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

Analytical and experimental study of stratification and liquid-ullage coupling, 1 June 1964 - 31 May 1965

Closed-form solution for stratification of subcooled fluids in containers subjected to heating, and for liquid-ullage vapor couplin

Dynamical Effects from Asteroid Belts for Planetary Systems

The orbital evolution and stability of planetary systems with interaction from the belts is studied using the standard phase-plane analysis. In addition to the fixed point which corresponds to the Keplerian orbit, there are other fixed points around the inner and outer edges of the belt. Our results show that for the planets, the probability to move stably around the inner edge is larger than the one to move around the outer edge. It is also interesting that there is a limit cycle of semi-attractor for a particular case. Applying our results to the Solar System, we find that our results could provide a natural mechanism to do the orbit rearrangement for the larger Kuiper Belt Objects and thus successfully explain the absence of these objects beyond 50 AU.Comment: accepted by International Journal of Bifurcation and Chaos in Aug. 2003, AAS Latex, 27 pages with 6 color 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

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

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

Mapping functions and critical behavior of percolation on rectangular domains

The existence probability $E_p$ and the percolation probability $P$ of the bond percolation on rectangular domains with different aspect ratios $R$ are studied via the mapping functions between systems with different aspect ratios. The superscaling behavior of $E_p$ and $P$ for such systems with exponents $a$ and $b$, respectively, found by Watanabe, Yukawa, Ito, and Hu in [Phys. Rev. Lett. \textbf{93}, 190601 (2004)] can be understood from the lower order approximation of the mapping functions $f_R$ and $g_R$ for $E_p$ and $P$, respectively; the exponents $a$ and $b$ can be obtained from numerically determined mapping functions $f_R$ and $g_R$, respectively.Comment: 17 pages with 6 figure

Vibrational energy relaxation in proteins

An overview of theories related to vibrational energy relaxation (VER) in proteins is presented. VER of a selected mode in cytochrome c is studied using two theoretical approaches. One is the equilibrium simulation approach with quantum correction factors, and the other is the reduced model approach which describes the protein as an ensemble of normal modes interacting through nonlinear coupling elements. Both methods result in estimates of the VER time (sub ps) for a CD stretching mode in the protein at room temperature. The theoretical predictions are in accord with the experimental data of Romesberg's group. A perspective on future directions for the detailed study of time scales and mechanisms for VER in proteins is presented.Comment: 12 pages, 4 figures, accepted for publication in PNA

Geometric and impurity effects on quantum rings in magnetic fields

We investigate the effects of impurities and changing ring geometry on the energetics of quantum rings under different magnetic field strengths. We show that as the magnetic field and/or the electron number are/is increased, both the quasiperiodic Aharonov-Bohm oscillations and various magnetic phases become insensitive to whether the ring is circular or square in shape. This is in qualitative agreement with experiments. However, we also find that the Aharonov-Bohm oscillation can be greatly phase-shifted by only a few impurities and can be completely obliterated by a high level of impurity density. In the many-electron calculations we use a recently developed fourth-order imaginary time projection algorithm that can exactly compute the density matrix of a free-electron in a uniform magnetic field.Comment: 8 pages, 7 figures, to appear in PR

`St\"uckelberg interferometry' with ultracold molecules

We report on the realization of a time-domain `St\"uckelberg interferometer', which is based on the internal state structure of ultracold Feshbach molecules. Two subsequent passages through a weak avoided crossing between two different orbital angular momentum states in combination with a variable hold time lead to high-contrast population oscillations. This allows for a precise determination of the energy difference between the two molecular states. We demonstrate a high degree of control over the interferometer dynamics. The interferometric scheme provides new possibilities for precision measurements with ultracold molecules.Comment: 4 pages, 5 figure