Quantum Statistical Calculations and Symplectic Corrector Algorithms

The quantum partition function at finite temperature requires computing the trace of the imaginary time propagator. For numerical and Monte Carlo calculations, the propagator is usually split into its kinetic and potential parts. A higher order splitting will result in a higher order convergent algorithm. At imaginary time, the kinetic energy propagator is usually the diffusion Greens function. Since diffusion cannot be simulated backward in time, the splitting must maintain the positivity of all intermediate time steps. However, since the trace is invariant under similarity transformations of the propagator, one can use this freedom to "correct" the split propagator to higher order. This use of similarity transforms classically give rises to symplectic corrector algorithms. The split propagator is the symplectic kernel and the similarity transformation is the corrector. This work proves a generalization of the Sheng-Suzuki theorem: no positive time step propagators with only kinetic and potential operators can be corrected beyond second order. Second order forward propagators can have fourth order traces only with the inclusion of an additional commutator. We give detailed derivations of four forward correctable second order propagators and their minimal correctors.Comment: 9 pages, no figure, corrected typos, mostly missing right bracket

An investigation of the lower Permian middle Ecca ammonite locality at Alleta, Natal

Main articeThe problematic ammonite Paraceltites bowdeni Teichert & Rilett has been recorded only from the Alleta iron-ore mine near Dundee in Natal. It is unique in the Early Permian Ecca Series as it suggests a normal salinity for the depositional environment of sediments that have yielded no other clearly marine fossils. An investigation of the matrix of the specimen slabs, however, yields information which is incompatible with equivalent data from the Alleta mine and the Ecca sediments in general. The matrix contains the distinctive pollen Classopollis which is not known from elsewhere in the world in deposits older than Late Triassic. Comparative tests of the degree of thermal diagenesis of the contained organic material suggests that the ammonite specimens have not been subjected to the same degree of alteration as the sediments at the Alleta mine. Results of other tests have not been definitive but do not contradict the suggestion that the ammonites were mistakenly accredited to the A1leta mine. It is concluded that the ammonites derived originally from sediments of Late Triassic to Early Jurassic age at an unknown locality outside of South Africa.Non

PROBLEMATIC MICROFOSSILS FROM THE LOWER KARROO BEDS IN SOUTH AFRICA

A problematic group of microfossils has recently been recovered from strata of Permian age, in the northern part of the Karroo basin in South Africa. This article attempts to present the information that is presently known about them. They have been found in a wide variety of sediments in the Lower, Middle and Upper Ecca stages, and in carbonaceous sediments in the lower part of the Beaufort series. The external morphology of the microfossils is extremely varied, but they are characterized by a regular cup-shaped organ. They closely resemble forms called Anellotubulata by O. Wetzel (1959), who described them from the Upper Lias (c) of Germany. Other workers have recovered, but not described, similar microfossils from Permian, Triassic and Cretaceous strata in Australia. In this paper, the microfossils are referred to as anellotubulates. They are remarkable in a number of respects, the most extraordinary of which is their composition. Electron-microprobe and X-ray diffraction tests have shown the shell to consist of a non-crystalline mineral or minerals, composed mainly of iron and phosphorus, with minor calcium. It has not been possible to demonstrate clearly whether this is the original shell composition, or whether it has resulted from replacement. The available information, including that provided by associated fossils, which has bearing on the palaeoenvironment of the anellotubulates, is discussed. It is hoped that, when more information is available, these problematic microfossils will contribute towards a better understanding of the depositional environment of the sediments in which they occur

Higher-order splitting algorithms for solving the nonlinear Schr\"odinger equation and their instabilities

Since the kinetic and the potential energy term of the real time nonlinear Schr\"odinger equation can each be solved exactly, the entire equation can be solved to any order via splitting algorithms. We verified the fourth-order convergence of some well known algorithms by solving the Gross-Pitaevskii equation numerically. All such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved. A detail error analysis reveals that the noise, or elementary excitations of the nonlinear Schr\"odinger, obeys the Bogoliubov spectrum and the instability is due to the exponential growth of high wave number noises caused by the splitting process. For a continuum wave function, this instability is unavoidable no matter how small the time step. For a discrete wave function, the instability can be avoided only for \dt k_{max}^2{<\atop\sim}2 \pi, where $k_{max}=\pi/\Delta x$.Comment: 10 pages, 8 figures, submitted to Phys. Rev.

Dynamical Multiple-Timestepping Methods for Overcoming the Half-Period Time Step Barrier

Current molecular dynamic simulations of biomolecules using multiple time steps to update the slowingly changing force are hampered by an instability occuring at time step equal to half the period of the fastest vibrating mode. This has became a critical barrier preventing the long time simulation of biomolecular dynamics. Attemps to tame this instability by altering the slowly changing force and efforts to damp out this instability by Langevin dynamics do not address the fundamental cause of this instability. In this work, we trace the instability to the non-analytic character of the underlying spectrum and show that a correct splitting of the Hamiltonian, which render the spectrum analytic, restores stability. The resulting Hamiltonian dictates that in additional to updating the momentum due to the slowly changing force, one must also update the position with a modified mass. Thus multiple-timestepping must be done dynamically.Comment: 10 pages, 2 figures, submitted to J. Chem. Phy

Any-order propagation of the nonlinear Schroedinger equation

We derive an exact propagation scheme for nonlinear Schroedinger equations. This scheme is entirely analogous to the propagation of linear Schroedinger equations. We accomplish this by defining a special operator whose algebraic properties ensure the correct propagation. As applications, we provide a simple proof of a recent conjecture regarding higher-order integrators for the Gross-Pitaevskii equation, extend it to multi-component equations, and to a new class of integrators.Comment: 10 pages, no figures, submitted to Phys. Rev.