The Compton-Schwarzschild correspondence from extended de Broglie relations

The Compton wavelength gives the minimum radius within which the mass of a particle may be localized due to quantum effects, while the Schwarzschild radius gives the maximum radius within which the mass of a black hole may be localized due to classial gravity. In a mass-radius diagram, the two lines intersect near the Planck point $(l_P,m_P)$, where quantum gravity effects become significant. Since canonical (non-gravitational) quantum mechanics is based on the concept of wave-particle duality, encapsulated in the de Broglie relations, these relations should break down near $(l_P,m_P)$. It is unclear what physical interpretation can be given to quantum particles with energy $E \gg m_Pc^2$, since they correspond to wavelengths $\lambda \ll l_P$ or time periods $T \ll t_P$ in the standard theory. We therefore propose a correction to the standard de Broglie relations, which gives rise to a modified Schr{\" o}dinger equation and a modified expression for the Compton wavelength, which may be extended into the region $E \gg m_Pc^2$. For the proposed modification, we recover the expression for the Schwarzschild radius for $E \gg m_Pc^2$ and the usual Compton formula for $E \ll m_Pc^2$. The sign of the inequality obtained from the uncertainty principle reverses at $m \approx m_P$, so that the Compton wavelength and event horizon size may be interpreted as minimum and maximum radii, respectively. We interpret the additional terms in the modified de Broglie relations as representing the self-gravitation of the wave packet.Comment: 40 pages, 7 figures, 2 appendices. Published version, with additional minor typos corrected (v3

Accurate and efficient calculation of response times for groundwater flow

We study measures of the amount of time required for transient flow in heterogeneous porous media to effectively reach steady state, also known as the response time. Here, we develop a new approach that extends the concept of mean action time. Previous applications of the theory of mean action time to estimate the response time use the first two central moments of the probability density function associated with the transition from the initial condition, at $t=0$, to the steady state condition that arises in the long time limit, as $t \to \infty$. This previous approach leads to a computationally convenient estimation of the response time, but the accuracy can be poor. Here, we outline a powerful extension using the first $k$ raw moments, showing how to produce an extremely accurate estimate by making use of asymptotic properties of the cumulative distribution function. Results are validated using an existing laboratory-scale data set describing flow in a homogeneous porous medium. In addition, we demonstrate how the results also apply to flow in heterogeneous porous media. Overall, the new method is: (i) extremely accurate; and (ii) computationally inexpensive. In fact, the computational cost of the new method is orders of magnitude less than the computational effort required to study the response time by solving the transient flow equation. Furthermore, the approach provides a rigorous mathematical connection with the heuristic argument that the response time for flow in a homogeneous porous medium is proportional to $L^2/D$, where $L$ is a relevant length scale, and $D$ is the aquifer diffusivity. Here, we extend such heuristic arguments by providing a clear mathematical definition of the proportionality constant.Comment: 22 pages, 3 figures, accepted version of paper published in Journal of Hydrolog

New homogenization approaches for stochastic transport through heterogeneous media

The diffusion of molecules in complex intracellular environments can be strongly influenced by spatial heterogeneity and stochasticity. A key challenge when modelling such processes using stochastic random walk frameworks is that negative jump coefficients can arise when transport operators are discretized on heterogeneous domains. Often this is dealt with through homogenization approximations by replacing the heterogeneous medium with an $\textit{effective}$ homogeneous medium. In this work, we present a new class of homogenization approximations by considering a stochastic diffusive transport model on a one-dimensional domain containing an arbitrary number of layers with different jump rates. We derive closed form solutions for the $k$th moment of particle lifetime, carefully explaining how to deal with the internal interfaces between layers. These general tools allow us to derive simple formulae for the effective transport coefficients, leading to significant generalisations of previous homogenization approaches. Here, we find that different jump rates in the layers gives rise to a net bias, leading to a non-zero advection, for the entire homogenized system. Example calculations show that our generalized approach can lead to very different outcomes than traditional approaches, thereby having the potential to significantly affect simulation studies that use homogenization approximations.Comment: 9 pages, 2 figures, accepted version of paper published in The Journal of Chemical Physic

Rapid calculation of maximum particle lifetime for diffusion in complex geometries

Diffusion of molecules within biological cells and tissues is strongly influenced by crowding. A key quantity to characterize diffusion is the particle lifetime, which is the time taken for a diffusing particle to exit by hitting an absorbing boundary. Calculating the particle lifetime provides valuable information, for example, by allowing us to compare the timescale of diffusion and the timescale of reaction, thereby helping us to develop appropriate mathematical models. Previous methods to quantify particle lifetimes focus on the mean particle lifetime. Here, we take a different approach and present a simple method for calculating the maximum particle lifetime. This is the time after which only a small specified proportion of particles in an ensemble remain in the system. Our approach produces accurate estimates of the maximum particle lifetime, whereas the mean particle lifetime always underestimates this value compared with data from stochastic simulations. Furthermore, we find that differences between the mean and maximum particle lifetimes become increasingly important when considering diffusion hindered by obstacles.Comment: 10 pages, 1 figur

Accuracy of Intensity and Inclinometer Output of Three Activity Monitors for Identification of Sedentary Behavior and Light-Intensity Activity

Purpose. To examine the accuracy of intensity and inclinometer output of three physical activity monitors during various sedentary and light-intensity activities. Methods. Thirty-six participants wore three physical activity monitors (ActiGraph GT1M, ActiGraph GT3X+, and StepWatch) while completing sedentary (lying, sitting watching television, sitting using computer, and standing still) light (walking 1.0 mph, pedaling 7.0 mph, pedaling 15.0 mph) intensity activities under controlled settings. Accuracy for correctly categorizing intensity was assessed for each monitor and threshold. Accuracy of the GT3X+ inclinometer function (GT3X+Incl) for correctly identifying anatomical position was also assessed. Percentage agreement between direct observation and the monitor recorded time spent in sedentary behavior and light intensity was examined. Results. All monitors using all thresholds accurately identified over 80% of sedentary behaviors and 60% of light-intensity walking time based on intensity output. The StepWatch was the most accurate in detecting pedaling time but unable to detect pedal workload. The GT3X+Incl accurately identified anatomical position during 70% of all activities but demonstrated limitations in discriminating between activities of differing intensity. Conclusions. Our findings suggest that all three monitors accurately measure most sedentary and light-intensity activities although choice of monitors should be based on study-specific needs

Simplified models of diffusion in radially-symmetric geometries

We consider diffusion-controlled release of particles from $d$-dimensional radially-symmetric geometries. A quantity commonly used to characterise such diffusive processes is the proportion of particles remaining within the geometry over time, denoted as $P(t)$. The stochastic approach for computing $P(t)$ is time-consuming and lacks analytical insight into key parameters while the continuum approach yields complicated expressions for $P(t)$ that obscure the influence of key parameters and complicate the process of fitting experimental release data. In this work, to address these issues, we develop several simple surrogate models to approximate $P(t)$ by matching moments with the continuum analogue of the stochastic diffusion model. Surrogate models are developed for homogeneous slab, circular, annular, spherical and spherical shell geometries with a constant particle movement probability and heterogeneous slab, circular, annular and spherical geometries, comprised of two concentric layers with different particle movement probabilities. Each model is easy to evaluate, agrees well with both stochastic and continuum calculations of $P(t)$ and provides analytical insight into the key parameters of the diffusive transport system: dimension, diffusivity, geometry and boundary conditions.Comment: 22 pages, 3 figures, submitte

State and parameter estimation techniques for stochastic systems

This thesis documents research undertaken on state and parameter estimation techniques for stochastic systems in a maintenance context. Two individual problem scenarios are considered. For the first scenario, we are concerned with complex systems and the research involves an investigation into the ability to identify and quantify the occurrence of fault injection during routine preventive maintenance procedures. This is achieved using an appropriate delay time modelling specification and maximum-likelihood parameter estimation techniques. The delay time model of the failure process is parameterised using objective information on the failure times and the number of faults removed from the system during preventive maintenance. We apply the proposed modelling and estimation process to simulated data sets in an attempt to recapture specified parameters and the benefits of improving maintenance processes are demonstrated for the particular example. We then extend the modelling of the system in a predictive manner and combine it with a stochastic filtering approach to establish an adaptive decision model. The decision model can be used to schedule the subsequent maintenance intervention during the course of an operational cycle and can potentially provide an improvement on fixed interval maintenance policies. The second problem scenario considered is that of an individual component subject to condition monitoring such as, vibration analysis or oil-based contamination. The research involves an investigation into techniques that utilise condition information that we assume is related stochastically to the underlying state of the component, taken here to be the residual life. The techniques that we consider are the proportional hazards model and a probabilistic stochastic filtering approach. We investigate the residual life prediction capabilities of the two techniques and construct relevant replacement decision models. The research is then extended to consider multiple indicators of condition obtained simultaneously at monitoring points. We conclude with a brief investigation into the use of stochastic filtering techniques in specific scenarios involving limited computational power and variable underlying relationships between the monitored information and the residual life.EThOS - Electronic Theses Online ServiceGBUnited Kingdo