given by the stability condition (22)
Summary
Consider a system of N molecules, modeled as identical spherical particles, enclosed in a bounded region B of R3. In Theorem 2 we give a necessary and sufficient condition (cf (22)), in terms of the intermolecular potential, for this equilibrium point to be a (local) minimizer of the energy functional This condition leads to a set of values A for the area parameter A for which the equilateral triangle is a stable configuration. The equilibrium configurations in this case are given as solutions to a nonlinear system of seven equations in eight unknowns (cf (40)) We treat this system as a bifurcation problem with the parameter V as a bifurcation parameter, and the set of equilateral tetrahedrons as the trivial solution branch. Using some of the machinery of equivariance theory as in [16], we can construct suitable reduced problems in each of these two cases, which enables us to establish the existence of non-equilateral equilibrium configurations and to get a full description of the symmetries of the bifurcating branches (cf Theorems 5–7). If the variables in ~F are given by (~x, ~y), D~x~F denotes the derivative of ~F with respect to the vector of variables ~x, i.e., the matrix of the partial derivatives of the components of ~F with respect to the variables corresponding to ~x
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