Bulletin of the American Physical Society
2005 TSAPS/AAPT/SPS Joint Fall Meeting
Thursday–Saturday, October 20–22, 2005; Houston, TX
Session E3: Monte Carlo |
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Room: Beverly Hilton 218 |
Saturday, October 22, 2005 10:30AM - 10:42AM |
E3.00001: Feynman-Kac Path Integral Convergence Enhancement by Rational Curve Fitting John Hopkins, Bruce Miller By creating a mapping between a quantum system and a classical polymer chain of p beads, the Feynman-Kac path integral provides a well-established formalism for representing the system density matrix. In the limit of large p, it is accurately represented by an integral over all possible paths of the Euclidean action. Few cases can be worked out analytically, so the integral is usually performed numerically using Monte Carlo techniques. Limits must be taken for large numbers of Monte Carlo samples and large p. Systematic errors associated with large p make it is desirable to determine this limit without the direct calculation of integrals for large p. We have numerical evidence that a rational function curve-fitting technique applied to integrations for several relatively small p values gives reasonable answers for the large p limit for a simple system where the analytic solution is known--a single quantum particle in a one-dimensional periodic lattice of atoms. This algorithm is similar to other convergence enhancing techniques, such as the Bulirsch-Stoer algorithm for finding numerical solutions to differential equations. [Preview Abstract] |
Saturday, October 22, 2005 10:42AM - 10:54AM |
E3.00002: Comparison of Multifractal Behavior in One Dimensional Models of Hierarchical Expansion Bruce Miller, Jean-Louis Rouet Observations of galaxies over large scales reveals the possibility of a fractal distribution of their positions. The assumed source of fractal behavior is the lack of a length scale in the two body gravitational interaction. However, even with new, larger, sample sizes from recent studies, it is difficult to extract information concerning fractal properties with confidence. Similarly, simulations with a billion particles only provide a thousand particles per dimension, far too small for accurate conclusions. With one dimensional ``toy models'' we can overcome these limitations by carrying out simulations with on the order of a quarter of a million particles. Here we compare the multifractal analysis of a group of one dimensional models which incorporate different features of the equations governing the evolution of a matter dominated universe. The results share some similarities with galaxy observations, such as apparent bifractal geometry. They also provide insights concerning possible constraints on length and time scales for fractal geometry. [Preview Abstract] |
Saturday, October 22, 2005 10:54AM - 11:06AM |
E3.00003: Pattern Formation in a Two Dimensional Network of Interacting Neurons Bill Maier, Bruce N. Miller We employ a realistic mathematical model of the neuron to study neural networks as dynamical systems. The nonlinear model, which consists of two coupled phase variables, was introduced by Izhikevich and is able to simulate the major types of nerve cells through an appropriate choice of parameters. The dynamics of a pair of interacting neurons was simulated to investigate their mutual properties. In order to gain insight into the propagation of signals in the brain, we constructed a two dimensional, two layer, grid of 90,000 neurons consisting of both excitatory and inhibitory elements. Simulations were carried out under various conditions to explore different types of firing patterns. Under special circumstances regimes of strong synchronization, reminiscent of seizure, were observed. [Preview Abstract] |
Saturday, October 22, 2005 11:06AM - 11:18AM |
E3.00004: (Quasi)-Convexity Formulation of a Multi-Minima, Variational, Quantization Method Carlos Handy Barta's ground state energy bounding theorem states that the E$_{L}$ = Inf ( H$\Phi $(x)/ $\Phi $ (x)) and E$_{U}$ = Sup ( H$\Phi $(x)/ $\Phi $ (x)) for an arbitrary, positive, bounded, configuration, $\Phi $(x), define lower and upper bounds, respectively, to the bosonic ground state energy: E$_{L} <$ E$_{ground} <$ E$_{U}$. Searching for the x-values corresponding to the infimum (Inf) and supremum (Sup) is a multi-extrema plagued process, particularly in multidimensions. We can reformulate Barta's configuration space analysis in terms of the \textit{Moment Problem}, via a Generalized Eigenvalue Problem representation. This removes all multi-extrema considerations, recasting the original variational problem as one involving (quasi)-convexity optimization. We outline the procedure, and apply it to some, illustrative problems. [Preview Abstract] |
Saturday, October 22, 2005 11:18AM - 11:30AM |
E3.00005: Vibrationally Averaged Properties Using Monte Carlo Methods Steven Alexander, R.L. Coldwell Using explicitly correlated wavefunctions and variational Monte Carlo methods we have computed potential energy surfaces for several small molecules. These surfaces include both relativistic corrections and non-Born Oppenheimer corrections. From this surface we then computed several of the lowest rotational and vibrational energies and wavefunctions again using Monte Carlo methods. With our molecular wavefunctions we next evaluated a number of properties for each molecule as a function of the nuclear positions. When we finally integrated these properties over the rotational/vibrational wavefunctions, the result is a set of vibrationally averaged properties. [Preview Abstract] |
Saturday, October 22, 2005 11:30AM - 11:42AM |
E3.00006: Scaling of the Step Position Distribution of Stepped Crystal Surfaces Howard L. Richards, Amber N. Benson, T.L. Einstein Both Monte Carlo simulations and Scanning Tunneling Microscope images of stepped crystal surfaces can only include some finite length, $\Delta y$, along average direction of steps. This has important consequences, because the variance of the Step Position Distribution (SPD), $\sigma^2_Q(\Delta y)$, calculated from these ``snapshots'' depends on $\Delta y$. For $\Delta y \! < \! \xi_Q$, where $\xi_Q$ is the correlation length of the steps, $\sigma^2_Q(\Delta y) \! \propto \! (\Delta y)^{0.8}$; for $\Delta y \! > \! 4 \xi_Q$, $\sigma^2_Q(\Delta y)/\sigma^2_{Q, {\rm W}} \! \approx \! 0.158+ 0.318 ln(\Delta y)$, where $\sigma^2_{Q, {\rm W}}$ is the finite value of $\sigma^2_Q(\infty)$ predicted by the two-step approximation which yields the generalized Wigner distribution (GWD) for the Terrace Width Distribution (TWD). A ``Wigner length'', $L_{\rm W}$, can be defined by $\sigma^2_Q(L_{\rm W}) \! = \! \sigma^2_{Q, {\rm W}}$. It appears that $L_{\rm W} \! \approx \! 14.2 \xi_Q$ independent of the magnitude of step interaction. This is very close to a length which must be introduced to reproduce the GWD from an ensemble averge of Gruber-Mullins approximations of the TWD. [Preview Abstract] |
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