201 research outputs found
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
Explicit symplectic integrators for solving non-separable Hamiltonians
By exploiting the error functions of explicit symplectic integrators for
solving separable Hamiltonians, I show that it is possible to develop explicit,
time-reversible symplectic integrators for solving non-separable Hamiltonians
of the product form. The algorithms are unusual in that they of fractional
order.Comment: 8 pages, 3 figure
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
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