864 research outputs found
Coarse-grained spin density-functional theory: infinite-volume limit via the hyperfinite
Coarse-grained spin density functional theory (SDFT) is a version of SDFT
which works with number/spin densities specified to a limited resolution ---
averages over cells of a regular spatial partition --- and external potentials
constant on the cells. This coarse-grained setting facilitates a rigorous
investigation of the mathematical foundations which goes well beyond what is
currently possible in the conventional formualation. Problems of existence,
uniqueness and regularity of representing potentials in the coarse-grained SDFT
setting are here studied using techniques of (Robinsonian) nonstandard
analysis. Every density which is nowhere spin-saturated is V-representable, and
the set of representing potentials is the functional derivative, in an
appropriate generalized sense, of the Lieb interal energy functional.
Quasi-continuity and closure properties of the set-valued representing
potentials map are also established. The extent of possible non-uniqueness is
similar to that found in non-rigorous studies of the conventional theory,
namely non-uniqueness can occur for states of collinear magnetization which are
eigenstates of .Comment: corrected non-propagating error in Cor. 4.2 and adopted better
methods in Sec. 6
In search of the Hohenberg-Kohn theorem
The Hohenberg-Kohn theorem, a cornerstone of electronic density functional
theory, concerns uniqueness of external potentials yielding given ground
densities of an -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 -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 under less stringent conditions.Comment: 21 pages, version accepted by Journal of Mathematical Physic
Good ultrafilters and highly saturated models: a friendly explanation
Highly saturated models are a fundamental part of the model-theoretic
machinery of nonstandard analysis. Of the two methods for producing them,
ultrapowers constructed with the aid of -good ultrafilters seems by
far the less popular. Motivated by the hypothesis that this is partly due to
the standard exposition being somewhat dense, a presentation is given which is
designed to be easier to digest.Comment: 8 pages. Comments welcome
Drift-diffusion processes from elimination of fast variables under inhomogeneous conditions
The problem of eliminating fast-relaxing variables to obtain an effective
drift-diffusion process in position is solved in a uniform and straightforward
way for models with velocity a function jointly of position and fast variables.
A more unified view is thereby obtained of the effect of environmental
inhomgeneity on the motion of a diffusing particle, in particular, whether a
drift is induced, covering both passive and active particles. Infinitesimal
generators (equivalently, drift-diffusion fields) for the contracted processes
are worked out in detail for several models.Comment: 10 pages. Abstract, introduction, and conclusion have been rewritten
to improve clarit
Efficacy of self-phoretic colloids and microswimmers
Within a unified formulation, encompassing self-electrophoresis,
self-diffusiophoresis, and self-thermophoresis, we provide a simple integral
kernel transforming the relevant surface flux to particle velocity for any
spheroid with axisymmetric surface activity and uniform phoretic mobility. We
define efficacy, a dimensionless efficiency-like quantity expressing the speed
resulting from unit absolute flux density on the surface, which allows a
meaningful comparison of the performance of different motor designs. For
bipartite designs with piecewise uniform flux over complementary surface
regions, the efficacy is mapped out over the entire range of geometry (discotic
through sphere to rod-like) and of bipartitioning, and intermediate aspect
ratios that maximize efficacy are identified. Comparison is made to
experimental data from the literature
Extrapolation theory for Stokes flow past a deformed sphere
We formulate a method for computing Stokes flow past a highly deformed sphere
with arbitrarily defined surface velocity. The fundamental ingredient is an
explicit extrapolation operator extending a velocity field from the surface of
a sphere, which is expressed in terms of a complete set of basis Stokes fields
for the pressure and velocity derived from scalar and vector spherical
harmonics. We present a matrix algebra packaging suitable for numerical
computation to arbitrary order in the deformation amplitude (deviation from
sphericity). The hydrodynamic force and torque on a deformed sphere with
arbitrary surface velocity are expressed in terms of basis field amplitudes,
and for the classic problem of a rotating and translating rigid body, we
compute explicitly the first order in deformation corrections to the flow field
as well as the hydrodynamic force and torque.Comment: 10 pages, 1 figur
How Much Phase Coherence Does a Pseudogap Need?
It has been suggested that the ``pseudogap'' regime in cuprate
superconductors, extending up to hudreds of degrees into the normal phase,
reflects an incoherent d-wave pairing, with local superconducting order
coherent over a finite length scale , insufficient to establish
superconductivity. We calculate the single-particle spectral density in such a
state from a minimal phenomenological disordered BCS model. When the
phase-coherence length exceeds the Cooper pair size, a clear pseudogap appears.
The pseudogap regime, however, is found only over a relatively narrow range of
phase stiffnesses, hence is not expected to extend more than about 20% above
.Comment: RevTeX, 4 pages, 5 embedded PostScript figure
Bypassing slip velocity: rotational and translational velocities of autophoretic colloids in terms of surface flux
A standard approach to propulsion velocities of autophoretic colloids with
thin interaction layers uses a reciprocity relation applied to the slip
velocity. But the surface flux (chemical, electrical, thermal, etc.), which is
the source of the field driving the slip is often more accessible. We show how,
under conditions of low Reynolds number and a field obeying the Laplace
equation in the outer region, the slip velocity can be bypassed in velocity
calculations. In a sense, the actual slip velocity and a normal field
proportional to the flux density are equivalent for this type of calculation.
Using known results for surface traction induced by rotating or translating an
inert particle in a quiescent fluid, we derive simple and explicit integral
formulas for translational and rotational velocities of arbitrary spheroidal
and slender-body autophoretic colloids.Comment: 11 page
Triangular Ising antiferromagnet through a fermionic lens, part 2: information-theoretic aspects of zero-temperature states on cylinders
A classical lattice spin model wrapped on a cylinder is profitably viewed as
a chain of rings of spins. From that perspective, mutual information between
ring configurations plays much the same role as spin-spin correlation functions
in simpler settings. We study zero-temperature states of triangular lattice
Ising antiferromagnet (TIAFM) systems from this point of view using a fermionic
representation presented in a companion paper (Part 1). On infinite cylinders,
ring-to-ring mutual information falls off asymptotically at a rate which
decreases smoothly with cylinder circumference, but the end-to-end mutual
information for finite cylinders depends strongly on the residue class modulo 3
of the circumference as well as on whether spin periodicity or antiperiodicity
is imposed in the circumferential direction. In some cases, the falloff is only
as the inverse square of the cylinder length. These features, puzzling within
the original spin formulation, are easily understood and calculated within the
fermionic formulation
Gaussian Memory in Kinematic Matrix Theory for Self-Propellers
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.Comment: 8 pages, 4 figure
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