arXiv subjects

Cristian Predescu

Publications and source records attributed to Cristian Predescu.

27 records · Page 2Linked to original sources

arXiv2003

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.

physics.chem-ph↗

arXiv2003

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.

cond-mat.stat-mech↗

arXiv2003

Heat capacity estimators for random series path-integral methods by finite-difference schemes

Previous heat capacity estimators used in path integral simulations either have large variances that grow to infinity with the number of path variables or require the evaluation of first and second order derivatives of the potential. In the present paper, we show that the evaluation of the total energy by the T-method estimator and of the heat capacity by the TT-method estimator can be implemented by a finite difference scheme in a stable fashion. As such, the variances of the resulting estimators are finite and the evaluation of the estimators requires the potential function only. By comparison with the task of computing the partition function, the evaluation of the estimators requires k + 1 times more calls to the potential, where k is the order of the difference scheme employed. Quantum Monte Carlo simulations for the Ne_13 cluster demonstrate that a second order central-difference scheme should suffice for most applications.

cond-mat.stat-mech↗

arXiv2003

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.

math-ph↗

arXiv2003

Local Variational Principle

A generalization of the Gibbs-Bogoliubov-Feynman inequality for spinless particles is proven and then illustrated for the simple model of a symmetric double-well quartic potential. The method gives a pointwise lower bound for the finite-temperature density matrix and it can be systematically improved by the Trotter composition rule. It is also shown to produce groundstate energies better than the ones given by the Rayleigh-Ritz principle as applied to the groundstate eigenfunctions of the reference potentials. Based on this observation, it is argued that the Local Variational Principle performs better than the equivalent methods based on the centroid path idea and on the Gibbs-Bogoliubov-Feynman variational principle, especially in the range of low temperatures.

cond-mat.stat-mech↗

arXiv2003

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.

cond-mat.stat-mech↗

arXiv2003

Reweighted techniques: definition and asymptotic convergence

I define and characterize the reweighted methods, which are techniques used in conjunction with the random series implementation of the Feynman-Kac formula. I prove several convergence results valid for all series representations and then I specialize the results for the Levy-Ciesielski and Wiener-Fourier series. As opposed to the partial averaging method on which they are based, the reweighted techniques do not involve any modification of the physical potential. Rather, the underlying idea is to develop some specialized constructions of the Brownian bridge that enters the Feynman-Kac formula, so as to simulate the partial averaging effect. For the Levy-Ciesielski series representation, I develop a reweighted technique which has o(1/n^2) convergence for potentials having first order Sobolev derivatives. It is suggested that the asymptotic convergence may reach O(1/n^3) for potentials having second order Sobolev derivatives. The method preserves the favorable log(n) scaling for the time necessary to compute a path at a given discretization point. For the Wiener-Fourier series representation, the particular reweighted method designed in the present article is shown to have O(1/n^3) convergence if the potential has second order Sobolev derivatives. The convergence constant has superior dependence with the inverse temperature as compared to the partial averaging method for the same series. Because the expression of the convergence constant does not actually involve the second order derivatives of the potential, it is conjectured that the O(1/n^3) convergence extends to the potentials having first order Sobolev derivatives only.

cond-mat.stat-mech↗

arXiv2003

Asymptotic convergence of the partial averaging technique

We study the asymptotic convergence of the partial averaging method, a technique used in conjunction with the random series implementation of the Feynman-Kac formula. We prove asymptotic bounds valid for most series representations in the case when the potential has first order Sobolev derivatives. If the potential has also second order Sobolev derivatives, we prove a sharper theorem which gives the exact asymptotic behavior of the density matrices. The results are then specialized for the Wiener-Fourier series representation. It is found that the asymptotic behavior is o(1/n^2) if the potential has first order Sobolev derivatives. If the potential has second order Sobolev derivatives, the convergence is shown to be O(1/n^3) and we give the exact expressions for the convergence constants of the density matrices.

cond-mat.stat-mech↗

arXiv2002

Optimal series representations for numerical path integral simulations

By means of the Ito-Nisio theorem, we introduce and discuss a general approach to series representations of path integrals. We then argue that the optimal basis for both ``primitive'' and partial averaged approaches is the Wiener sine-Fourier basis. The present analysis also suggests a new approach to improving the convergence of primitive path integral methods. Current work indicates that this new technique, the ``reweighted'' method, converges as the cube of the number of path variables for ``smooth'' potentials. The technique is based on a special way of approximating the Brownian bridge which enters the Feynman-Kac formula and it does not require the Gaussian transform of the potential for its implementation.

physics.chem-ph↗