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The goal of this thesis is to contribute to the ambitious program of the foundation of developing statistical physics using chaos. We build a deterministic model of Brownian motion and provide a microscpoic derivation of the Fokker-Planck equation. Since the Brownian motion of a particle is the result of the competing processes of diffusion and dissipation, we create a model where both diffusion and dissipation originate from the same deterministic mechanism - the deterministic interaction of that particle with its environment. We show that standard diffusion which is the basis of the Fokker-Planck equation rests on the Central Limit Theorem, and, consequently, on the possibility of deriving it from a deterministic process with a quickly decaying correlation function. The sensitive dependence on initial conditions, one of the defining properties of chaos insures this rapid decay. We carefully address the problem of deriving dissipation from the interaction of a particle with a fully deterministic nonlinear bath, that we term the booster. We show that the solution of this problem essentially rests on the linear response of a booster to an external perturbation. This raises a long-standing problem concerned with Kubo's Linear Response Theory and the strong criticism against it by van Kampen. Kubo's theory is based on a perturbation treatment of the Liouville equation, which, in turn, is expected to be totally equivalent to a first-order perturbation treatment of single trajectories. Since the boosters are chaotic, and chaos is essential to generate diffusion, the single trajectories are highly unstable and do not respond linearly to weak external perturbation. We adopt chaotic maps as boosters of a Brownian particle, and therefore address the problem of the response of a chaotic booster to an external perturbation. We notice that a fully chaotic map is characterized by an invariant measure which is a continuous function of the control parameters of the map. Consequently if the external perturbation is made to act on a control parameter of the map, we show that the booster distribution undergoes slight modifications as an effect of the weak external perturbation, thereby leading to a linear response of the mean value of the perturbed variable of the booster. This approach to linear response completely bypasses the criticism of van Kampen. The joint use of these two phenomena, diffusion and friction stemming from the interaction of the Brownian particle with the same booster, makes the microscopic derivation of a Fokker-Planck equation and Brownian motion, possible.

This thesis consists of three self-contained parts, each with its own abstract, body, references and page numbering. Part I, 'Potential theory, path integrals and the Laplacian of the indicator', finds the transition density of absorbed or reflected Brownian motion in a d-dimensional domain as a Feynman-Kac functional involving the Laplacian of the indicator, thereby relating the hitherto unrelated fields of classical potential theory and path integrals. Part II, 'The problem of alternatives', considers parallel investment in alternative technologies or drugs developed over time, where there can be only one winner. Parallel investment accelerates the search for the winner, and increases the winner's expected performance, but is also costly. To determine which candidates show sufficient performance and/or promise, we find an integral equation for the boundary of the optimal continuation region. Part III, 'Optimal support for renewable deployment', considers the role of government subsidies for renewable technologies. Rapidly diminishing subsidies are cheaper for taxpayers, but could prematurely kill otherwise successful technologies. By contrast, high subsidies are not only expensive but can also prop up uneconomical technologies. To analyse this trade-off we present a new model for technology learning that makes capacity expansion endogenous. There are two reasons for this standalone structure. First, the target readership is divergent. Part I concerns mathematical physics, Part II operations research, and Part III policy. Readers interested in specific parts can thus read these in isolation. Those interested in the thesis as a whole may prefer to read the three introductions first. Second, the separate parts are only partially interconnected. Each uses some theory from the preceding part, but not all of it; e.g. Part II uses only a subset of the theory from Part I. The quickest route to Part III is therefore not through the entirety of the preceding parts. Furthermore, those instances where results from previous parts are used are clearly indicated.

Call number: LD2668 .T4 1985 C49 / Master of Science

After an introduction to Brownian motion, Hausdorff dimension, nonstandard analysis and Loeb measure theory, we explore the notion of a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, we find that Hausdorff dimension can be computed through a counting argument rather than the traditional way. This formulation is then applied to obtain simple proofs of some of the dimensional properties of Brownian motion, such as the doubling of the dimension of a set of dimension smaller than 1/2 under Brownian motion, by utilising Anderson's formulation of Brownian motion as a hyperfinite random walk. We also use the technique to refine a theorem of Orey and Taylor's on the Hausdorff dimension of the rapid points of Brownian motion. The result is somewhat stronger than the original. Lastly, we give a corrected proof of Kaufman's result that the rapid points of Brownian motion have similar Hausdorff and Fourier dimensions, implying that they constitute a Salem set. / Mathematical Sciences / D. Phil. (Mathematical Sciences)

This dissertation presents and explores a technique to confine and manipulate single and multiple nano-objects in solution by exploiting the thermophoretic interactions with local temperature gradients. The method named thermophoretic trap uses an all-optically controlled heating via plasmonic absorption by a gold nano-structure designed for this purpose. The dissipation of absorbed laser light to thermal energy generates a localized temperature field. The spatial localization of the heat source thereby leads to strong temperature gradients that are used to drive a particle or molecule into a desired direction. The behavior of nano-objects confined by thermal inhomogeneities is explored experimentally as well as theoretically. The monograph treats three major experimental stages of development, which essentially differ in the way the heating laser beam is shaped and controlled. In a first generation, a static heating of an appropriate gold structure is used to induce a steady temperature profile that exhibits a local minimum in which particles can be confined. This simple realization illustrates the working principle best. In a second step, the static heating is replaced. A focused laser beam is used to heat a smaller spatial region. In order to confine a particle, the beam is steered in circles along a circular gold structure. The trapping dynamics are studied in detail and reveal similarities to the well-established Paul trap. The largest part of the thesis is dedicated to the third generation of the trap. While the hardware is identical to the second generation, using the real-time information on the position of the trapped object to heat only particular sites of the gold structure strongly increases the efficiency of the trap compared to the earlier versions. Beyond that, the optical feedback control allows for an active shaping of the effective virtual trapping potential by applying modified feedback rules, including e.g. a double-well or a box-like potential. This transforms the formerly pure trapping device to a versatile technique for micro and nano-fluidic manipulation. The physical and technical contributions to the limits of the method are explored. Finally, the feasibility of trapping single macro-molecules is demonstrated by the confinement of lambda-DNA for extended time periods over which the molecules center-of-mass motion as well as its conformational dynamics can be studied.

Texas’ Mass Emission Cap and Trade program is a mandatory Nitrous Oxide (NOx) abatement program for medium and large stationary sources located in the Houston-Galveston ozone non-attainment area. Effected companies are required to upgrade equipment to meet the current best achievable NOx control technology (BACT) standards or to purchase emission credits in sufficient quantity to cover the difference in emissions between existing equipment and equipment meeting the BACT standard. With over 260 participating companies, the market for emission credits is ever changing, making it difficult to evaluate whether the lowest cost decision is to upgrade equipment or to purchase NOx emission credits. Because equipment upgrades are capital investments, a well informed, rational decision can have a significant impact on the corporate balance sheet. The objective of this research is to aid the decision maker by predicting credit prices based on a Geometric Brownian Motion model based on historical NOx emission credit transactions. The predicted credit price is useful in evaluating the likelihood of the equipment upgrade option being a favorable or unfavorable decision. For the examined cases, modeled results indicate that equipment upgrade is the more cost effective option. / text

Diffusion processes play an important role in describing systems in many fields of science, as in physics, biology, finance and social science. One of the most famous examples of the diffusion process is the Brownian motion.    At mesoscopic scale, the Brownian theory describes the very irregular and animated motion of a particle suspended in a fluid. In this thesis, the dynamics and thermodynamics of diffusion processes driven out of equilibrium, at mesoscopic scale, are investigated.    For dynamics, the theory of Brownian motion for a particle which is able to rotate and translate in three dimensions is presented.  Moreover, it is presented how to treat diffusion process on n-dimensional Riemann manifolds defining the Kolmogorov forward equation on such manifold.   For thermodynamics, this thesis describes how to define thermodynamics quantities at mesoscopic scale using the tools of Brownian theory. The theory of stochastic energetics and how to compute entropy production along a trajectory are presented introducing the new field of stochastic thermodynamics. Moreover, the "anomalous entropy production" is introduced. This anomaly in the entropy production arises when diffusion processes are driven out of equilibrium by space dependent temperature field. The presence of this term expresses the fallacy of the overdamped approximation in computing thermodynamic quantities.    In the first part of the thesis the translational and rotational motion of an ellipsoidal particle in a heterogeneous thermal environment, with a space-dependent temperature field, is analyzed from the point of view of stochastic thermodynamics.    In the final part of the thesis, the motion of a Brownian rigid body three-dimensional space in a homogeneous thermal environment under the presence of an external force field is analyzed, using multiscale method and homogenization. / <p>QC 20170515</p>

Diffusion processes provide a natural way of modelling a variety of physical and economic phenomena. It is often the case that one is unable to observe a diffusion process directly, and must instead rely on noisy observations that are discretely spaced in time. Given these discrete, noisy observations, one is faced with the task of inferring properties of the underlying diffusion process. For example, one might be interested in inferring the current state of the process given observations up to the present time (this is known as the filtering problem). Alternatively, one might wish to infer parameters governing the time evolution the diffusion process. In general, one cannot apply Bayes’ theorem directly, since the transition density of a general nonlinear diffusion is not computationally tractable. In this thesis, we investigate a novel method of simplifying the problem. The stochastic differential equation that describes the diffusion process is replaced with a simpler ordinary differential equation, which has a random driving noise that approximates Brownian motion. We show how one can exploit this approximation to improve on standard methods for inferring properties of nonlinear diffusion processes.

This dissertation examines the pricing of the same credit risk in two currencies, and hence the valuation of credit-contingent foreign exchange products. Such pricing hinges upon the dependence of the credit risk and the foreign exchange rate. We recall the reduced-form model proposed by Ehlers (2007), which allows credit-currency dependence through correlation between the Brownian motions driving the default intensity and the exchange rate, and through a jump in the exchange rate at the default time. Four basic specifications of this model are considered. Two of these specifications have not previously appeared in the literature and one of these, based on a lognormal process for the default intensity, proves to be especially useful and tractable. The problem of hedging defaultable claims in one currency with similar claims in another is briefly considered, and it is shown that hedging against the default event and against credit spread movements are not in general equivalent.