Hybrid Monte Carlo Algorithm Research Articles

Simulated Annealing Using Hybrid Monte Carlo

We propose a variant of the simulated annealing method for optimization in the multivariate analysis of differentiable functions. The method uses global actualizations via the hybrid Monte Carlo algorithm in their generalized version for the proposal of new configurations. We show how this choice can improve upon the performance of simulated annealing methods (mainly when the number of variables is large) by allowing a more effective searching scheme and a faster annealing schedule.

Monte Carlo simulations of water clusters on a parallel computer using an ab initio potential

We have performed a simulation of a cluster of five water molecules at 300 K in which the potential at each step was calculated at the Hartree-Fock level using a 6–31G ∗ basis set. A parallel big move (hybrid) Monte Carlo algorithm was used to conduct this modelling. We compare the results obtained for this 'quantum' system with those obtained from a conventional simulation employing an effective pair potential. We also estimate properties of the quantum system in terms of the configurations generated by the conventional simulation by employing the appropriate Boltzmann weighting. These estimates are in good agreement with those obtained from the full quantum simulation.

A polynomial hybrid Monte Carlo algorithm

We present a simulation algorithm for dynamical fermions that combines the multiboson technique with the Hybrid Monte Carlo algorithm. We find that the algorithm gives a substantial gain over the standard methods in practical simulations. We point out the ability of the algorithm to treat fermion zeromodes in a clean and controllable manner.

Fast fermion Monte Carlo

Three possibilities to speed up the Hybrid Monte Carlo algorithm are investigated. Changing the step-size adaptively brings no practical gain. On the other hand, substantial improvements result from using an approximate Hamiltonian or a preconditioned action.

Decorrelating the topology in full QCD

We investigate the performance of the hybrid Monte Carlo algorithm in updating non-trivial global topological structures. We find that the hybrid Monte Carlo algorithm has serious problems decorrelating the global topological charge. This represents a warning which must be seriously considered when simulating full QCD, regardless of the number and type of fermions, with this or any similar algorithm. Simulated tempering is examined as a means of accelerating the decorrelation.

Hybrid Monte Carlo without pseudofermions

We introduce a dynamical fermion algorithm which is based on the hybrid Monte Carlo (HMC) algorithm, but without pseudofermions. The molecular dynamics steps in HMC are retained except the derivatives with respect to the gauge fields are calculated with Z 2 noise. The determinant ratios are estimated with the Pade - Z 2 method. Finally, we use the Kennedy-Kuti linear accept/reject test for the Monte Carlo step which is shown to respect detailed balance. We comment on the comparison of this algorithm with the pseudofermion algorithm/

Exact local bosonic algorithm for dynamical quarks

We present an exact local bosonic algorithm for the simulation of dynamical fermions in lattice QCD. We show that this algorithm is a viable alternative to the Hybrid Monte Carlo algorithm.

Can we do better than hybrid Monte Carlo in lattice QCD?

The Hybrid Monte Carlo algorithm for the simulation of QCD with dynamical staggered fermions is compared with Kramers equation algorithm. We find substantially different autocorrelation times for local and nonlocal observables. The calculations have been performed on the parallel computer CRAY T3D.

Implementation of Symanzik's improvement program for simulations of dynamical Wilson fermions in lattice QCD

We discuss the implementation of a Sheikholeslami-Wohlert term for simulations of lattice QCD with dynamical Wilson fermions as required by Symanzik's improvement program. We show that for the Hybrid Monte Carlo or Kramers equation algorithm standard even-odd preconditioning can be maintained. We design tests of the implementation using analytically and numerically computed cumulant expansions. We find that, for situations where the average number of Conjugate Gradient iterations exceeds 200, the overhead is only about 20%.

Instabilities and non-reversibility of molecular dynamics trajectories

The theoretical justification of the Hybrid Monte Carlo algorithm depends upon the molecular dynamics trajectories within it being exactly reversible. If computations were carried out with exact arithmetic then it would be easy to ensure such reversibility, but the use of approximate floating point arithmetic inevitably introduces violations of reversibility. In the absence of evidence to the contrary, we are usually prepared to accept that such rounding errors can be made small enough to be innocuous, but in certain circumstances they are exponentially amplified and lead to blatantly erroneous results. We show that there are two types of instability of the molecular dynamics trajectories which lead to this behavior, instabilities due to insufficiently accurate numerical integration of Hamilton's equations, and intrinsic chaos in the underlying continuous fictitious time equations of motion themselves. We analyze the former for free field theory, and show that it is essentially a finite-volume effect. For the latter we propose a hypothesis as to how the Liapunov exponent describing the chaotic behavior of the fictitious time equations of motion for an asymptotically free quantum field theory behaves as the system is taken to its continuum limit, and explain why this means that instabilities in molecular dynamics trajectories are not a significant problem for Hybrid Monte Carlo computations. We present data for pure SU(3) gauge theory and for QCD with dynamical fermions on small lattices to illustrate and confirm some of our results.

Hybrid Monte Carlo and topological modes of full QCD

We investigate the performance of the hybrid Monte Carlo algorithm, the standard algorithm used for lattice QCD simulations involving fermions, in updating non-trivial global topological structures. We find that the hybrid Monte Carlo algorithm has serious problems decorrelating the global topological charge. This represents a warning which must be seriously considered when simulating full QCD by hybrid Monte Carlo.

Critical dynamics in the two-dimensional classical XY model: A spin-dynamics study.

Using spin-dynamics techniques we have performed large-scale computer simulations of the dynamic behavior of the classical three component XY model (i.e., the anisotropic limit of an easy-plane Heisenberg ferromagnet), on square lattices of size up to ${192}^{2}$, for several temperatures below, at, and above ${\mathit{T}}_{\mathrm{KT}}$. The temporal evolution of spin configurations was determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method, with initial spin configurations generated by a hybrid Monte Carlo algorithm. The neutron-scattering function S(q,\ensuremath{\omega}) was calculated from the resultant space-time displaced spin-spin correlation function. Pronounced spin-wave peaks were found both in the in-plane and the out-of-plane scattering function over a wide range of temperatures. The in-plane scattering function ${\mathit{S}}^{\mathit{xx}}$ also has a large number of clear but weak additional peaks, which we interpret to come from two-spin-wave scattering. In addition, we observed a small central peak in ${\mathit{S}}^{\mathit{xx}}$, even at temperatures well below the phase transition. We used dynamic finite-size scaling theory to extract the dynamic critical exponent z. We find z=1.00(4) for all T\ensuremath{\le}${\mathit{T}}_{\mathrm{KT}}$, in excellent agreement with theoretical predictions, although the shape of S(q,\ensuremath{\omega}) is not well described by current theory. \textcopyright{} 1996 The American Physical Society.

Molecular dynamics, Langevin and hydrid Monte Carlo simulations in a multicanonical ensemble

We demonstrate that the multicanonical approach is not restricted to Monte Carlo simulations, but can also be applied to simulation techniques such as the molecular dynamics, Langevin and hybrid Monte Carlo algorithms. The effectiveness of the methods is tested with an energy function for the protein folding problem. Simulations in the multicanonical ensemble by the three methods are performed for a penta peptide, Met-enkephalin. For each algorithm, it is shown that from only one simulation run one cannot only find the global minimum energy conformation but also obtain probability distributions in the canonical ensemble at any temperature, which allows the calculation of any thermodynamic quantity as a function of temperature.

Casimir effect in critical systems: A Monte Carlo simulation.

If a critical system is confined to a finite geometry, critical fluctuations of the order parameter generate long-ranged forces between the system boundaries. These forces, commonly known as Casimir forces, are characterized by universal amplitudes and scaling functions. A hybrid Monte Carlo algorithm has been devised and used to measure the Casimir amplitudes directly and accurately. We apply the algorithm to a critical $q$-state Potts model confined to a rectangular $M\ifmmode\times\else\texttimes\fi{}L$ geometry in $d=2$ dimensions and to a critical Ising model confined to a ${M}^{2}\ifmmode\times\else\texttimes\fi{}L$ geometry in $d=3$ dimensions. We find good agreement with rigorous results in $d=2$ and compare our results with field-theoretic estimates of the Casimir amplitude in $d=3$.

Tuning the generalized Hybrid Monte Carlo algorithm

We discuss the analytic computation of autocorrelation functions for the generalized Hybrid Monte Carlo algorithm applied to free field theory and compare the results with numerical results for the $O(4)$ spin model in two dimensions. We explain how the dynamical critical exponent $z$ for some operators may be reduced from two to one by tuning the amount of randomness introduced by the updating procedure, and why critical slowing down is not a problem for other operators.

Interplay of monopoles and chiral symmetry breaking in noncompact lattice QED.

Non-compact lattice QED is simulated for various numbers of fermion species $N_f$ ranging from 8 through 40 by the exact Hybrid Monte Carlo algorithm. Over this range of $N_f$, chiral symmetry breaking is found to be strongly correlated with the effective monopoles in the theory. For $N_f$ between 8 and 16 the chiral symmetry breaking and monopole percolation transitions are second order and coincident. Assuming powerlaw critical behavior, the correlation length exponent for the chiral transition is identical to that of monopole percolation. This result supports the conjecture that monopole percolation ``drives" the nontrivial chiral transition. For $N_f$ between 20 and 32, the monopoles experience a first order condensation transition coincident with a first order chiral transition. For $N_f$ as large as 40 both transitions are strongly suppressed. The data at large $N_f (N_f \mathrel {\mathpalette \vereq >} 20)$ is interpreted in terms of a strongly interacting monopole gas-liquid transition.

Kramers equation algorithm for simulations of QCD with two flavors of Wilson fermions and gauge group SU (2 )

We compare the Hybrid Monte Carlo (HMC) and the Kramers equation algorithms for simulations of QCD with two flavors of dynamical Wilson fermions and gauge group SU(2). The results for the performance of both algorithms are obtained on 6 312, 12 4 and 16 4 lattices at a pion to ϱ meson mass ratio of m π / m ϱ ∼ 0.9. We find that the Kramers equation algorithm gives an equally good performance as the HMC algorithm. We demonstrate that the classical equations of motion used in these algorithms lack reversibility in practical simulations and behave like those of a chaotic dynamical system with a Liapunov exponent ν ∼ 0.75.

Failure of the Regge approach in two dimensional quantum gravity

Regge's method for regularizing euclidean quantum gravity is applied to two dimensional gravity. We use two different strategies to simulate the Regge path integral at a fixed value of the total area: A standard Metropolis simulation combined with a histogramming method and a direct simulation using a Hybrid Monte Carlo algorithm. Using topologies with genus zero and two and a scale invariant integration measure, we show that the Regge method does not reproduce the value of the string susceptibility of the continuum model. We show that the string susceptibility depends strongly on the choice of the measure in the path integral. We argue that the failure of the Regge approach is due to spurious contributions of reparametrization degrees of freedom to the path integral.

FULL QCD ON APE100 MACHINES

We present the first tests and results from a study of QCD with two flavours of dynamical Wilson fermions using the Hybrid Monte Carlo Algorithm (HMCA) on APE100 machines. The simulations have been performed on 64 lattice for the pure gauge HMCA and on 84, 123×32 lattices for full QCD configurations. We discuss the inversion algorithm for the fermionic operator, the methods used to overcome the problems arising using a 32 bit machine and the implementation of a new random number generator for APE100 machines. We propose different scenarios for the simulation of physical observables, with respect to the memory capacity and speed of different APE100 configurations.

Hybrid Monte Carlo: An efficient algorithm for condensed matter simulation

AbstractA detailed comparison has been made of the performance of molecular dynamics and hybrid Monte Carlo simulation algorithms for calculating thermodynamic properties of 2D Lennard‐Jonesium. The hybrid Monte Carlo simulation required an order of magnitude fewer steps than the molecular dynamics simulation to calculate reproducible values of the specific heat. The ergodicity of the two algorithms was compared via the use of intermediate scattering functions. For classical systems the intermediate scattering functions should be real; however, a simple analysis demonstrates that this function will have a significant imaginary component when ergodicity breaks down. For q vectors near the zone boundary, the scattering functions are real for both algorithms. However, for q vectors near the zone center (i.e., harmonic, weakly coupled modes), the scattering function calculated via molecular dynamics had a significantly larger imaginary component than that calculated using hybrid Monte Carlo. Therefore, the hybrid Monte Carlo algorithm is more ergodic and samples phase space more efficiently than molecular dynamics for simulations of 2D Lennard‐Jonesium. © 1994 by John Wiley & Sons, Inc.