Fourth Order Algorithms for Solving the Multivariable Langevin Equation and the Kramers Equation

We develop a fourth order simulation algorithm for solving the stochastic Langevin equation. The method consists of identifying solvable operators in the Fokker-Planck equation, factorizing the evolution operator for small time steps to fourth order and implementing the factorization process numerically. A key contribution of this work is to show how certain double commutators in the factorization process can be simulated in practice. The method is general, applicable to the multivariable case, and systematic, with known procedures for doing fourth order factorizations. The fourth order convergence of the resulting algorithm allowed very large time steps to be used. In simulating the Brownian dynamics of 121 Yukawa particles in two dimensions, the converged result of a first order algorithm can be obtained by using time steps 50 times as large. To further demostrate the versatility of our method, we derive two new classes of fourth order algorithms for solving the simpler Kramers equation without requiring the derivative of the force. The convergence of many fourth order algorithms for solving this equation are compared.Comment: 19 pages, 2 figure

Terahertz magneto-spectroscopy of transient plasmas in semiconductors

Using synchronized near-infrared (NIR) and terahertz (THz) lasers, we have performed picosecond time-resolved THz spectroscopy of transient carriers in semiconductors. Specifically, we measured the temporal evolution of THz transmission and reflectivity after NIR excitation. We systematically investigated transient carrier relaxation in GaAs and InSb with varying NIR intensities and magnetic fields. Using this information, we were able to determine the evolution of the THz absorption to study the dynamics of photocreated carriers. We developed a theory based on a Drude conductivity with time-dependent density and density-dependent scattering lifetime, which successfully reproduced the observed plasma dynamics. Detailed comparison between experimental and theoretical results revealed a linear dependence of the scattering frequency on density, which suggests that electron-electron scattering is the dominant scattering mechanism for determining the scattering time. In InSb, plasma dynamics was dramatically modified by the application of a magnetic field, showing rich magneto-reflection spectra, while GaAs did not show any significant magnetic field dependence. We attribute this to the small effective masses of the carriers in InSb compared to GaAs, which made the plasma, cyclotron, and photon energies all comparable in the density, magnetic field, and wavelength ranges of the current study.Comment: 8 pages, 9 figures, submitted to Phys. Rev.

The Complete Characterization of Fourth-Order Symplectic Integrators with Extended-Linear Coefficients

The structure of symplectic integrators up to fourth-order can be completely and analytical understood when the factorization (split) coefficents are related linearly but with a uniform nonlinear proportional factor. The analytic form of these {\it extended-linear} symplectic integrators greatly simplified proofs of their general properties and allowed easy construction of both forward and non-forward fourth-order algorithms with arbitrary number of operators. Most fourth-order forward integrators can now be derived analytically from this extended-linear formulation without the use of symbolic algebra.Comment: 12 pages, 2 figures, submitted to Phys. Rev. E, corrected typo

A new broken U(1)-symmetry in extreme type-II superconductors

A phase transition within the molten phase of the Abrikosov vortex system without disorder in extreme type-II superconductors is found via large-scale Monte-Carlo simulations. It involves breaking a U(1)-symmetry, and has a zero-field counterpart, unlike vortex lattice melting. Its hallmark is the loss of number-conservation of connected vortex paths threading the entire system {\it in any direction}, driving the vortex line tension to zero. This tension plays the role of a generalized ``stiffness'' of the vortex liquid, and serves as a probe of the loss of order at the transition, where a weak specific heat anomaly is found.Comment: 5 pages, 3 figure

Higher Order Force Gradient Symplectic Algorithms

We show that a recently discovered fourth order symplectic algorithm, which requires one evaluation of force gradient in addition to three evaluations of the force, when iterated to higher order, yielded algorithms that are far superior to similarly iterated higher order algorithms based on the standard Forest-Ruth algorithm. We gauge the accuracy of each algorithm by comparing the step-size independent error functions associated with energy conservation and the rotation of the Laplace-Runge-Lenz vector when solving a highly eccentric Kepler problem. For orders 6, 8, 10 and 12, the new algorithms are approximately a factor of $10^3$, $10^4$, $10^4$ and $10^5$ better.Comment: 23 pages, 10 figure

On the construction of high-order force gradient algorithms for integration of motion in classical and quantum systems

A consequent approach is proposed to construct symplectic force-gradient algorithms of arbitrarily high orders in the time step for precise integration of motion in classical and quantum mechanics simulations. Within this approach the basic algorithms are first derived up to the eighth order by direct decompositions of exponential propagators and further collected using an advanced composition scheme to obtain the algorithms of higher orders. Contrary to the scheme by Chin and Kidwell [Phys. Rev. E 62, 8746 (2000)], where high-order algorithms are introduced by standard iterations of a force-gradient integrator of order four, the present method allows to reduce the total number of expensive force and its gradient evaluations to a minimum. At the same time, the precision of the integration increases significantly, especially with increasing the order of the generated schemes. The algorithms are tested in molecular dynamics and celestial mechanics simulations. It is shown, in particular, that the efficiency of the new fourth-order-based algorithms is better approximately in factors 5 to 1000 for orders 4 to 12, respectively. The results corresponding to sixth- and eighth-order-based composition schemes are also presented up to the sixteenth order. For orders 14 and 16, such highly precise schemes, at considerably smaller computational costs, allow to reduce unphysical deviations in the total energy up in 100 000 times with respect to those of the standard fourth-order-based iteration approach.Comment: 23 pages, 2 figures; submitted to Phys. Rev.

A Scanning Electron Microscope Study of Fiowers of Carambola, Durian and Rambutan

Fresh specimens of flowers of carambola (Averrhoa carambola L.), durian (Durio zibethinus Murr.) and rambutan (Nephelium lappaceum L.) were examined using the scanning electron microscope (SEM). Whole flowers and floral parts were described and illustrated in the micrographs. Floral structures and their differences within species were highlighted, their relationship and functions are discussed with special reference to pollination mechanisms and pollen morphology

Phase diagram and universality of the Lennard-Jones gas-liquid system

The gas-liquid phase transition of the three-dimensional Lennard-Jones particles system is studied by molecular dynamics simulations. The gas and liquid densities in the coexisting state are determined with high accuracy. The critical point is determined by the block density analysis of the Binder parameter with the aid of the law of rectilinear diameter. From the critical behavior of the gas-liquid coexsisting density, the critical exponent of the order parameter is estimated to be $\beta = 0.3285(7)$. Surface tension is estimated from interface broadening behavior due to capillary waves. From the critical behavior of the surface tension, the critical exponent of the correlation length is estimated to be $\nu = 0.63 (4)$. The obtained values of $\beta$ and $\nu$ are consistent with those of the Ising universality class.Comment: 8 pages, 8 figures, new results are adde