Hagai Meirovitch

Computer simulation of self‐avoiding walks: Testing the scanning method

The scanning method is a computer simulation technique for macromolecules suggested recently. The method is described here in detail and its applicability (in contrast to other simulation techniques) to a wide range of chain models is discussed. It is argued that for most of these models the scanning method constitutes the most efficient tool for estimating the entropy. The method is applied to self‐avoiding walks (SAWs) (of N≤399 steps) on both a three‐choice square lattice and a five‐choice simple cubic lattice and the results for the entropy, the end‐to‐end distance, the radius of gyration, and other quantities of interest are found to be in very good agreement with the results obtained by other numerical techniques. In particular, our calculations support the law of divergence for the persistence length of SAWs on two‐dimensional lattices suggested recently by Grassberger. However, for the simple cubic lattice, the persistence length is found by us to be constant.

Computer simulation of long polymers adsorbed on a surface. II. Critical behavior of a single self‐avoiding walk

The scanning simulation method is applied to a model of polymer adsorption in which a single self‐avoiding walk is terminally attached to an attracting impenetrable surface on a simple cubic lattice. Relatively long chains are studied, of up to 1000 steps, which enable us to obtain new estimates for the reciprocal transition temperature ‖e‖/kBTa=θa =0.291±0.001 (e is the interaction energy of a monomer with the surface), the crossover exponent φ=0.530±0.007 and the free energy exponents at Ta, γ1SB =1.304±0.006 and γ11SB =0.805±0.015. At T=∞ we obtain, γ1=0.687±0.005, γ11=−0.38±0.02, and the effective coordination number q=4.6839±0.0001, which are in good agreement with estimates obtained by other methods. At T>Ta we demonstrate the existence of strong correction to scaling for the perpendicular part of the mean‐square end‐to‐end distance 〈R2〉⊥ and for the monomer concentration profile ρ(z) (z is the distance from the surface). At T=∞ the leading correction to scaling term for 〈R2〉⊥ is c/Nψ, where c≊−0.9 ...

A new method for simulation of real chains: scanning future steps

A new method for simulating real polymer chains is developed and applied to self-avoiding walks (SAWs) of length 49-599 on a three-choice square lattice. Very good results for the entropy are obtained which deviate from the series expansion estimates by 0.1-2%. The author also discusses how to extend the method to models of polymer chains with both excluded volume and finite interactions (attractive or repulsive). The authors method is expected to be more efficient than other simulation methods for treating self-interacting SAWs and chains which are subject to various lattice constraints.

Computer simulation of long polymers adsorbed on a surface. I. Corrections to scaling in an ideal chain

This paper is the first in a series of papers in which polymer adsorption on a surface is studied by computer simulation using the ‘‘scanning method.’’ This method is especially efficient to handle chain systems with finite interactions and geometrical constraints. Here we test the method by applying it to models of a single random walk (without excluded volume) on a simple cubic lattice, which are solved analytically; in the immediately following paper a self‐avoiding walk model is treated. The scanning method is found to be extremely efficient, where walks of up to N=105 steps can be simulated reliably, leading thereby to very precise estimates of transition temperatures and critical exponents. In particular we test carefully for a lattice model the range of validity of scaling functions developed by Eisenrigler, Kremer and Binder [J. Chem. Phys. 77, 6296 (1982)] for a continuous model. We pay a special attention to corrections to scaling and demonstrate that they are strong above the transition tempera...

On the zero fluctuation of the “microscopic free energy” and its potential use

The fluctuations of the “microscopic free energy” calculated with the ensemble probability are shown to be zero. We suggest that this result be used for estimating approximate free energies calculated on the basis of the minimum free energy principle. As an example the estimation is performed with respect to a certain computer simulation of the square Ising lattice. The zero fluctuations also can be used to obtain relations among fluctuations with the accurate ensemble probability distribution.

A Monte Carlo study of the entropy, the pressure, and the critical behavior of the hard-square lattice gas

An approximate technique for estimating the entropyS with computer simulation methods, suggested recently by Meirovitch, is applied here to the Metropolis Monte Carlo (MC) simulation of the hard-square lattice gas in both the grand canonical and the canonical ensembles. The chemical potentialμ, calculated by Widoms method, andS enable one to obtain also the pressureP. The MC results are compared with results obtained with Padé approximants (PA) and are found to be very accurate; for example, at the critical activityzc the MC and the PA estimates forS deviate by 0.5%. Beyondzc this deviation decreases to 0.01% and comparable accuracy is detected forP. We argue that close tozc our results forS, μ, andP are more accurate than the PA estimates. Independent of the entropy study, we also calculate the critical exponents by applying Fishers finite-size scaling theory to the results for the long-range order, the compressibility and the staggered compressibility, obtained for several lattices of different size at zc. The data are consistent with the critical exponents of the plane Ising latticeβ=1/8,ν=1,γ=7/4, andα=0. Our values forβ and ν agree with series expansion and renormalization group results, respectively,α=0 has been obtained also by matrix method studies; it differs, however, from the estimate of Baxteret al. α=0.09 ± 0.05. As far as we knowγ has not been calculated yet.

Computer simulation of a lattice model of nematic liquid crystal

Abstract A lattice version of the Maier-Saupe model of anematic liquid crystal is investigated with the help of the Stochastic Models (SM) method which is an approximate computer simulation technique different from the usual Metropolis Monte Carlo method. The model consists of red-like molecules which are restricted to be on a simple cubic lattice. The interaction energy Ek,l = −e( 3 2 cos2 θk,l − 1 2 ) is only between nearest neighbors molecules. We find a first order phase transition at e/kBT = 0.90+- 0.003 with latent heat 0.12 ± 0.04 and discontinuity in the order parameter 0.27 ± 0.04. These results are very close to Lebwohl and Lashers results, who first studied this model using the Metropolis Monte Carlo method.

The stochastic models method applied to the critical behavior of Ising lattices

The stochastic models (SM) computer simulation method for treating manybody systems in thermodynamic equilibrium is investigated. The SM method, unlike the commonly used Metropolis Monte Carlo method, is not of a relaxation type. Thus an equilibrium configuration is constructed at once by adding particles to an initiallyempty volume with the help of a model stochastic process. The probability of the equilibrium configurations is known and this permits one to estimate the entropy directly. In the present work we greatly improve the accuracy of the SM method for the two and three-dimensional Ising lattices and extend its scope to calculate fluctuations, and hence specific heat and magnetic susceptibility, in addition to average thermodynamic quantities like energy, entropy, and magnetization. The method is found to be advantageous near the critical temperature. Of special interest are the results at the critical temperature itself, where the Metropolis method seems to be impractical. At this temperature, the average thermodynamic quantities agree well with theoretical values, for both the two and three-dimensional lattices. For the two-dimensional lattice the specific heat exhibits the expected logarithmic dependence on lattice size; the dependence of the susceptibility on lattice size is also satisfactory, leading to a ratio of critical exponentsγ/ν=1.85 ±0.08. For the three-dimensional lattice the dependence of the specific heat, long-range order, and susceptibility on lattice size leads to similarly satisfactory exponents:α=0.12 ±0.03,β=0.30 ±0.03, andγ=1.32 ±0.05 (assuming ν=2/3).

An approximate stochastic process for computer simulation of the Ising model at equilibrium

Develops the stochastic models (SM) method which is an approximate computer simulation technique for treating many-body systems in thermodynamic equilibrium, suggested by Alexandrowicz (1971, 1972). The method is applied as a test to the critical region of the square Ising lattice. First the authors define the exact transition probabilities (TP), which are parameter independent; since their calculation is impractical for a large lattice they also define approximate TP (based on two parameters) which can be improved systematically. The saving in computer time, compared with previous work, (where 10 parameters have been used) is significant, and the accuracy of the free energy is increased by a factor of 4-100. The results for chi , the magnetic susceptibility, are also improved, in particular for T<Tc. Using the finite size scaling theory and assuming v=1, the authors estimate at Tc, the exponent gamma to be 1.80+or-0.12.