arXiv subjects

Christopher N. Angstmann

Publications and source records attributed to Christopher N. Angstmann.

11 recordsLinked to original sources

arXiv2025

First-Passage Times for the Space-Fractional Spectral Fokker-Planck Equation

We extend the random walk framework to include compounded steps, providing first-passage time (FPT) properties for a new class of superdiffusive processes, which are governed by the space-fractional spectral Fokker-Planck equation. This first-passage process leads to novel FPT properties, different from L\'evy flights, that account for space dependent forces and hitting boundaries throughout the path of a jump. The FPT distribution can be derived for different types of barriers and potentials, for which we also provide specific examples. For the one-sided absorbing boundary with no potential on the semi-infinite line, we find that the FPT density scales asymptotically as $t^{-1/(2\alpha)-1}$ for large times, where the parameter $\alpha \in (0,1]$ relates to the power-law behavior for the distribution of the number of compounded steps. This is in agreement with the method of images but different to the Sparre-Andersen scaling $t^{-3/2}$ for corresponding L\'evy flights of order $2\alpha$. In this case, there exists an optimal space-fractional exponent $\alpha$ to minimize the mean FPT.

cond-mat.stat-mech↗

arXiv2025

The Global Diffusion Limit for the Space Dependent Variable-Order Time-Fractional Diffusion Equation

The diffusion equation and its time-fractional counterpart can be obtained via the diffusion limit of continuous time random walks with exponential and heavy-tailed waiting time distributions. The space-dependent variable-order time-fractional diffusion equation is a generalization of the time-fractional diffusion equation with a fractional exponent that varies over space, modelling systems with spatial heterogeneity. However, there has been limited work on defining a global diffusion limit and an underlying random walk for this macroscopic governing equation, which is needed to make meaningful interpretations of the parameters for applications. Here, we introduce continuous time and discrete time random walk models that limit to the variable-order fractional diffusion equation via a global diffusion limit and space- and time-continuum limits. From this, we show how the master equation of the discrete time random walk can be used to provide a numerical method for solving the variable-order fractional diffusion equation. The results in this work provide underlying random walks and an improved understanding of the diffusion limit for the variable-order fractional diffusion equation, which is critical for the development, calibration and validation of models for diffusion in spatially inhomogeneous media with traps and obstacles.

cond-mat.stat-mech↗

arXiv2024

A compounded random walk for space-fractional diffusion on finite domains

We formulate a compounded random walk that is physically well defined on both finite and infinite domains, and samples space-dependent forces throughout jumps. The governing evolution equation for the walk limits to a space-fractional Fokker-Planck equation valid on bounded domains, and recovers the well known superdiffusive space-fractional diffusion equation on infinite domains. We describe methods for numerical approximation and Monte Carlo simulations and demonstrate excellent correspondence with analytical solutions. This compounded random walk, and its associated fractional Fokker-Planck equation, provides a major advance for modeling space-fractional diffusion through potential fields and on finite domains.

cond-mat.stat-mech↗

arXiv2024

Delay Infectivity and Delay Recovery SIR model

We have derived the governing equations for an SIR model with delay terms in both the infectivity and recovery of the disease. The equations are derived by modelling the dynamics as a continuous time random walk, where individuals move between the classic SIR compartments. With an appropriate choice of distributions for the infectivity and recovery processes delay terms are introduced into the governing equations in a manner that ensures the physicality of the model. This provides novel insight into the underlying dynamics of an SIR model with time delays. The SIR model with delay infectivity and recovery allows for a more diverse range of dynamical behaviours. The model accounts for an incubation effect without the need to introduce new compartments.

math.DS↗

arXiv2024

Delay compartment models from a stochastic process

Compartment models with delay terms are widely used across a range of disciplines. The motivation to include delay terms varies across different contexts. In epidemiological and pharmacokinetic models, the delays are often used to represent an incubation period. In this work, we derive a compartment model with delay terms from an underlying non-Markov stochastic process. Delay terms arise when waiting times are drawn from a delay exponential distribution. This stochastic process approach allows us to preserve the physicality of the model, gaining understanding into the conditions under which delay terms can arise. By providing the conditions under which the delay exponential function is a probability distribution, we establish a critical value for the delay terms. An exact stochastic simulation method is introduced for the generalized model, enabling us to utilize the simulation in scenarios where intrinsic stochasticity is significant, such as when the population size is small. We illustrate the applications of the model and validate our simulation algorithm on examples drawn from epidemiology and pharmacokinetics.

math.DS↗

arXiv2024

Finite Time Blowup of Integer- and Fractional-Order Time-Delayed Diffusion Equations

In this work, exact solutions are derived for an integer- and fractional-order time-delayed diffusion equation with arbitrary initial conditions. The solutions are obtained using Fourier transform methods in conjunction with the known properties of delay functions. It is observed that the solutions do not exhibit infinite speed of propagation for smooth initial conditions that are bounded and positive. Sufficient conditions on the initial condition are also established such that the finite time blowup of the solutions can be explicitly calculated. Examples are provided that highlight the contrasting behaviours of these exact solutions with the known dynamics of solutions to the standard diffusion equation.

math.AP↗

arXiv2024

Exact Solutions of Time-Delay Integer- and Fractional-Order Advection Equations

Transport phenomena play a vital role in various fields of science and engineering. In this work, exact solutions are derived for advection equations with integer- and fractional-order time derivatives and a constant time-delay in the spatial derivative. Solutions are obtained, for arbitrary separable initial conditions, by incorporating recently introduced delay functions in a separation of variables approach. Examples are provided showing oscillatory and translatory behaviours that are fundamentally different to standard propagating wave solutions.

math.AP↗

arXiv2023

A Stochastic Simulation Method for Fractional Order Compartment Models

Our study focuses on fractional order compartment models derived from underlying physical stochastic processes, providing a more physically grounded approach compared to models that use the dynamical system approach by simply replacing integer-order derivatives with fractional order derivatives. In these models, inherent stochasticity becomes important, particularly when dealing with the dynamics of small populations far from the continuum limit of large particle numbers. The necessity for stochastic simulations arises from deviations of the mean states from those obtained from the governing equations in these scenarios. To address this, we introduce an exact stochastic simulation algorithm designed for fractional order compartment models, based on a semi-Markov process. We have considered a fractional order resusceptibility SIS model and a fractional order recovery SIR model as illustrative examples, highlighting significant disparities between deterministic and stochastic dynamics when the total population is small. Beyond its modeling applications, the algorithm presented serves as a versatile tool for solving fractional order differential equations via Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2021

The Lawnmower: an autonomous, protein-based artificial molecular motor

Inspired by biology, great progress has been made in creating artificial molecular motors. However, the dream of harnessing proteins - the building blocks selected by Nature - to design autonomous motors has so far remained elusive. Here we report the synthesis and characterization of the Lawnmower, an autonomous, protein-based artificial molecular motor comprised of a spherical hub decorated with proteases. Its "burnt-bridge" motion is directed by cleavage of a peptide lawn, promoting motion towards unvisited substrate. We find that Lawnmowers exhibit directional motion with average speeds of up to 80 nm/s, comparable to biological motors. By selectively patterning the peptide lawn on microfabricated tracks, we furthermore show that the Lawnmower is capable of track-guided motion. Our work opens an avenue towards nanotechnology applications of artificial protein motors.

physics.bio-ph↗

arXiv2015

Precision isotope shift measurements in Ca$^+$ using highly sensitive detection schemes

We demonstrate an efficient high-precision optical spectroscopy technique for single trapped ions with non-closed transitions. In a double-shelving technique, the absorption of a single photon is first amplified to several phonons of a normal motional mode shared with a co-trapped cooling ion of a different species, before being further amplified to thousands of fluorescence photons emitted by the cooling ion using the standard electron shelving technique. We employ this extension of the photon recoil spectroscopy technique to perform the first high precision absolute frequency measurement of the $^{2}$D$_{3/2}$ $\rightarrow$ $^{2}$P$_{1/2}$ transition in $^{40}$Ca$^{+}$, resulting in a transition frequency of $f=346\, 000\, 234\, 867(96)$ kHz. Furthermore, we determine the isotope shift of this transition and the $^{2}$S$_{1/2}$ $\rightarrow$ $^{2}$P$_{1/2}$ transition for $^{42}$Ca$^{+}$, $^{44}$Ca$^{+}$ and $^{48}$Ca$^{+}$ ions relative to $^{40}$Ca$^{+}$ with an accuracy below 100 kHz. Improved field and mass shift constants of these transitions as well as changes in mean square nuclear charge radii are extracted from this high resolution data.

physics.atom-ph↗

arXiv2013

Pattern Formation on Networks with Reactions: A Continuous Time Random Walk Approach

We derive the generalized master equation for reaction-diffusion on networks from an underlying stochastic process, the continuous time random walk (CTRW). The non-trivial incorporation of the reaction process into the CTRW is achieved by splitting the derivation into two stages. The reactions are treated as birth-death processes and the first stage of the derivation is at the single particle level, taking into account the death process, whilst the second stage considers an ensemble of these particles including the birth process. Using this model we have investigated different types of pattern formation across the vertices on a range of networks. Importantly, the CTRW defines the Laplacian operator on the network in a non \emph{ad-hoc} manner and the pattern formation depends on the structure of this Laplacian. Here we focus attention on CTRWs with exponential waiting times for two cases; one in which the rate parameter is constant for all vertices and the other where the rate parameter is proportional to the vertex degree. This results in nonsymmetric and symmetric CTRW Laplacians respectively. In the case of symmetric Laplacians, pattern formation follows from the Turing instability. However in nonsymmetric Laplacians, pattern formation may be possible with or without a Turing instability.

math.DS↗