335 research outputs found
Rational Construction of Stochastic Numerical Methods for Molecular Sampling
In this article, we focus on the sampling of the configurational
Gibbs-Boltzmann distribution, that is, the calculation of averages of functions
of the position coordinates of a molecular -body system modelled at constant
temperature. We show how a formal series expansion of the invariant measure of
a Langevin dynamics numerical method can be obtained in a straightforward way
using the Baker-Campbell-Hausdorff lemma. We then compare Langevin dynamics
integrators in terms of their invariant distributions and demonstrate a
superconvergence property (4th order accuracy where only 2nd order would be
expected) of one method in the high friction limit; this method, moreover, can
be reduced to a simple modification of the Euler-Maruyama method for Brownian
dynamics involving a non-Markovian (coloured noise) random process. In the
Brownian dynamics case, 2nd order accuracy of the invariant density is
achieved. All methods considered are efficient for molecular applications
(requiring one force evaluation per timestep) and of a simple form. In fully
resolved (long run) molecular dynamics simulations, for our favoured method, we
observe up to two orders of magnitude improvement in configurational sampling
accuracy for given stepsize with no evident reduction in the size of the
largest usable timestep compared to common alternative methods
Generating Generalized Distributions from Dynamical Simulation
We present a general molecular-dynamics simulation scheme, based on the Nose'
thermostat, for sampling according to arbitrary phase space distributions. We
formulate numerical methods based on both Nose'-Hoover and Nose'-Poincare'
thermostats for two specific classes of distributions; namely, those that are
functions of the system Hamiltonian and those for which position and momentum
are statistically independent. As an example, we propose a generalized variable
temperature distribution that designed to accelerate sampling in molecular
systems.Comment: 10 pages, 3 figure
A molecular-dynamics algorithm for mixed hard-core/continuous potentials
We present a new molecular-dynamics algorithm for integrating the equations
of motion for a system of particles interacting with mixed continuous/impulsive
forces. This method, which we call Impulsive Verlet, is constructed using
operator splitting techniques similar to those that have been used successfully
to generate a variety molecular-dynamics integrators. In numerical experiments,
the Impulsive Verlet method is shown to be superior to previous methods with
respect to stability and energy conservation in long simulations.Comment: 18 pages, 6 postscript figures, uses rotate.st
Ab initio mass tensor molecular dynamics
Mass tensor molecular dynamics was first introduced by Bennett [J. Comput.
Phys. 19, 267 (1975)] for efficient sampling of phase space through the use of
generalized atomic masses. Here, we show how to apply this method to ab initio
molecular dynamics simulations with minimal computational overhead. Test
calculations on liquid water show a threefold reduction in computational effort
without making the fixed geometry approximation. We also present a simple
recipe for estimating the optimal atomic masses using only the first
derivatives of the potential energy.Comment: 19 pages, 5 figure
On the long-time integration of stochastic gradient systems
This article addresses the weak convergence of numerical methods for Brownian dynamics. Typical analyses of numerical methods for stochastic differential equations focus on properties such as the weak order which estimates the asymptotic (stepsize h â 0) convergence behavior of the error of finite time averages. Recently it has been demonstrated, by study of Fokker-Planck operators, that a non-Markovian numerical method [Leimkuhler and Matthews, 2013] generates approximations in the long time limit with higher accuracy order (2nd order) than would be expected from its weak convergence analysis (finite-time averages are 1st order accurate). In this article we describe the transition from the transient to the steady-state regime of this numerical method by estimating the time-dependency of the coefficients in an asymptotic expansion for the weak error, demonstrating that the convergence to 2nd order is exponentially rapid in time. Moreover, we provide numerical tests of the theory, including comparisons of the efficiencies of the Euler-Maruyama method, the popular 2nd order Heun method, and the non-Markovian method
Active swarms on a sphere
Here we show that coupling to curvature has profound effects on collective
motion in active systems, leading to patterns not observed in flat space.
Biological examples of such active motion in curved environments are numerous:
curvature and tissue folding are crucial during gastrulation, epithelial and
endothelial cells move on constantly growing, curved crypts and vili in the
gut, and the mammalian corneal epithelium grows in a steady-state vortex
pattern. On the physics side, droplets coated with actively driven microtubule
bundles show active nematic patterns. We study a model of self-propelled
particles with polar alignment on a sphere. Hallmarks of these motion patterns
are a polar vortex and a circulating band arising due to the incompatibility
between spherical topology and uniform motion - a consequence of the hairy ball
theorem. We present analytical results showing that frustration due to
curvature leads to stable elastic distortions storing energy in the band.Comment: 5 pages, 4 figures plus Supporting Informatio
A Robust Numerical Method for Integration of Point-Vortex Trajectories in Two Dimensions
The venerable 2D point-vortex model plays an important role as a simplified
version of many disparate physical systems, including superfluids,
Bose-Einstein condensates, certain plasma configurations, and inviscid
turbulence. This system is also a veritable mathematical playground, touching
upon many different disciplines from topology to dynamic systems theory.
Point-vortex dynamics are described by a relatively simple system of nonlinear
ODEs which can easily be integrated numerically using an appropriate adaptive
time stepping method. As the separation between a pair of vortices relative to
all other inter-vortex length scales decreases, however, the computational time
required diverges. Accuracy is usually the most discouraging casualty when
trying to account for such vortex motion, though the varying energy of this
ostensibly Hamiltonian system is a potentially more serious problem. We solve
these problems by a series of coordinate transformations: We first transform to
action-angle coordinates, which, to lowest order, treat the close pair as a
single vortex amongst all others with an internal degree of freedom. We next,
and most importantly, apply Lie transform perturbation theory to remove the
higher-order correction terms in succession. The overall transformation
drastically increases the numerical efficiency and ensures that the total
energy remains constant to high accuracy.Comment: 21 pages, 4 figure
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