Showing papers by "Paul E. Lammert published in 2014"

Kinematic matrix theory and universalities in self-propellers and active swimmers.

Abstract: We describe an efficient and parsimonious matrix-based theory for studying the ensemble behavior of self-propellers and active swimmers, such as nanomotors or motile bacteria, that are typically studied by differential-equation-based Langevin or Fokker-Planck formalisms. The kinematic effects for elementary processes of motion are incorporated into a matrix, called the ``kinematrix,'' from which we immediately obtain correlators and the mean and variance of angular and position variables (and thus effective diffusivity) by simple matrix algebra. The kinematrix formalism enables us recast the behaviors of a diverse range of self-propellers into a unified form, revealing universalities in their ensemble behavior in terms of new emergent time scales. Active fluctuations and hydrodynamic interactions can be expressed as an additive composition of separate self-propellers.

Gaussian memory in kinematic matrix theory for self-propellers.

Abstract: We extend the kinematic matrix ("kinematrix") formalism [Phys. Rev. E 89, 062304 (2014)], which via simple matrix algebra accesses ensemble properties of self-propellers influenced by uncorrelated noise, to treat Gaussian correlated noises. This extension brings into reach many real-world biological and biomimetic self-propellers for which inertia is significant. Applying the formalism, we analyze in detail ensemble behaviors of a 2D self-propeller with velocity fluctuations and orientation evolution driven by an Ornstein-Uhlenbeck process. On the basis of exact results, a variety of dynamical regimes determined by the inertial, speed-fluctuation, orientational diffusion, and emergent disorientation time scales are delineated and discussed.

On F and E, in DFT

Abstract: Rigorous mathematical foundations of density functional theory are revisited, with some use of infinitesimal (nonstandard) methods. A thorough treatment is given of basic properties of internal energy and ground-state energy functionals along with several improvements and clarifications of known results.A simple metrizable topology is constructed on the space of densities using a hierarchy of spatial partitions. This topology is very weak, but supplemented by control of internal energy, it is, in a rough sense, essentially as strong as $L^1$. Consequently, the internal energy functional $F$ is lower semicontinuous with respect to it. With separation of positive and negative parts of external potentials, very badly behaved, even infinite, positive parts can be handled. Confining potentials are thereby incorporated directly into the density functional framework.

Hohenberg-Kohn redux

Abstract: The Hohenberg-Kohn theorem is a cornerstone of electronic density functional theory, yet completing its proof in the traditional way requires the {\em assumption} that ground state wavefunctions never vanish on sets of nonzero Lebesgue measure. This is an unsatisfactory situation, since DFT is supposed to obviate knowledge of many-body wavefunctions. We approach the issue from a more density-centric direction, allowing mild hypotheses on the density which can be regarded as checkable in a DFT context. By ordinary Hilbert space analysis, the following is proved: If the density $\rho$ is continuous and everywhere nonzero, then there can be at most one potential (modulo constants) expressible as a sum of a square-integrable and a bounded function (i.e., Kato-Rellich) with $\rho$ as a ground state density. In case $\rho$ is not nonzero everywhere, the theorem allows an independent constant on each connected component of the set where the density is positive, a weakening which can be reversed by requiring locally weak-$L^3$ potentials and calling on a unique continuation result of Schechter and Simon.

In search of the Hohenberg-Kohn theorem

Abstract: The Hohenberg-Kohn theorem, a cornerstone of electronic density functional theory, concerns uniqueness of external potentials yielding given ground densities of an ${\mathcal N}$-body system. The problem is rigorously explored in a universe of three-dimensional Kato-class potentials, with emphasis on trade-offs between conditions on the density and conditions on the potential sufficient to ensure uniqueness. Sufficient conditions range from none on potentials coupled with everywhere strict positivity of the density, to none on the density coupled with something a little weaker than local $3{\mathcal N}/2$-power integrability of the potential on a connected full-measure set. A second theme is localizability, that is, the possibility of uniqueness over subsets of ${\mathbb R}^3$ under less stringent conditions.