Session N5: Computational Physics

Cosmology in One Dimension: Vlasov Dynamics

Numerical simulations of self-gravitating systems are generally based on N-body codes, which solve the equations of motion of a large number of interacting particles. This approach suffers from poor statistical sampling in regions of low density. In contrast, Vlasov codes, by meshing the entire phase space, can reach higher accuracy irrespective of the density. Here, we performed one-dimensional Vlasov simulations of a long-standing cosmological problem, namely the fractal properties of an expanding Einstein-De Sitter universe in Newtonian gravity. The N-body results were confirmed for high-density regions and extended to regions of low matter density, where the N-body approach usually fails.   

Path integral Monte Carlo on a lattice II: bound states

The equilibrium properties of a single quantum particle (qp) interacting with a classical gas for a wide range of temperatures that explore the system's behavior in the classical as well as in the quantum regime is investigated. Both the qp and atoms are restricted to sites on a one-dimensional lattice. A path integral formalism developed within the context of the canonical ensemble is utilized, where the qp is represented by a closed, variable-step random walk on the lattice. Monte Carlo methods are employed to determine the system's properties. To test the usefulness of the path integral formalism, the Metropolis algorithm is employed to determine the equilibrium properties of the qp in the context of a square well potential, forcing the qp to occupy bound states. We consider a one-dimensional square well potential where all atoms on the lattice are occupied with one atom with an on-site potential except for a contiguous set of sites of various lengths centered at the middle of the lattice. Comparison of the potential energy, total energy, the energy fluctuations and the correlation function are made between the results of the Monte Carlo simulations and the analytical calculations.   

Numerical techniques to improve lattice QCD calculations

I will be talking about numerical techniques which improve lattice QCD disconnected loop calculations.~We use an evaluation technique where the lattice matrix is projected over the multiple noises to obtain solution vectors. This is done with linear equation solvers like GMRES-DR (Generalized Minimum RESidual algorithm-Deflated and Restarted) for the first noise, and GMRES-Proj (similar algorithm projected over eigenvectors) for remaining noises. Noise subtraction methods that we are working on deflate out eigenvectors and fit a polynomial to the quark propagator to speed up calculations. A combination of deflation and polynomial methods gives the best results so far.   

Continuing Studies of Hydrogenic Quantum Systems Using the Feynman-Kac Path Integral Method

The Feynman-Kac path integral method is applied to the atomic hydrogen quantum system for the purpose of evaluating eigenvalues. These are computed by random walk simulations on a discrete grid. The study provides the latest simulation analysis and includes the use of symmetry that allows higher order eigenstates to be computed. The method provides exact values in the limit of infinitesimal step size and infinite time for the lowest eigenstates.   

Analyzing theoretical calculations for the results of e- and e$+$ interactions

In particle physics certain theoretical calculations are done to calculate the probability of obtaining certain results from an interaction of electrons and positrons. The goal of this work is to improve the accuracy of the error bands. This improvement in the accuracy of the error bands will help experimentalists in interpreting their data when they are comparing it to the data they collect from their experiment. These calculations could be improved upon by using a variety of different methods to estimate the error; I will be showing results from multiple different methods that I investigated.   

A Consistent Model of Plasma- The Potential in a Glass Box

Numerical modeling has become a valuable diagnostic tool for experiments in the modern physical world. In modeling the dynamics of charged dust particles confined in a glass box placed on the lower electrode of a GEC cell, there are many interactions between the dust, plasma, and boundaries that need to be accounted for more accurately. The lower electrode affects the plasma conditions in the sheath, altering the electron and ion densities. These local variations in the plasma determine the charge accumulated on the surface of the glass box and the resulting electrostatic potential within it. This work describes the steps taken to build a consistent model of the relationship between the plasma conditions and the confining electric potential due to the glass box in order to more accurately model the charging and dynamics of dust clusters and strings.   

Regular and Chaotic Motion of a Gravitational Billiard in a Cone

We study the nonlinear dynamics of a three dimensional billiard in a constant gravitational field colliding elastically with a linear cone of half angle ?. We derive a two-dimensional Poincare map with two parameters, the half angle of the cone and l, the z-component of the billiard’s angular momentum. We demonstrate several integrable cases of the parameter values, and analytically compute the system’s fixed point, analyzing the stability of this orbit as a function of the parameters as well as its relation to the physical trajectory of the billiard. Next, we explore the phase space of the system numerically. We find that for small values of l the conic billiard exhibits behavior characteristic of two-degree-of-freedom Hamiltonian systems with a discontinuity, and the dynamics is qualitatively similar to that of the wedge billiard, although the correspondence is not exact. As we increase l the dynamics becomes, on the whole, less chaotic and the correspondence with the wedge billiard is lost. In common with the wedge billiard, we anticipate that modifications of the cone system will prove valuable for experimental investigation, both with atoms at low temperature and driven billiards.