Collective Coordinate Control of Density Distributions

Real collective density variables $C(\boldsymbol{k})$ [c.f. Eq.\ref{Equation3})] in many-particle systems arise from non-linear transformations of particle positions, and determine the structure factor $S(\boldsymbol{k})$, where $\bf k$ denotes the wave vector. Our objective is to prescribe $C({\boldsymbol k})$ and then to find many-particle configurations that correspond to such a target $C({\bf k})$ using a numerical optimization technique. Numerical results reported here extend earlier one- and two-dimensional studies to include three dimensions. In addition, they demonstrate the capacity to control $S(\boldsymbol{k})$ in the neighborhood of $|\boldsymbol{k}| =$ 0. The optimization method employed generates multi-particle configurations for which $S(\boldsymbol{k}) \propto |\boldsymbol{k}|^{\alpha}$, $|\boldsymbol{k}| \leq K$, and $\alpha =$ 1, 2, 4, 6, 8, and 10. The case $\alpha =$ 1 is relevant for the Harrison-Zeldovich model of the early universe, for superfluid $^{4}{He}$, and for jammed amorphous sphere packings. The analysis also provides specific examples of interaction potentials whose classical ground state are configurationally degenerate and disordered.Comment: 26 pages, 8 figure

Augmented Lagrangian Method for Constrained Nuclear Density Functional Theory

The augmented Lagrangiam method (ALM), widely used in quantum chemistry constrained optimization problems, is applied in the context of the nuclear Density Functional Theory (DFT) in the self-consistent constrained Skyrme Hartree-Fock-Bogoliubov (CHFB) variant. The ALM allows precise calculations of multidimensional energy surfaces in the space of collective coordinates that are needed to, e.g., determine fission pathways and saddle points; it improves accuracy of computed derivatives with respect to collective variables that are used to determine collective inertia; and is well adapted to supercomputer applications.Comment: 6 pages, 3 figures; to appear in Eur. Phys. J.

A Constrained Sequential-Lamination Algorithm for the Simulation of Sub-Grid Microstructure in Martensitic Materials

We present a practical algorithm for partially relaxing multiwell energy densities such as pertain to materials undergoing martensitic phase transitions. The algorithm is based on sequential lamination, but the evolution of the microstructure during a deformation process is required to satisfy a continuity constraint, in the sense that the new microstructure should be reachable from the preceding one by a combination of branching and pruning operations. All microstructures generated by the algorithm are in static and configurational equilibrium. Owing to the continuity constrained imposed upon the microstructural evolution, the predicted material behavior may be path-dependent and exhibit hysteresis. In cases in which there is a strict separation of micro and macrostructural lengthscales, the proposed relaxation algorithm may effectively be integrated into macroscopic finite-element calculations at the subgrid level. We demonstrate this aspect of the algorithm by means of a numerical example concerned with the indentation of an Cu-Al-Ni shape memory alloy by a spherical indenter.Comment: 27 pages with 9 figures. To appear in: Computer Methods in Applied Mechanics and Engineering. New version incorporates minor revisions from revie

Existence of Dyons in Minimally Gauged Skyrme Model via Constrained Minimization

We prove the existence of electrically and magnetically charged particlelike static solutions, known as dyons, in the minimally gauged Skyrme model developed by Brihaye, Hartmann, and Tchrakian. The solutions are spherically symmetric, depend on two continuous parameters, and carry unit monopole and magnetic charges but continuous Skyrme charge and non-quantized electric charge induced from the 't Hooft electromagnetism. The problem amounts to obtaining a finite-energy critical point of an indefinite action functional, arising from the presence of electricity and the Minkowski spacetime signature. The difficulty with the absence of the Higgs field is overcome by achieving suitable strong convergence and obtaining uniform decay estimates at singular boundary points so that the negative sector of the action functional becomes tractable.Comment: 24 page

Free energies, vacancy concentrations and density distribution anisotropies in hard--sphere crystals: A combined density functional and simulation study

We perform a comparative study of the free energies and the density distributions in hard sphere crystals using Monte Carlo simulations and density functional theory (employing Fundamental Measure functionals). Using a recently introduced technique (Schilling and Schmid, J. Chem. Phys 131, 231102 (2009)) we obtain crystal free energies to a high precision. The free energies from Fundamental Measure theory are in good agreement with the simulation results and demonstrate the applicability of these functionals to the treatment of other problems involving crystallization. The agreement between FMT and simulations on the level of the free energies is also reflected in the density distributions around single lattice sites. Overall, the peak widths and anisotropy signs for different lattice directions agree, however, it is found that Fundamental Measure theory gives slightly narrower peaks with more anisotropy than seen in the simulations. Among the three types of Fundamental Measure functionals studied, only the White Bear II functional (Hansen-Goos and Roth, J. Phys.: Condens. Matter 18, 8413 (2006)) exhibits sensible results for the equilibrium vacancy concentration and a physical behavior of the chemical potential in crystals constrained by a fixed vacancy concentration.Comment: 17 pages, submitted to Phys. Rev.

An Exactly Solvable Phase-Field Theory of Dislocation Dynamics, Strain Hardening and Hysteresis in Ductile Single Crystals

An exactly solvable phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals is developed. The theory accounts for: an arbitrary number and arrangement of dislocation lines over a slip plane; the long-range elastic interactions between dislocation lines; the core structure of the dislocations resulting from a piecewise quadratic Peierls potential; the interaction between the dislocations and an applied resolved shear stress field; and the irreversible interactions with short-range obstacles and lattice friction, resulting in hardening, path dependency and hysteresis. A chief advantage of the present theory is that it is analytically tractable, in the sense that the complexity of the calculations may be reduced, with the aid of closed form analytical solutions, to the determination of the value of the phase field at point-obstacle sites. In particular, no numerical grid is required in calculations. The phase-field representation enables complex geometrical and topological transitions in the dislocation ensemble, including dislocation loop nucleation, bow-out, pinching, and the formation of Orowan loops. The theory also permits the consideration of obstacles of varying strengths and dislocation line-energy anisotropy. The theory predicts a range of behaviors which are in qualitative agreement with observation, including: hardening and dislocation multiplication in single slip under monotonic loading; the Bauschinger effect under reverse loading; the fading memory effect, whereby reverse yielding gradually eliminates the influence of previous loading; the evolution of the dislocation density under cycling loading, leading to characteristic `butterfly' curves; and others