Dynamics of a Pair of Interacting Spins Coupled to an Environmental Sea

We solve for the dynamics of a pair of spins, coupled to each other and also to an environmental sea of oscillators. The environment mediates an indirect interaction between the spins, causing both mutual coherence effects and dissipation. This model describes a wide variety of physical systems, ranging from 2 coupled microscopic systems (eg., magnetic impurities, bromophores, etc), to 2 coupled macroscopic quantum systems. We obtain analytic results for 3 regimes, viz., (i) The locked regime, where the 2 spins lock together; (ii) The correlated relaxation regime (mutually correlated incoherent relaxation); and (iii) The mutual coherence regime, with correlated damped oscillations. These results cover most of the parameter space of the system.Comment: 49 pages, To appear in Int J. Mod. Phys.

Computational applications in stochastic operations research

Several computational applications in stochastic operations research are presented, where, for each application, a computational engine is used to achieve results that are otherwise overly tedious by hand calculations, or in some cases mathematically intractable. Algorithms and code are developed and implemented with specific emphasis placed on achieving exact results and substantiated via Monte Carlo simulation. The code for each application is provided in the software language utilized and algorithms are available for coding in another environment. The topics include univariate and bivariate nonparametric random variate generation using a piecewise-linear cumulative distribution, deriving exact statistical process control chart constants for non-normal sampling, testing probability distribution conformance to Benford\u27s law, and transient analysis of M/M/s queueing systems. The nonparametric random variate generation chapters provide the modeler with a method of generating univariate and bivariate samples when only observed data is available. The method is completely nonparametric and is capable of mimicking multimodal joint distributions. The algorithm is black-box, where no decisions are required from the modeler in generating variates for simulation. The statistical process control chart constant chapter develops constants for select non-normal distributions, and provides tabulated results for researchers who have identified a given process as non-normal The constants derived are bias correction factors for the sample range and sample standard deviation. The Benford conformance testing chapter offers the Kolmogorov-Smirnov test as an alternative to the standard chi-square goodness-of-fit test when testing whether leading digits of a data set are distributed according to Benford\u27s law. The alternative test has the advantage of being an exact test for all sample sizes, removing the usual sample size restriction involved with the chi-square goodness-of-fit test. The transient queueing analysis chapter develops and automates the construction of the sojourn time distribution for the nth customer in an M/M/s queue with k customers initially present at time 0 (k ≥ 0) without the usual limit on traffic intensity, rho \u3c 1, providing an avenue to conduct transient analysis on various measures of performance for a given initial number of customers in the system. It also develops and automates the construction of the sojourn time joint probability distribution function for pairs of customers, allowing the calculation of the exact covariance between customer sojourn times

Broken Ergodicity and 1/f1/f Noise from Finite, Local Entropy Baths

abstract: Fluctuations with a power spectral density depending on frequency as $1/f^\alpha$ ($0<\alpha<2$) are found in a wide class of systems. The number of systems exhibiting $1/f$ noise means it has far-reaching practical implications; it also suggests a possibly universal explanation, or at least a set of shared properties. Given this diversity, there are numerous models of $1/f$ noise. In this dissertation, I summarize my research into models based on linking the characteristic times of fluctuations of a quantity to its multiplicity of states. With this condition satisfied, I show that a quantity will undergo $1/f$ fluctuations and exhibit associated properties, such as slow dynamics, divergence of time scales, and ergodicity breaking. I propose that multiplicity-dependent characteristic times come about when a system shares a constant, maximized amount of entropy with a finite bath. This may be the case when systems are imperfectly coupled to their thermal environment and the exchange of conserved quantities is mediated through their local environment. To demonstrate the effects of multiplicity-dependent characteristic times, I present numerical simulations of two models. The first consists of non-interacting spins in $0$-field coupled to an explicit finite bath. This model has the advantage of being degenerate, so that its multiplicity alone determines the dynamics. Fluctuations of the alignment of this model will be compared to voltage fluctuations across a mesoscopic metal-insulator-metal junction. The second model consists of classical, interacting Heisenberg spins with a dynamic constraint that slows fluctuations according to the multiplicity of the system's alignment. Fluctuations in one component of the alignment will be compared to the flux noise in superconducting quantum interference devices (SQUIDs). Finally, I will compare both of these models to each other and some of the most popular models of $1/f$ noise, including those based on a superposition of exponential relaxation processes and those based on power law renewal processes.Dissertation/ThesisDoctoral Dissertation Physics 201

Local geometry of random geodesics on negatively curved surfaces

It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting Poisson line process.Comment: This version extends the results of the previous version to surfaces with possibly variable negative curvatur

Markov chain models of instantaneously coupled intracellular calcium channels

Localized calcium elevations known as calcium puffs or sparks are cellular signals arising from cooperative activity of clusters of inositol 1,4,5-trisphosphate receptors (IP3Rs) or ryanodine receptors (RyRs) located at calcium release sites on the endoplasmic or sarcoplasmic reticulum membrane. When Markov chain models of these intracellular calcium-regulated calcium channels are coupled via a mathematical representation of the calcium microdomain, simulated calcium release sites may exhibit the phenomenon of stochastic calcium excitability where the IP3Rs or RyRs open and close in a concerted fashion. Although the biophysical theory relating the kinetics of single channels to the collective phenomena of puffs and sparks is only beginning to be developed, Markov chain models of coupled intracellular channels give insight into the dynamics of calcium puffs and sparks.;Interestingly, under some conditions simulated puffs and sparks can be observed even when the single channel model used does not include slow calcium inactivation or any long-lived closed state. In this case termination of the localized calcium elevation occurs when all of the intracellular channels at a release site simultaneously close through a process called stochastic attrition. This dissertation investigates the statistical properties of stochastic attrition viewed as an absorption time on a terminating Markov chain that represents a calcium release site composed of two-state channels that are activated by calcium. Assuming that the local calcium concentration experienced by a channel depends only on the number of open channels at the calcium release site, the probability distribution function for the time until stochastic attrition occurs is derived and an analytical formula for the expectation of this random variable is presented. Also explored is how the contribution of stochastic attrition to the termination of calcium puffs and sparks depends on the number of channels at a release site, the source amplitude of the channels, the background calcium concentration, channel kinetics, and the cooperativity of calcium binding.;This dissertation also studies whether single channel models with calcium inactivation are less sensitive to the details of release site ultrastructure than models that lack a slow calcium-inactivation process. Release site dynamics obtained from simulated calcium release sites composed of instantaneously coupled calcium-regulated calcium channels whose random spatial locations were chosen from a uniform distribution on a disc of specified radius are compared to simulations with channels arranged on hexagonal lattices. Analysis of puff/spark statistics confirms that puffs and sparks are less sensitive to the spatial organization of release sites when the single channel model includes a slow inactivation process. The validity of several different mean-field reductions that do not explicitly account for the details of release site ultrastructure is also investigated.;Calcium release site models are stochastic automata networks that involve many functional transitions, that is, the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. A Kronecker structured representation for calcium release site models is presented and benchmark stationary distribution calculations using both exact and approximate iterative numerical solution techniques that leverage this structure are performed. When it is possible to obtain an exact solution, response measures such as the number of channels in a particular state converge more quickly using the iterative numerical methods than occupation measures calculated via Monte Carlo simulation. When an exact solution is not feasible, iterative approximate methods based on the Power method may be used, with performance similar to Monte Carlo estimates

Stochastic models of ion channels

This thesis is concerned with models and inference for single ion channels. Molecular modelling studies are used as the basis for biologically realistic, large state-space gating models of the nicotinic acetylcholine receptor which enable single-channel kinetic behaviour to be characterized in terms of a small number of free parameters. A model is formulated which incorporates known structural information concerning pentameric subunit composition, interactions between neighbouring subunits and knowledge of the behaviour of agonist binding sites within the receptor-channel proteins. Expressions are derived for various channel properties and results are illustrated using numerical examples. The model is adapted and extended to demonstrate how properties of the calcium ion-activated potassium ion channel may be modelled. A two-state stochastic model for ion channels which incorporates time interval omission is examined. Two new methods for overcoming a non-identifiability problem induced by time interval omission are introduced and simulation studies are presented in support of these methods. A framework is presented for analysing the asymptotic behaviour of the method-of-moments estimators of the mean lengths of open and closed sojourns. This framework is used to clarify the origin of the non-identifiability and to construct confidence sets for the mean sojourn lengths. A conjecture concerning the number of solutions of the moment estimating equations is proved