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For systems that are in equilibrium, fluctuations can be understood through interactions with external heat reservoirs. For this reason these fluctuations are known as thermal noise, and they usually become vanishingly small in the thermodynamic limit. However, many systems comprising interacting constituents studied by physicists in recent years are both far from equilibrium, and sufficiently small so that they must be considered finite. The finite number of constituents gives rise to an inherent demographic noise in the system, a source of fluctuations that is always present in the stochastic dynamics. This thesis investigates the role of stochastic fluctuations in the macroscopically observable dynamical behaviour of non-equilibrium, finite systems. To facilitate such a study, we construct microscopic models using an individual based modelling approach, allowing the explicit form of the demographic noise to be identified. In many physical systems and theoretical models, absorbing states are a defining feature. Once a system enters one, it cannot leave. We study the dynamics of a system with two symmetric absorbing states, finding that the amplitude of the multiplicative noise can induce a transition between two universal modes of domain coarsening as the system evolves to one of the absorbing states. In biological and ecological systems, cycles are a ubiquitously observed phenomenon, but are di cult to predict analytically from stochastic models. We examine a potential mechanism for cycling behaviour due to the flow of probability currents, induced by the athermal nature of the demographic noise, in a single patch population comprising two competing species. We find that such a current by itself cannot generate macroscopic cycles, but when combined with deterministic dynamics which constrain the system to a closed circular manifold, gives rise to global quasicycles in the population densities. Finally, we examine a spatially extended system comprising many such patch populations, exploring the emergence of synchronisation between the different cycles. By a stability analysis of the global synchronised state, we probe the relationship between the synchronicity of the metapopulation and the magnitude of the coupling between patches due to species migration. In all cases, we conclude that the nature of the demographic noise can play a pivotal role in the macroscopically observed dynamical behaviour of the system.

This thesis addresses one area of major interest in reaction kinetics, the theoretical description of recombination in nonhomogeneous systems. The reaction of the highly reactive particles formed by the passage of ionising radiation through a medium is an important example of this type of system. Stochastic effects are apparent in every stage of the development of a radiolysis system: in the interaction between the radiation and the medium and in the diffusion and reaction processes that follow. Conventional models for nonhomogeneous kinetics in radiation chemistry are based upon a deterministic analysis. These models are appraised together with an alternative stochastic approach. Three stochastic methods are discussed: a full Monte Carlo simulation of the diffusion and reaction and two approximate models based upon an independent pairs approximation. It is important that any kinetic model accurately describes the system it purports to represent and this can only be assured by extensive validation. The stochastic models are developed to encompass the diffusion-controlled reactions of ions and radicals and to include the effects of a bulk electrolyte upon the reactions between ions. To model a realistic radiation chemistry reaction scheme, it is necessary to consider reactions that are not fully diffusion-controlled. The radiation boundary condition is introduced and used to extend stochastic modelling to partially diffusion-controlled reactions. Recombination in an anisotropic environment is also considered. Although a complete analysis of the chemistry of a radiolysis system requires a complex reaction scheme, the scheme can be simplified, in acid and in alkali, by the use of an instantaneous scavenging approximation. In acid, this approximation produces a simple three reaction mechanism based upon five species: H, OH, H₂ , H₂0 and H₂0₂ . The acid system is used to demonstrate the stochastic treatment of realistic kinetics. The instantaneous scavenging approximation is examined in detail and techniques are developed for the explicit modelling of reactions with a homogeneously distributed scavenger. A stochastic treatment of nonhomogeneous reaction kinetics requires a description of the initial spatial distribution of the reacting particles. A rudimentary Monte Carlo simulation is used to determine a simple distribution of clusters of reactive particles similar to that found along the path of a high energy electron in water. This distribution provides a suitable basis for kinetic simulation. The kinetics of a more detailed idealised track structure are also considered and the stochastic and deterministic kinetics of extended particle distributions are discussed.

1 Title: Statistical Physics of Frustrated Evolutionary Games Author: Miroslav Pištěk Department: Institute of Theoretical Physics Supervisor: RNDr. František Slanina, CSc. Supervisor's e-mail address: slanina@fzu.cz Abstract: In last two decades, the effort devoted to interdisciplinary research of bounded sources allocation is growing, examining complex phenomena as stock markets or traffic jams. The Minority Game is a multiple-agent model of inevitable frus- tration arising in such situations. It is analytically tractable using the replica method originated in statistical physics of spin glasses. We generalised the Mi- nority Game introducing heterogenous agents. This heterogeneity causes a con- siderable decrease of an average agent's frustration. For many configurations, we achieve even a positive-sum game, which is not possible in the original game variant. This result is in accordance with real stock market data. Keywords: frustrated evolutionary games, Minority Game, Replica method

Stochastic portfolio theory is a novel methodology, developed by Fernholz (2002), for analysing stock and portfolio behaviour, and equity market structure, constructing portfolios and understanding the structure of equity markets. It thus has immediate applications to equity portfolio management and performance measurement. This theory successfully generalises well-known models for the stock price to provide models for portfolios and markets, leading to a better and more precise understanding of equity market structure. The aim of this dissertation is to present an exhaustive review of stochastic portfolio theory by imitating the work done and contributions made by Fernholz (2002) thus far. A detailed discussion of stochastic portfolio theory as well as how the implications di er from the conclusions and results of classic portfolio theory will be provided. In this dissertation, we will undertake a thorough investigation into stochastic portfolio theory; by focusing on the central, innovative ideas of the excess growth rate, long-term stock market and portfolio behaviour, stock market diversity of equity markets, portfolio generating functions, the concept of how to select stocks by their rank and the existence of relative arbitrage opportunities within the context of stochastic portfolio theory. Thus, we shall review the central concepts of stochastic portfolio theory, this will include a detailed explanation of the excess growth rate, long-term behaviour of portfolios, stock market diversity, portfolio generating functions and stocks selected by rank. We will also present examples of portfolios and markets with a wide variety of di erent properties. We will also show how this new and fast-evolving theory can be applied, in particular, to equity management, by considering the performance of certain functionally generated portfolios. Furthermore, several results and implications of stochastic portfolio theory will be discussed, and in this dissertation, we shall examine these results in far greater depth. Keywords and Phrases: Stochastic portfolio theory, Portfolios, Stock market and portfolio behaviour, Stock market diversity, Portfolio generating functions, Functionally generated portfolios, Rank-dependent portfolio generating functions, Local time, Relative arbitrage opportunities, Performance of functionally generated portfolios.

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Wong Man Hong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 60-63). / Abstracts in English and Chinese. / Abstract --- p.i / Abstract in Chinese --- p.ii / Thesis Assessment Committee --- p.iii / Acknowledgement --- p.iv / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Our Work and Motivation --- p.1 / Chapter 1.2 --- Literature Review --- p.3 / Chapter 1.3 --- Thesis Structure --- p.5 / Chapter 2 --- Preliminary --- p.6 / Chapter 2.1 --- Calculus for Volume of Sphere --- p.6 / Chapter 2.2 --- Fractional Programming and Dinkelbach's Algorithm --- p.7 / Chapter 2.3 --- Nonlinear Programming and Interior Point Algorithm --- p.8 / Chapter 2.4 --- Second Order Cones and Conic Programming --- p.10 / Chapter 3 --- The Probability Model --- p.12 / Chapter 3.1 --- Derive the Chance Constraint --- p.12 / Chapter 3.2 --- Single Cluster Model --- p.18 / Chapter 3.3 --- Multi-clusters Model --- p.21 / Chapter 4 --- The Downside Risk Model --- p.24 / Chapter 4.1 --- Derive the Downside Risk Measure --- p.24 / Chapter 4.2 --- Calculate the First and Second Derivative of the Downside Risk --- p.27 / Chapter 4.3 --- Single Cluster Model and Numerical Algorithm --- p.29 / Chapter 4.4 --- Multi-clusters Model --- p.34 / Chapter 5 --- The Conditional Value-at-Risk Model --- p.37 / Chapter 5.1 --- Derive the Conditional Value at Risk --- p.37 / Chapter 5.2 --- Single Cluster Model and Numerical Algorithm --- p.41 / Chapter 5.3 --- Multi-clusters Model --- p.47 / Chapter 6 --- Numerical Results --- p.51 / Chapter 6.1 --- Data Set --- p.51 / Chapter 6.2 --- The Probability Model --- p.53 / Chapter 6.3 --- The Downside Risk Model --- p.53 / Chapter 6.4 --- The CVaR Model --- p.56 / Chapter 7 --- Conclusions --- p.58 / Bibliography --- p.60