Showing papers on "Quantum Monte Carlo published in 1970"

Soft‐Sphere Equation of State

Abstract: The pressure and entropy for soft‐sphere particles interacting with an inverse twelfth‐power potential are determined using the Monte Carlo method. The solid‐phase entropy is calculated in two ways: by integrating the single‐occupancy equation of state from the low density limit to solid densities, and by using solid‐phase Monte Carlo pressures to evaluate the anharmonic corrections to the lattice‐dynamics high‐density limit. The two methods agree, and the entropy is used to locate the melting transition. The computed results are compared with the predictions of the virial series, lattice dynamics, perturbation theories, and cell models. For the fluid phase, perturbation theory is very accurate up to two‐thirds of the freezing density. For the solid phase, a correlated cell model predicts pressures very close to the Monte Carlo results.

Letter to the Editor—A Monte Carlo Method for the Approximate Solution of Certain Types of Constrained Optimization Problems

Abstract: This paper considers the problem of minimizing a function F(x1, …, xn) over a closed, bounded region S in n-dimensional space under the assumption that there exists a unique minimizing point (z1, …, zn)ϵS. In a previous paper I represented the coordinates of the minimizing point as the limit of a ratio of integrals. The same type of ratio appears, in a different context, in statistical mechanics where a Monte Carlo method has been developed, by Metropolis et al., for its numerical evaluation. The purpose of this paper is to point out the connection of Metropolis's method with the above type of minimisation problem. The idea of the method is to associate with the minimization problem a Markov chain whose sample averages converge with probability one to (approximately) the minimizing point (z1, …, zn). The Markov chain should be easily realizable on a computer. An estimate of the error from sampling over a finite time period is given.

Test of the Monte Carlo Method: Fast Simulation of a Small Ising Lattice

Abstract: A very fast stochastic procedure is used to generate samples of configurations of a 4 × 4 periodic Ising lattice in zero field. Running on the IBM 7094, we require only 15 μsec to process each site (cf. Yang who required 300 μsec on the 7090) and hence we generate 4000 completely new configurations each second. Our main results are based on samples of 106 configurations at each of 10 temperatures; we also took samples of 107 configurations at three temperatures. The 4 × 4 lattice can be solved exactly without much difficulty. Hence our data give information about the Monte Carlo method itself, especially its rate of convergence. We define the statistical inefficiency (SI) in a variable as the limiting ratio of the observed variance of its long‐term averages to their expected (Gaussian) variance. (Thus, if the SI is 3, then averages of the variable taken over Monte Carlo runs of three million configurations will be as accurate as averages over one million configurations drawn randomly from the true ensemble. The SI accounts for correlations between configurations closely following one another in the Monte Carlo run.) We find that for this lattice the SI in energy never exceeds 12, and that its maximum occurs slightly above the temperature which is critical for the infinite lattice. We confirm earlier statements that the influence of the initial configuration is lost very quickly, except when two phases coexist.

NpT‐Ensemble Monte Carlo Calculations for the Hard‐Disk Fluid

Abstract: The NpT‐ensemble Monte Carlo method previously described, in which the hard‐disk interaction is represented by a soft pseudopotential in a reduced configuration space with a fixed periodic enclosure, is used to calculate the equation of state and radial distribution function in the high‐density fluid phase. The equation of state agrees within statistical error with the (3, 3) Pade approximant to the six‐term virial expansion and, for large enough systems, with Salsburg's analysis of the N dependence. The results also agree with recent Monte Carlo calculations by Chae, Ree, and Ree using the NVT ensemble. Some qualitative results for systems of 48 and 90 disks which lend further support to the now‐accepted presence of a solid–fluid phase transition are also presented.

Andymg3: the basic program of a series of monte carlo programs for time- dependent transport of particles and photons.

Abstract: ANDYMG3 is the basic program of a aeries of Monte Carlo programs designed to solve applied problems in timedependent particle and photon transport for general geometries. Particle or photon type and energy are identified ty multlenersy-proup number. Cross sections are read in SJJ format viMi scattering pattern components up to Pj. Particle splitting and termination routines permit negative weights. Particle and photon emission can be delayed. A library of reaction i:ross sections is provided so that practically useful results are computed after execution of the Monte Carlo. 1'he AHDYMG3 program, in FORTRAK-IV for the CDC-66OO and UNIVAC-1108 computers, requires. 0,5-2 msec per collision (CDC-66OO) and a field length less than 60gK for most problems.

Feasibility study of Monte Carlo modelling techniques for distributed-parameter systems. Part 2: Search problem involving a distributed-parameter system solved by Monte Carlo methods

Abstract: The theory given in Pt. 1 of the paper is illustrated. The paper takes, as an example of the use of Monte Carlo solutions of partial differential equations, a linear model for the heating of steel billets, and solves an optimal-control problem of achieving uniform temperature in minimum time.

Exact Analytic Solution for First‐Order Fluorescent Buildup in One Dimension

Abstract: The problem of fluorescence in slab geometries is solved exactly and analytically to first order. Analytic expressions, whose direct evaluation should be faster and more accurate than current numerical methods, are derived for differential and integral dose and for differential reflection and transmission coefficients, for the single slab case. The basic expressions are then used to obtain the general solution for the dose profile of the multislab problem. The limitations of the first‐order approximation are discussed and illustrated by comparison with Monte Carlo results. It is found that even in a situation especially chosen to ensure the validity of the first‐order approximation, a Monte Carlo calculation reveals the presence of higher‐order effects.