arXiv subjects

Alexander R. Klotz

Publications and source records attributed to Alexander R. Klotz.

26 records · Page 2Linked to original sources

arXiv2021

Flatness and Intrinsic Curvature of Linked-Ring Membranes

Recent experiments have elucidated the physical properties of kinetoplasts, which are chain-mail-like structures found in the mitochondria of trypanosome parasites formed from catenated DNA rings. Inspired by these studies, we use Monte Carlo simulations to examine the behavior of two-dimensional networks ("membranes") of linked rings. For simplicity, we consider only identical rings that are circular and rigid and that form networks with a regular linking structure. We find that the scaling of the eigenvalues of the shape tensor with membrane size are consistent with the behavior of the flat phase observed in self-avoiding covalent membranes. Increasing ring thickness tends to swell the membrane. Remarkably, unlike covalent membranes, the linked-ring membranes tend to form concave structures with an intrinsic curvature of entropic origin associated with local excluded-volume interactions. The degree of concavity increases with increasing ring thickness and is also affected by the type of linking network. The relevance of the properties of linked-ring model membranes to those observed in kinetoplasts is discussed.

cond-mat.soft↗

arXiv2021

The Ropelength of Complex Knots

The ropelength of a knot is the minimum contour length of a tube of unit radius that traces out the knot in three dimensional space without self-overlap, colloquially the minimum amount of rope needed to tie a given knot. Theoretical upper and lower bounds have been established for the asymptotic relationship between crossing number and ropelength and stronger bounds have been conjectured, but numerical bounds have only been calculated exhaustively for knots and links with up to 11 crossings, which are not sufficiently complex to test these conjectures. Existing ropelength measurements also have not established the complexity required to reach asymptotic scaling with crossing number. Here, we investigate the ropelength of knots and links beyond the range of tested crossing numbers, past both the 11-crossing limit as well as the 16-crossing limit of the standard knot catalog. We investigate torus knots up to 1023 crossings, establishing a stronger upper bound for T(p,2) knots and links and demonstrating power-law scaling in T(p,p+1) below the proven limit. We investigate satellite knots up to 42 crossings to determine the effect of a systematic crossing-increasing operation on ropelength, finding that a satellite knot typically has thrice the ropelength of its companion. We find that ropelength is well described by a model of repeated Hopf links in which each component adopts a minimal convex hull around the cross-section of the others, and derive formulae that heuristically predict the crossing-ropelength relationship of knots and links without free parameters.

math.GT↗

arXiv2020

Trapping in Self-Avoiding Walks with Nearest-Neighbor Attraction

The statistics of self-avoiding random walks have been used to model polymer physics for decades. A self-avoiding walk that grows one step at a time on a lattice will eventually trap itself, which occurs after an average of 71 steps on a square lattice. Here, we consider the effect of nearest-neighbor attractive interactions on the growing self-avoiding walk, and examine the effect that self-attraction has both on the statistics of trapping as well as on chain statistics through the transition between expanded and collapsed walks at the theta point. We find the trapping length increases exponentially with the nearest-neighbor contact energy, but that there is a local minimum in trapping length for weakly self-attractive walks. While it has been controversial whether growing self-avoiding walks have the same asymptotic behavior as traditional self-avoiding walks, we find that the theta point is not at the same location for growing self-avoiding walks, and that the persistence length converges much more rapidly to a smaller value.

cond-mat.stat-mech↗

arXiv2017

Relativistic minimization with constraints: A smooth trip to Alpha Centauri

We consider the relativistic generalization of the problem of the "least uncomfortable" linear trajectory from point A to point B. The traditional problem minimizes the time-integrated squared acceleration (termed the "discomfort"), and there is a universal solution for all distances and durations. This universality fails when the maximum speed of the trajectory becomes relativistic, and we consider the more general case of minimizing the squared proper acceleration over a proper time. The least uncomfortable relativistic trajectory has a rapidity that evolves like the motion of a particle in a hyperbolic sine potential, agreeing with the classical solution at low velocities. We consider the special case of a hypothetical trip to Alpha Centauri, and compare the minimal-discomfort trajectory to one with uniform Earth-like acceleration.

physics.class-ph↗

arXiv2016

Waves of DNA: Propagating Excitations in Extended Nanoconfined Polymers

We use a nanofluidic system to investigate the emergence of thermally driven collective phenomena along a single polymer chain. In our approach, a single DNA molecule is confined in a nanofluidic slit etched with arrays of embedded nanocavities; the cavity lattice is designed so that a single chain occupies multiple cavities. Fluorescent video-microscopy data shows that waves of excess fluorescence propagate across the cavity-straddling molecule, corresponding to propagating fluctuations of contour overdensity in the cavities. The waves are quantified by examining the correlation in intensity fluctuations between neighbouring cavities. Correlations grow from an anti-correlated minimum to a correlated maximum before decaying, corresponding to a transfer of contour between neighbouring cavities at a fixed transfer time-scale. The observed dynamics can be modelled using Langevin dynamics simulations and a minimal lattice model of coupled diffusion. This study shows how confinement-based sculpting of the polymer equilibrium configuration, by renormalizing the physical system into a series of discrete cavity states, can lead to new types of dynamic collective phenomena.

cond-mat.soft↗

arXiv2015

A Guided Tour of Planetary Interiors

We explore the gravitational dynamics of falling through planetary interiors. Two trajectory classes are considered: a straight cord between two surface points, and the brachistochrone path that minimizes the falling time between two points. The times taken to fall along these paths, and the shapes of the brachistochrone paths, are examined for the Moon, Mars, Earth, Saturn, and the Sun, based on models of their interiors. A toy model of the internal structure, a power-law gravitational field, characterizes the dynamics with one parameter, the exponent of the power-law, with values from -2 for a point-mass to +1 for a uniform sphere. Smaller celestial bodies behave like a uniform sphere, while larger bodies begin to approximate point-masses, consistent with an effective exponent describing their interior gravity.

physics.pop-ph↗

arXiv2013

The Gravity Tunnel in a Non-Uniform Earth

How long does it take to fall down a tunnel through the center of the Earth to the other side? Assuming a uniformly dense Earth, it would take 42 minutes, but this assumption has not been validated. This paper examines the gravity tunnel without this restriction, using the internal structure of the Earth as ascertained by seismic data, and the dynamics are solved numerically. The time taken to fall along the diameter is found to be 38 rather than 42 minutes. The time taken to fall along a straight line between any two points is no longer independent of distance, but interpolates between 42 minutes for short trips and 38 minutes for long trips. The brachistochrone path (minimizing the fall time between any two points) is similar to the uniform density solution, but tends to reach a greater maximum depth and takes less time to traverse. Although the assumption of uniform density works well in many cases, the simpler assumption of a constant gravitational field serves as a better approximation to the true results.

physics.class-ph↗

arXiv2012

Bubble Dynamics in N Dimensions

Cavitation and bubble dynamics are central concepts in engineering, the natural sciences, and the mathematics of fluid mechanics. Due to the nonlinear nature of their dynamics, the governing equations are not fully solvable. Here, the dynamics of a spherical bubble in an N-dimensional fluid are discussed in the hope that examining bubble behavior in N dimensions will add insight to their behavior in three dimensions. Several canonical results in bubble dynamics are re-derived, including the Rayleigh collapse time, the Rayleigh-Plesset equation, and the Minnaert frequency. Numerical simulations are used to examine the onset of nonlinear behavior. Overall, the dynamics of bubbles are faster at higher dimensions, with nonlinear behavior occurring at lower amplitudes. Several features are found to be unique to three dimensions, including the trend of nonlinear behaviour and apparent coincidences in timescales.

physics.flu-dyn↗