Browsing by Subject "equations"
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Item Application Of The Cubed-Sphere Grid To Tilted Black Hole Accretion Disks(2009-01) Fragile, P. Chris; Lindner, Christopher C.; Anninos, Peter; Salmonson, Jay D.; Lindner, Christopher C.In recent work we presented the first results of global general relativistic magnetohydrodynamic (GRMHD) simulations of tilted (or misaligned) accretion disks around rotating black holes. The simulated tilted disks showed dramatic differences from comparable untilted disks, such as asymmetrical accretion onto the hole through opposing "plunging streams" and global precession of the disk powered by a torque provided by the black hole. However, those simulations used a traditional spherical-polar grid that was purposefully under-resolved along the pole, which prevented us from assessing the behavior of any jets that may have been associated with the tilted disks. To address this shortcoming we have added a block-structured "cubed-sphere" grid option to the Cosmos++ GRMHD code, which will allow us to simultaneously resolve the disk and polar regions. Here we present our implementation of this grid and the results of a small suite of validation tests intended to demonstrate that the new grid performs as expected. The most important test in this work is a comparison of identical tilted disks, one evolved using our spherical-polar grid and the other with the cubed-sphere grid. We also demonstrate an interesting dependence of the early-time evolution of our disks on their orientation with respect to the grid alignment. This dependence arises from the differing treatment of current sheets within the disks, especially whether or not they are aligned with symmetry planes of the grid.Item Construction of coarse-grained order parameters in nonequilibrium systems(2009-06) Reynolds, David E.; Reynolds, David E.We develop a renormalization-group (RG) procedure that includes important system-specific features. The key ingredient is to systematize the coarse-graining procedure that generates the RG flow. The coarse-graining technology comes from the control and operator theoretic model reduction. The resulting "generalized" RG is a consistent generalization of the Wilsonian RG. We apply the procedure to a deterministic nonlinear wave equation (NLWE) with probabilistic initial conditions. We derive the form of the projection operator from the dynamics of the NLWE and then use it to generate the RG flow for the distribution of initial conditions. The probability density of the initial conditions is chosen to be a Boltzmann weight that is quartic in the field variables. In our calculation, we find that in contrast to conventional implementations of the RG, naive power counting breaks down. We also show that the resulting RG equations are different from those derived from the conventional RG.Item Excess-entropy scaling of dynamics for a confined fluid of dumbbell-shaped particles(2010-10) Chopra, Ravi; Truskett, Thomas M.; Errington, Jeffrey R.; Truskett, Thomas M.We use molecular simulation to study the ability of excess entropy scaling relationships to describe the kinetic properties of a confined molecular system. We examine a model for a confined fluid consisting of dumbbell-shaped molecules that interact with atomistically detailed pore walls via a Lennard-Jones potential. We obtain kinetic, thermodynamic, and structural properties of the system at three wall-fluid interaction strengths and over a temperature range that includes sub-and super-critical conditions. Four dynamic properties are considered: translational and rotational diffusivities, a characteristic relaxation time for rotational motion, and a collective relaxation time stemming from analysis of the coherent intermediate scattering function. We carefully consider the reference state used to define the excess entropy of a confined fluid. Three ideal-gas reference states are considered, with the cases differentiated by the extent to which one-body spatial and orientational correlations are accounted for in the reference state. Our results indicate that a version of the excess entropy that includes information related to the one-body correlations in a confined fluid serves as the best scaling variable for dynamic properties. When adopting such a definition for the reference state, to a very good approximation, bulk and confined data for a specified dynamic property at a given temperature collapse onto a common curve when plotted against the excess entropy.Item Harmonic moment dynamics in Laplacian growth(2010-01) Leshchiner, Alexander; Thrasher, Matthew; Mineev-Weinstein, Mark B.; Swinney, Harry L.; Leshchiner, Alexander; Thrasher, Matthew; Swinney, Harry L.Harmonic moments are integrals of integer powers of z=x+iy over a domain. Here, the domain is an exterior of a bubble of air growing in an oil layer between two horizontal closely spaced plates. Harmonic moments are a natural basis for such Laplacian growth phenomena because, unlike other representations, these moments linearize the zero surface tension problem [S. Richardson, J. Fluid Mech. 56, 609 (1972)], so that all moments except the lowest one (the area of the bubble) are conserved in time. In our experiments, we directly determine the harmonic moments and show that for nonzero surface tension, all moments (except the lowest one) decay in time rather than exhibiting the divergences of other representations. Further, we derive an expression that relates the derivative of the k(th) harmonic moment M(k) to measurable quantities (surface tension, viscosity, the distance between the plates, and a line integral over the contour encompassing the growing bubble). The laboratory observations are in good accord with the expression we derive for dM(k)/dt, which is proportional to the surface tension; thus in the zero surface tension limit, the moments (above k=0) are all conserved, in accord with Richardson's theory. In addition, from the measurements of the time evolution of the harmonic moments we obtain a value for the surface tension that is within 20% of the accepted value. In conclusion, our analysis and laboratory observations demonstrate that an interface dynamics description in terms of harmonic moments is physically realizable and robust.Item Shooting for Success: an Analysis of Predictive Basketball Analytics(2023-05) Geelhoed, DevinBasketball has changed greatly over recent years, thanks to the data-driven revolution in the way the game is played. Models to predict player and team performance are increasingly popular for team personnel to focus on what they are most successful at, for analysts to break down where advantages and disadvantages are had for different players or teams, and for viewers to create their own opinions on the players or teams they want to succeed or fail and inform betting decisions. This thesis seeks to define where current predictive analytics are lacking with a multimodal examination of three ways we analyze the game: equation-based prediction, machine learning prediction, and human prediction. The thesis focuses on each of these three methods of forecasting in turn, noting their strengths and weaknesses. Additionally, with equation-based prediction, the thesis provides an example model of Expected Points to project the outcome of a certain shot from a certain player. Lastly, the thesis focuses on future developments in predictive analytics and the ways they are shaping the basketball viewing experience.