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

High-order Path Integral Monte Carlo methods for solving quantum dot problems

The conventional second-order Path Integral Monte Carlo method is plagued with the sign problem in solving many-fermion systems. This is due to the large number of anti-symmetric free fermion propagators that are needed to extract the ground state wave function at large imaginary time. In this work, we show that optimized fourth-order Path Integral Monte Carlo methods, which use no more than 5 free-fermion propagators, can yield accurate quantum dot energies for up to 20 polarized electrons with the use of the Hamiltonian energy estimator.Comment: 14 pages, 4 figures, submitted to PRE - revised with a new figure and added larger N calculation

The physics of symplectic integrators: perihelion advances and symplectic corrector algorithms

Symplectic integrators evolve dynamical systems according to modified Hamiltonians whose error terms are also well-defined Hamiltonians. The error of the algorithm is the sum of each error Hamiltonian's perturbation on the exact solution. When symplectic integrators are applied to the Kepler problem, these error terms cause the orbit to precess. In this work, by developing a general method of computing the perihelion advance via the Laplace-Runge-Lenz vector even for non-separable Hamiltonians, I show that the precession error in symplectic integrators can be computed analytically. It is found that at each order, each paired error Hamiltonians cause the orbit to precess oppositely by exactly the same amount after each period. Hence, symplectic corrector, or process integrators, which have equal coefficients for these paired error terms, will have their precession errors exactly cancel after each period. Thus the physics of symplectic integrators determines the optimal algorithm for integrating long time periodic motions.Comment: 18 pages, 5 figures, 1 tabl

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

Fourth-Order Algorithms for Solving the Imaginary Time Gross-Pitaevskii Equation in a Rotating Anisotropic Trap

By implementing the exact density matrix for the rotating anisotropic harmonic trap, we derive a class of very fast and accurate fourth order algorithms for evolving the Gross-Pitaevskii equation in imaginary time. Such fourth order algorithms are possible only with the use of {\it forward}, positive time step factorization schemes. These fourth order algorithms converge at time-step sizes an order-of-magnitude larger than conventional second order algorithms. Our use of time-dependent factorization schemes provides a systematic way of devising algorithms for solving this type of nonlinear equations.Comment: 14 pages with 3 figures, revised figures with the use of the Lambert W-function for doing the self-consistent iterations. Published versio