77 research outputs found
The incomplete beta function law for parallel tempering sampling of classical canonical systems
We show that the acceptance probability for swaps in the parallel tempering
Monte Carlo method for classical canonical systems is given by a universal
function that depends on the average statistical fluctuations of the potential
and on the ratio of the temperatures. The law, called the incomplete beta
function law, is valid in the limit that the two temperatures involved in swaps
are close to one another. An empirical version of the law, which involves the
heat capacity of the system, is developed and tested on a Lennard-Jones
cluster. We argue that the best initial guess for the distribution of
intermediate temperatures for parallel tempering is a geometric progression and
we also propose a technique for the computation of optimal temperature
schedules. Finally, we demonstrate that the swap efficiency of the parallel
tempering method for condensed-phase systems decreases naturally to zero at
least as fast as the inverse square root of the dimensionality of the physical
system.Comment: 11 pages, 4 figures; minor changes; to appear in J. Chem. Phy
Sampling diffusive transition paths
We address the problem of sampling double-ended diffusive paths. The ensemble
of paths is expressed using a symmetric version of the Onsager-Machlup formula,
which only requires evaluation of the force field and which, upon direct time
discretization, gives rise to a symmetric integrator that is accurate to second
order. Efficiently sampling this ensemble requires avoiding the well-known
stiffness problem associated with sampling infinitesimal Brownian increments of
the path, as well as a different type of stiffness associated with sampling the
coarse features of long paths. The fine-feature sampling stiffness is
eliminated with the use of the fast sampling algorithm (FSA), and the
coarse-feature sampling stiffness is avoided by introducing the sliding and
sampling (S&S) algorithm. A key feature of the S&S algorithm is that it enables
massively parallel computers to sample diffusive trajectories that are long in
time. We use the algorithm to sample the transition path ensemble for the
structural interconversion of the 38-atom Lennard-Jones cluster at low
temperature.Comment: 13 pages 5 figure
The fast sampling algorithm for Lie-Trotter products
A fast algorithm for path sampling in path integral Monte Carlo simulations
is proposed. The algorithm utilizes the Levy-Ciesielski implementation of
Lie-Trotter products to achieve a mathematically proven computational cost of
n*log_2(n) with the number of time slices n, despite the fact that each path
variable is updated separately, for reasons of optimality. In this respect, we
demonstrate that updating a group of random variables simultaneously results in
loss of efficiency.Comment: 4 pages, 1 figure; fast rejection from Phys. Rev. Letts; transfered
to PRE as a Rapid Communication. Eq. 6 to 10 contained some inconsistencies
that have been repaired in the present version; A sample code implementing
the algorithm for LJ clusters is available from the author upon reques
Numerical implementation of some reweighted path integral methods
The reweighted random series techniques provide finite-dimensional
approximations to the quantum density matrix of a physical system that have
fast asymptotic convergence. We study two special reweighted techniques that
are based upon the Levy-Ciesielski and Wiener-Fourier series, respectively. In
agreement with the theoretical predictions, we demonstrate by numerical
examples that the asymptotic convergence of the two reweighted methods is cubic
for smooth enough potentials. For each reweighted technique, we propose some
minimalist quadrature techniques for the computation of the path averages.
These quadrature techniques are designed to preserve the asymptotic convergence
of the original methods.Comment: 15 pages, 10 figures, submitted to JC
The Partial Averaging method
The partial averaging technique is defined and used in conjunction with the
random series implementation of the Feynman-Kac formula. It enjoys certain
properties such as good rates of convergence and convergence for potentials
with coulombic singularities. In this work, I introduce the reader to the
technique and I analyze the basic mathematical properties of the method. I show
that the method is convergent for all Kato-class potentials that have finite
Gaussian transform.Comment: 9 pages, no figures; one reference correcte
Thermodynamics and equilibrium structure of Ne_38 cluster: Quantum Mechanics versus Classical
The equilibrium properties of classical LJ_38 versus quantum Ne_38
Lennard-Jones clusters are investigated. The quantum simulations use both the
Path-Integral Monte-Carlo (PIMC) and the recently developed
Variational-Gaussian-Wavepacket Monte-Carlo (VGW-MC) methods. The PIMC and the
classical MC simulations are implemented in the parallel tempering framework.
The VGW method is used to locate and characterize the low energy states of
Ne_38, which are then further refined by PIMC calculations. Unlike the
classical case, the ground state of Ne_38 is a liquid-like structure. Among the
several liquid-like states with energies below the two symmetric states (O_h
and C_5v), the lowest two exhibit strong delocalization over basins associated
with at least two classical local minima. Because the symmetric structures do
not play an essential role in the thermodynamics of Ne_38, the quantum heat
capacity is a featureless curve indicative of the absence of any structural
transformations. Good agreement between the two methods, VGW and PIMC, is
obtained.Comment: 13 pages, 9 figure
Energy estimators for random series path-integral methods
We perform a thorough analysis on the choice of estimators for random series
path integral methods. In particular, we show that both the thermodynamic
(T-method) and the direct (H-method) energy estimators have finite variances
and are straightforward to implement. It is demonstrated that the agreement
between the T-method and the H-method estimators provides an important
consistency check on the quality of the path integral simulations. We
illustrate the behavior of the various estimators by computing the total,
kinetic, and potential energies of a molecular hydrogen cluster using three
different path integral techniques. Statistical tests are employed to validate
the sampling strategy adopted as well as to measure the performance of the
parallel random number generator utilized in the Monte Carlo simulation. Some
issues raised by previous simulations of the hydrogen cluster are clarified.Comment: 15 pages, 1 figure, 3 table
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