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Physical Review E

Ising-like criticality derived from the theory of fluids

В. Н. Бондарев

Published May 16, 2008• 13 citations
Publisher

Abstract

It is shown that the conventional theory based on the integral equations for the correlation functions of fluid allows for a reasonable scaling analysis of the critical phenomena. The calculated critical exponents are in rather good agreement with the experimental data for real media, but some of the exponents (first of all, Fisher's exponent eta ) differ from the values predicted by the three-dimensional Ising model. The possibility to obtain Ising-like criticality from the statistical theory of fluids means that the latter does not disagree with the Kadanoff-Wilson-Fisher renormalization group approach.

DOI
10.1103/physreve.77.050103

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It is shown that the conventional theory based on the integral equations for the correlation functions of fluid allows for a reasonable scaling analysis of the critical phenomena. The calculated critical exponents are in rather good agreement with the experimental data for real media, but some of the exponents (first of all, Fisher's exponent eta ) differ from the values predicted by the three-dimensional Ising model. The possibility to obtain Ising-like criticality from the statistical theory of fluids means that the latter does not disagree with the Kadanoff-Wilson-Fisher renormalization group approach.
Discoveries in physics often hinge on a disagreement between theory and experiment of a percent or less. Because of such a discrepancy, many workers in statistical mechanics have been concerned about the complete validity of the scaling and renormalization-group theories of critical phenomena. For the three-dimensional Ising model, the experimentally derived values for the critical exponents in fluids have agreed to three significant figures with those predicted by the renormalization group. However, the predictions disagreed by a couple of percent with exponents calculated by series expansions, and the claimed accuracy of these expansion estimates was much less than a percent.
The theory of critical phenomena in systems at equilibrium is reviewed at an introductory level with special emphasis on the values of the critical point exponents α, β, γ,..., and their interrelations. The experimental observations are surveyed and the analogies between different physical systems - fluids, magnets, superfluids, binary alloys, etc. - are developed phenomenologically. An exact theoretical basis for the analogies follows from the equivalence between classical and quantal `lattice gases' and the Ising and Heisenberg-Ising magnetic models. General rigorous inequalities for critical exponents at and below T c are derived. The nature and validity of the `classical' (phenomenological and mean field) theories are discussed, their predictions being contrasted with the exact results for plane Ising models, which are summarized concisely. Padé approximant and ratio techniques applied to appropriate series expansions lead to precise critical-point estimates for the three-dimensional Heisenberg and Ising models (tables of data are presented). With this background a critique is presented of recent theoretical ideas: namely, the `droplet' picture of the critical point and the `homogeneity' and `scaling' hypotheses. These lead to a `law of corresponding states' near a critical point and to relations between the various exponents which suggest that perhaps only two or three exponents might be algebraically independent for any system.
Critical finite-size scaling functions for the order-parameter distribution of the two- and three-dimensional Ising model are investigated. Within a recently introduced classification theory of phase transitions the universal part of critical finite-size scaling functions has been derived by employing a scaling limit which differs from the traditional finite-size scaling limit. In this paper the analytical predictions are compared with Monte Carlo simulation results. We find good agreement between the analytical expression and the simulation results. The agreement is consistent with the possibility that the functional form of the critical finite-size scaling function for the order-parameter distribution is determined uniquely by only a few universal parameters, most notably the equation of state exponent.
The primary goal of modern theories of critical phenomena is the evaluation of critical exponents from first principles, implying the need to get beyond mean field theories which predict the same exponents independent of the nature of the system. This chapter focuses on a new approach emerged in the 1960s based on two ideas—scaling and renormalization. It presents the physical picture developed by L. P. Kadanoff that provides a conceptual basis for scaling and which has become part of the standard language of critical phenomena. Renormalization provides a way to calculate the free energy not involving a direct evaluation of the partition function—it's a new paradigm in statistical mechanics. The chapter analyzes a simplified, approximate set of recursion relations valid at high temperature for a model with nearest and next-nearest-neighbor interactions. In 1971 K. G. Wilson published two articles marking the introduction of the renormalization group to the study of critical phenomena.
In a recent article we have shown that, by applying sophisticated summation methods to Wilson-Fisher's ε-expansion, it is possible from the presently known terms of the series to obtain accurate values of critical exponents for the 0 ( n ) symmetric n-vector model : these values are consistent with the best estimates obtained from three-dimensional Renormalization Group calculations and, in the case of Ising-like systems, with the exactly known two-dimensional values of the Ising model. The controversial conjecture has been recently formulated that some fractal lattices could interpolate regular lattices in non-integer dimensions. Numerical calculations have been done for the Ising model. To allow for direct comparison with Renormalization Group values, we present here estimates for exponents in non-integer dimensions d(1
A pair-correlation function ansatz, previously used in the derivation of the Ising model critical exponent η for the square lattice from a criticality equation within the i-δ approximation, is investigated further. Various methods are used to calculate the multispin correlation functions entering the criticality equation. The calculations are extended to the three-dimensional cubic lattices. It is found that the values obtained for η are relatively insensitive to the specific method used. In two dimensions, η takes the values 0.2506 (i-δ method), 0.2471, and 0.2490 (renormalized Hamiltonian methods), thereby remaining within 1.2% of the exact value 1/4. In three dimensions, η ranges from 0.030 to 0.075, being of the same order of magnitude as the series value 0.041 ± 0.01.