In this thesis, I propose a general stochastic simulation algorithm for particle based lattice model using the concepts of Gillespie's stochastic simulation algorithm, which was originally designed for well-stirred systems. I describe the details about this method and analyze its complexity compared with the StochSim algorithm, another simulation algorithm originally proposed to simulate stochastic lattice model. I compare the performance of both algorithms with application to two different examples: the May-Leonard model and Ziff-Gulari-Barshad model. Comparison between the simulation results from both algorithms has validate our claim that our new proposed algorithm is comparable to the StochSim in simulation accuracy. I also compare the efficiency of both algorithms using the CPU cost of each code and conclude that the new algorithm is as efficient as the StochSim in most test cases, while performing even better for certain specific cases. / Computer simulation has been developed for almost one century. Stochastic lattice model, which follows the physics concept of lattice, is defined as a kind of system in which individual entities live on grids and demonstrate certain random behaviors according to certain specific rules. It is mainly studied using computer simulations. The most widely used simulation method to for stochastic lattice systems is the StochSim algorithm, which just randomly pick an entity and then determine its behavior based on a set of specific random rules. Our goal is to develop new simulation methods so that it is more convenient to simulate and analyze stochastic lattice system. In this thesis I propose another type of simulation methods for the stochastic lattice model using totally different concepts and procedures. I developed a simulation package and applied it to two different examples using both methods, and then conducted a series of numerical experiment to compare their performance. I conclude that they are roughly equivalent and our new method performs better than the old one in certain special cases.

This report combines a series of papers to trace progression in the area of type-2 topoisomerase research. First, Deibler et al. show that knotted DNA is harmful to cells. Knots can block both transcription and replication, and can also act as a catalyst for mutation. Despite the fact that type-2 topoisomerases perform the important functions of unknotting and decatenating DNA, the mechanism by which they accomplish this task is still unknown. Buck and Zechiedrich propose a model in which the enzyme uses local geometry to infer global topology, and thus where to perform segment passage in order to obtain the desired results. In two articles, Liu et al. evaluate this theory quantitatively for the decatenation and unknotting problems. In both cases it is shown that the presence of certain juxtapositions is strongly correlated with global topology. This correlation is not enough, however, and Liu et al. go on to show that when segment passage operations designed to mimic type-2 topoisomerase action are performed at hooked juxtapositions, the overwhelming tendency is towards unknotting and decatenation. / text

One of the most important problems in statistical mechanics is the measurement of free energies, these being the quantities that determine the direction of chemical reactions and--the concern of this thesis--the location of phase transitions. While Monte Carlo (MC) computer simulation is a well-established and invaluable aid in statistical mechanical calculations, it is well known that, in its most commonly-practised form (where samples are generated from the Boltzmann distribution), it fails if applied directly to the free energy problem. This failure occurs because the measurement of free energies requires a much more extensive exploration of the system's configuration space than do most statistical mechanical calculations: configurations which have a very low Boltzmann probability make a substantial contribution to the free energy, and the important regions of configuration space may be separated by potential barriers. We begin the thesis with an introduction, and then give a review of the very substantial literature that the problem of the MC measurement of free energy has produced, explaining and classifying the various different approaches that have been adopted. We then proceed to present the results of our own investigations. First, we investigate methods in which the configurations of the system are sampled from a distribution other than the Boltzmann distribution, concentrating in particular on a recently developed technique known as the multicanonical ensemble. The principal difficulty in using the multicanonical ensemble is the difficulty of constructing it: implicit in it is at least partial knowledge of the very free energy that we are trying to measure, and so to produce it requires an iterative process. Therefore we study this iterative process, using Bayesian inference to extend the usual method of MC data analysis, and introducing a new MC method in which inferences are made based not on the macrostates visited by the simulation but on the transitions made between them. We present a detailed comparison between the multicanonical ensemble and the traditional method of free energy measurement, thermodynamic integration, and use the former to make a high-accuracy investigation of the critical magnetisation distribution of the 2d Ising model from the scaling region all the way to saturation. We also make some comments on the possibility of going beyond the multicanonical ensemble to `optimal' MC sampling. Second, we investigate an isostructural solid-solid phase transition in a system consisting of hard spheres with a square-well attractive potential. Recent work, which we have confirmed, suggests that this transition exists when the range of the attraction is very small (width of attractive potential/ hard core diameter ~ 0.01). First we study this system using a method of free energy measurement in which the square-well potential is smoothly transformed into that of the Einstein solid. This enables a direct comparison of a multicanonical-like method with thermodynamic integration. Then we perform extensive simulations using a different, purely multicanonical approach, which enables the direct connection of the two coexisting phases. It is found that the measurement of transition probabilities is again advantageous for the generation of the multicanonical ensemble, and can even be used to produce the final estimators. Some of the work presented in this thesis has been published or accepted for publication: the references are G. R. Smith & A. D. Bruce, A Study of the Multicanonical Monte Carlo Method, J. Phys. A. 28, 6623 (1995). [reference details doi:10.1088/0305-4470/28/23/015] G. R. Smith & A. D. Bruce, Multicanonical Monte Carlo Study of a Structural Phase Transition, to be published in Europhys. Lett. [reference details Europhys. Lett. 34, 91 (1996) doi:10.1209/epl/i1996-00421-1] G. R. Smith & A. D. Bruce, Multicanonical Monte Carlo Study of Solid-Solid Phase Coexistence in a Model Colloid, to be published in Phys. Rev. E [reference details Phys. Rev. E 53, 6530–6543 (1996) doi:10.1103/PhysRevE.53.6530].

Abstract The softening effects of ferromagnetic magnon on some ferromagnetic semiconductors and colossal magnetoresistance manganites have attracted much attention. Such effect can be calculated from the single orbital ferromagnetic Kondo lattice model in proper conducting carrier numbers utilizing the equation of motion method with one magnon excitation and Random Phase approximations. However, if we take into account the Coulomb repulsion and use the Gutzwiller projection method to transfer this repulsion force to conducting bandwidth modulation, the softening effect disappear. This paper describes qualitively the effect of softening on properties of different colossal manganites.

The presented work highlights the remarkable potential of physical discretization – lattice model FyDiK in three-dimensional modelling of non-linear problems in structural mechanics. To achieve the objectives a software application, that implements the model FyDiK along with the 3D graphical user interface has been developed and thus is able to assemble a spring network model. Such a model was used for modelling the formation of cracks and fracture in the concrete specimens and also to model a plastic behaviour of steel I-beam. The calculations were performed by a massive parallelization on CUDA platform. In the first part the basic principles on which the work is based are introduced. Subsequently, a detailed description of individual parts of the model and the issue of parallelization by graphics cards are presented. In the next part the creation of the required software and improving of the model properties of mentioned materials are described. That is followed by evaluation of the achieved results with the comparison of other modelling software. The conclusion summarizes the achievements and suggestions for the further development possibilities of the presented method of modelling.

The methods of statistical physics are increasingly being employed in a range of interdisciplinary areas. In particular, aspects of complex biological processes have been elucidated by bringing the problems down to the level of simple interactions studied in a statistical sense. In nonequilibrium statistical physics, a one dimensional lattice model known as the totally asymmetric simple exclusion processes (TASEP) has become prominent as a tool for modelling various cellular transport processes. Indeed the context in which the TASEP was first introduced (MacDonald et. al., 1968) was to model ribosome motion along mRNA during protein synthesis. In this work I study a variation of the TASEP in which particles hop along a one dimensional lattice which extends as they reach the end. We introduce this model to describe the unique growth dynamics of filamentous fungi, whereby a narrow fungal filament extends purely from its tip region while being supplied with growth materials from behind the tip. We find that the steady state behaviour of our model reflects that of the TASEP, however there is an additional phase where a dynamic shock is present in the system. I show through Monte Carlo simulation and theoretical analysis that the qualitative behaviour of this model can be predicted with a simple mean-field approximation, while the details of the phase behaviour are accurate only in a refined approximation which takes into account some correlations. I also discuss a further refined mean-field approximation and give a heuristic argument for our results. Next I present an extension of the model which allows the particles to interact with a second lattice, on which they diffuse in either direction. A first order meanfield continuum approximation suggests that the steady states of this system will exhibit some novel behaviour. Through Monte Carlo simulation I discuss the qualitative changes that arise due to the on-off dynamics. Finally I study a model for a second biological phenomenon: the length fluctuations of microtubules. The model describes stochastic polymerisation events at the tip of a microtubule. Using a mean-field theory, we find a transition between regimes where the microtubule grows on average, and where the length remains finite. For low rates of polymerisation and depolymerisation, the transition is in good agreement with Monte Carlo simulation.

We present a novel modelling approach for the morphology formation from ternary mixtures upon evaporation, which takes into consideration two different length scales of the interaction, and further allows for these length scales to be altered. A quantitative study of the interfacial energy hints towards the existence of a mesoscopic regime - further research is needed to verify the validity of this claim however. We also demonstrate that the solvent initially follows a Fickian law of diffusion, then deviates from this behaviour, presumably due to the phase separated regions produced by the two remaining (active) components. We also attempt to bridge the gap between this work and a hypothetical three-dimensional model by considering a top-down view of the system. Here, we observe domain growth dominated by Ostwald ripening, with some coalescence. The domain growth was further characterised using Fourier image analysis. / Vi presenterar en ny modellansats för morfologiformation från trekomponentsblandningar under avdunstning, som tar hänsyn till två olika längdskalor hos interaktionen, samt möjliggör förändring av dessa längdskalor. En kvantitativ studie av energin vid domängränserna tyder på att det finns en mesoskopisk regim - ytterligare forskning är dock nödvändig för att verifiera giltigheten av detta påstående. Vi visar också att lösningsmedlet ursprungligen följer en Fickiansk diffusionslag, för att senare avvika från detta beteende, förmodligen på grund av de tydliga domänerna som produceras av de två återstående (aktiva) komponenterna. Vi försöker också minska klyftan mellan det här arbetet och en hypotetisk tredimensionell modell genom att behandla systemet uppifrån. Här observerar vi domäntillväxt dominerad av 'Ostwald ripening', med viss koalescens. Domäntillväxten karakteriserades vidare med hjälp av Fourier-bildanalys.

No description available.

Lookback option is a well-known path-dependent option where its payoff depends on the historical extremum prices. The thesis focuses on the binomial pricing of the American floating strike lookback put options with payoff at time $t$ (if exercise) characterized by \[ \max_{k=0, \ldots, t} S_k - S_t, \] where $S_t$ denotes the price of the underlying stock at time $t$. Build upon the idea of \hyperlink{RBCV}{Reiner Babbs Cheuk and Vorst} (RBCV, 1992) who proposed a transformed binomial lattice model for efficient pricing of this class of option, this thesis extends and enhances their binomial recursive algorithm by exploiting the additional combinatorial properties of the lattice structure. The proposed algorithm is not only computational efficient but it also significantly reduces the memory constraint. As a result, the proposed algorithm is more than 1000 times faster than the original RBCV algorithm and it can compute a binomial lattice with one million time steps in less than two seconds. This algorithm enables us to extrapolate the limiting (American) option value up to 4 or 5 decimal accuracy in real time.

No description available.