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).

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[2]  Z. Alexandrowicz,et al.  On the zero fluctuation of the “microscopic free energy” and its potential use , 1976 .

[3]  Michael E. Fisher,et al.  Scaling Theory for Finite-Size Effects in the Critical Region , 1972 .

[4]  C. Domb.  On co-operative effects in finite assemblies , 1965 .

[5]  K. Binder,et al.  Dynamic properties of the Monte Carlo method in statistical mechanics , 1973 .

[6]  J. Gibbs.  Elementary Principles in Statistical Mechanics , 1902 .

[7]  R. Kikuchi.  A Theory of Cooperative Phenomena , 1951 .

[8]  Z. Alexandrowicz.  Stochastic Models for the Statistical Description of Lattice Systems , 1971 .

[9]  J. W. Essam,et al.  Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point , 1963 .

[10]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[11]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[12]  H. Bethe.  Statistical Theory of Superlattices , 1935 .

[13]  David P. Landau,et al.  Finite-size behavior of the Ising square lattice , 1976 .

[14]  M. Fisher.  The theory of equilibrium critical phenomena , 1967 .

[15]  J. Gunton.  Finite size effects at the critical point , 1968 .

[16]  G. F. Newell,et al.  On the Theory of the Ising Model of Ferromagnetism , 1953 .

[17]  H. Müller-Krumbhaar,et al.  A „self consistent” monte carlo method for the heisenberg ferromagnet , 1972 .

[18]  P. C. Hohenberg,et al.  Surface effects on magnetic phase transitions , 1974 .

[19]  Z. Alexandrowicz,et al.  Stochastic Model for a Fluid of Hard Cubes with Attractive Potential , 1972 .

[20]  I. R. Mcdonald,et al.  The study of simple liquids by computer simulation , 1970 .

[21]  D. S. McKenzie,et al.  Specific heat of a three dimensional Ising ferromagnet above the Curie temperature. II , 1972 .

[22]  D. S. Gaunt,et al.  High temperature series for the susceptibility of the Ising model. II. Three dimensional lattices , 1972 .

[23]  K. Binder.  Statistical mechanics of finite three-dimensional Ising models , 1972 .

[24]  Kurt Binder,et al.  Monte Carlo computer experiments on critical phenomena and metastable states , 1974 .

[25]  M. Fisher,et al.  Bounded and Inhomogeneous Ising Models. I. Specific-Heat Anomaly of a Finite Lattice , 1969 .

[26]  B. Alder,et al.  Studies in Molecular Dynamics. I. General Method , 1959 .