arXiv subjects

Paul E. Lammert

Publications and source records attributed to Paul E. Lammert.

At least 19 recordsLinked to original sources

arXiv2026

Existence of Kraus decomposition in infinite dimension via strongly-convergent direct process tomography

An algorithm is presented for Kraus decomposition of a completely positive operator over separable (countably-infinite-dimensional) Hilbert spaces, together with an elementary proof that the generated sum convergences in strong-operator topology. This improves on the standard, nonconstructive, proof by fusing the abstract problem with practical process tomography. Kraus operators are generated one-by-one, each having one more guaranteed zero matrix entry than the previous one. In this way, the stream of outputs of the algorithm provides a coherent family of Kraus decompositions of restrictions of the target CP map to ever-larger subspaces.

quant-ph↗

arXiv2026

Notes on completely positive maps and continuous-time Markovian CP evolution. A geometry-flavored perspective

These notes provide a detailed and self-contained exposition of basic theory of CP maps and continuous-time Markovian evolution.The infinite-dimensional (separable) setting is handled as an extension of the finite-dimensional one.The treatment stands on two legs.For the finite-dimensional part, a basis-free version of the Choi-Jamiolkowski isomorphism called simply Jamiolkowski transform.And, for the extension, the ground matrix element topology (GMET), which does for the superoperators on trace-class operators what the weak-operator topology does for bounded operators on a Hilbert space. Background in open quantum systems or quantum information theory is not assumed.

quant-ph↗

arXiv2025

Classical triangular lattice antiferromagnetic Ising model as a free-fermion/superconductor system

We present a treatment of the triangular lattice antiferromagnetic Ising model (TAFIM) based on a small number of elementary ideas common to statistical and solid-state physics. The TAFIM is represented as a reduced BCS model in one space, one (imaginary) time dimension. The representation is approximate for nonzero temperature, but allows quick derivation of asymptotically exact thermodynamic functions, and the divergence of the spin-spin correlation length. The fermionic representation is exact at zero temperature. We demonstrate the existence of a two-dimensional continuum of zero-temperature equilibrium macrostates characterized by satisfied bond fractions of the three different orientations, and calculate their entropy densities.

cond-mat.stat-mech↗

arXiv2023

Kohn-Sham computation and the bivariate view of density functional theory

Informed by an abstraction of Kohn-Sham computation called a KS machine, a functional analytic perspective is developed on mathematical aspects of density functional theory. A natural semantics for the machine is bivariate, consisting of a sequence of potentials paired with a ground density. Although the question of when the KS machine can converge to a solution (where the potential component matches a designated target) is not resolved here, a number of related ones are. For instance: Can the machine progress toward a solution? Barring presumably exceptional circumstances, yes in an energetic sense, but using a potential-mixing scheme rather than the usual density-mixing variety. Are energetic and function space distance notions of proximity-to-solution commensurate? Yes, to a significant degree. If the potential components of a sequence of ground pairs converges to a target density, do the density components cluster on ground densities thereof? Yes, barring particle number drifting to infinity.

cond-mat.other↗

arXiv2022

Cluster expansion methods from physical concepts

The cluster expansion formalism used in materials science is reconstructed on an axiomatic basis with the aims of clarifying underlying concepts and improving computational procedures, and without using conventional cluster functions. Instead, cluster components of configuration functions are defined in an intrinsic manner, which can be viewed as Moebius inversion of conditional expectation. The associated method for fitting a model to a configurational sample is grounded entirely in Hilbert space geometry. By constructing models directly from the given data, we avoid an underdetermination problem to which the conventional approach is subject. Tensor observables are treated on an equal footing with scalar observables.

cond-mat.mtrl-sci↗

arXiv2021

Infinite-dimensional analyticity in quantum physics

A study is made, of families of Hamiltonians parameterized over open subsets of Banach spaces in a way which renders many interesting properties of eigenstates and thermal states analytic functions of the parameter. Examples of such properties are charge/current densities. The apparatus can be considered a generalization of Kato's theory of analytic families of type B insofar as the parameterizing spaces are infinite dimensional. It is based on the general theory of holomorphy in Banach spaces and an identification of suitable classes of sesquilinear forms with operator spaces associated with Hilbert riggings. The conditions of lower-boundedness and reality appropriate to proper Hamiltonians is thus relaxed to sectoriality, so that holomorphy can be used. Convenient criteria are given to show that a parameterization $x \mapsto {\mathsf{h}}_x$ of sesquilinear forms is of the required sort ({\it regular sectorial families}). The key maps ${\mathcal R}(ζ,x) = (ζ- H_x)^{-1}$ and ${\mathcal E}(β,x) = e^{-βH_x}$, where $H_x$ is the closed sectorial operator associated to ${\mathsf {h}}_x$, are shown to be analytic. These mediate analyticity of the variety of state properties mentioned above. A detailed study is made of nonrelativistic quantum mechanical Hamiltonians parameterized by scalar- and vector-potential fields and two-body interactions.

math-ph↗

arXiv2020

A bivariate view of Kohn-Sham iteration and the case for potential mixing

A bivariate perspective on Kohn-Sham density functional theory is proposed, treating potential and density as simultaneous independent variables, and used to make fruitful connection between Lieb's rigorous foundational framework and practical Kohn-Sham computation. Support is found for potential-mixing schemes, but not for more standard density-mixing. Under presumably-generic conditions, total energy can be lowered from one iteration to the next. Density, intrinsic and total energy are analytic functions of the noninteracting potential on the open set of potentials having a single isolated ground spin-multiplet.

cond-mat.mtrl-sci↗

arXiv2019

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.

cond-mat.stat-mech↗

arXiv2018

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.

cond-mat.stat-mech↗

arXiv2018

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 ${\mathcal N}$-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.

cond-mat.other↗

arXiv2018

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.

cond-mat.soft↗

arXiv2017

Communicating through a geometrically frustrated channel

We propose an intuitively appealing formulation of the zero-temperature triangular lattice Ising antiferromagnet (TIAFM) on a cylinder as a model of noninteracting fermions hopping on a ring and evolving in imaginary time with pair annihilation events. Among the features of the model which can then be related to local semi-conservation of particle number are: infinite-range influence of boundary conditions, multiple "zero temperature pure phases" in the infinite-length limit differing in entropy density, sensitivity of the asymptotic rate of decay (with respect to length) of mutual information between end configurations to circumference modulo 3, and even the known power-law falloff of the spin-spin correlator on an infinite plane. The ability of boundary conditions to determine the bulk entropy density enables communication between the two far-separated ends of such a cylinder even when all pair-wise spin-spin correlations between the ends are vanishingly small. In the fermionic language, breakdown of these zero-temperature phenomena at positive temperature is understood as passage of the system into a superconducting phase.

cond-mat.stat-mech↗

arXiv2017

Distinguishing Advective and Powered Motion in Self-Propelled Colloids

Self-powered motion in catalytic colloidal particles provides a compelling example of active matter, i.e. systems that engage in single-particle and collective behavior far from equilibrium. The long-time, long-distance behavior of such systems is of particular interest, since it connects their individual micro-scale behavior to macro-scale phenomena. In such analyses, it is important to distinguish motion due to subtle advective effects -- which also has long time scales and length scales -- from phenomena that derive from intrinsically powered motion. Here, we develop a methodology to analyze the statistical properties of the translational and rotational motions of powered colloids to distinguish, for example, active chemotaxis from passive advection by bulk flow.

cond-mat.soft↗

arXiv2016

Spiral diffusion of rotating self-propellers with stochastic perturbation

Translationally diffusive behavior arising from the combination of orientational diffusion and powered motion at microscopic scales is a known phenomenon, but the peculiarities of the evolution of expected position conditioned on initial position and orientation have been neglected. A theory is given of the spiral motion of the mean trajectory depending upon propulsion speed, angular velocity, orientational diffusion and rate of random chirality reversal. We demonstrate the experimental accessibility of this effect using both tadpole-like and Janus sphere dimer rotating motors. Sensitivity of the mean trajectory to the kinematic parameters suggest that it may be a useful way to determine those parameters.

cond-mat.soft↗

arXiv2016

On F and E, in DFT

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.

cond-mat.other↗

arXiv2016

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.

cond-mat.soft↗

arXiv2016

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.

cond-mat.soft↗

arXiv2015

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 $κ^+$-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.

math.LO↗