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Europhysics Letters (EPL)

Method to estimate critical exponents using numerical studies

Ehud Perlsman, Shlomo Havlin

Published Apr 1, 2002• 14 citations
Publisher

Abstract

A novel method to estimate the critical point and critical exponents of physical models from numerical studies is presented. The method utilizes linear approximation to compute the values of the characteristic variables in the near vicinity of the critical point from numerical results obtained for only one point in that vicinity. The method is applied to two models: Two-dimensional directed percolation, and one-dimensional reaction-diffusion model. In both cases the critical point and critical exponents are determined with higher accuracy than achieved in former studies. In the reaction-diffusion model, the results strongly suggest simple rational values of 1/3, 1/12, etc., for the characteristic exponents.

DOI
10.1209/epl/i2002-00621-7

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A novel method to estimate the critical point and critical exponents of physical models from numerical studies is presented. The method utilizes linear approximation to compute the values of the characteristic variables in the near vicinity of the critical point from numerical results obtained for only one point in that vicinity. The method is applied to two models: Two-dimensional directed percolation, and one-dimensional reaction-diffusion model. In both cases the critical point and critical exponents are determined with higher accuracy than achieved in former studies. In the reaction-diffusion model, the results strongly suggest simple rational values of 1/3, 1/12, etc., for the characteristic exponents.
The author introduces a new method of calculating critical amplitude ratios using series, which is both simple and powerful. This method, which gives estimates for the amplitude ratios that are neither biased by the values of the critical points nor by the critical exponents, is applied to several models. It is shown that this method produces results where no reliable estimates from series expansion exist. In particular one finds 0.025+or-0.001 for AT'/B 2 for the 3D Ising model and 220+or-10 for C + /C - for the two-dimensional percolation model in agreement with, and with more accuracy than, values obtained by other methods.
Pad\'e approximants have been used to estimate both the critical temperature and the critical-point exponents of the Ising model. In the standard Pad\'e analysis the approximants near the diagonal are given more credence. In the present study of two-dimensional models more credence is given to those entries for which the estimated critical temperature is close to the exact value. For three-dimensional models an analogous method is employed using the "exact" critical temperature estimated from the high-temperature compressibility series. The rationale for selecting these particular approximants is based on the fact that one is estimating critical-point behavior by extrapolating from a series representation. Thus, if the estimated critical point is shifted from its true value, one should expect a corresponding shift in the estimated exponent. Indeed, this is what is found.
The use of series expansions to determine the nature of critical point singularities, especially the values of the critical exponents, is reviewed.Ratio techniques, based on extrapolation of the ratios of successive expansion coefficients, and Padé approximant methods, utilizing logarithmic derivatives, are explained, illustrated, and compared. Introduction.In this lecture I will survey briefly the phenomena observed near the critical point of a system.The range of behavior is very varied and we shall only examine one or two typical examples.The main aim will be to establish the existence of the so-called "critical point exponents" which describe the nature of the mathematical singularities exhibited by the different thermodynamic functions as the critical point is approached.The theory of these singularities may be developed via suitable statistical-mechanical models which, while caricatures of a realistic physical system, retain such vital features as the dimensionality of space, the finite range of the interactions, and certain basic symmetry properties.These models in turn are usually intractable as regards exact mathematical analysis.However, general methods can be developed which yield power series expansions for many functions of interest; frequently between 8 to 30 leading coefficients of such power series can be obtained.The problem is then a numerical one of extrapolating the series in such a way as to estimate the nature of the particular critical point singularity and the behavior of the function in its vicinity.I will show how this can, in favorable cases, be handled by studying the ratios of successive expansion coefficients.The method was pioneered by Domb and his coworkers, especially Sykes and Fisher [1], [2].The reliability of the techniques can be tested against the relatively few closed form mathematical results, in particular, Onsager's renowned solution of the two-dimensional Ising models [3].In less favorable examples this approach fails, and then, as first demonstrated by Baker [3], Padé approximants have a vital role to play.In a number of important cases the intelligent application of Padé approximants has been strikingly successful [3], [4].Such techniques have thus established a central place for themselves in the
Abstract Until the early 1970s all calculations of critical exponents were made either by exactly solving a model for its thermodynamic properties and then examining its behaviour in the critical regime (as with the Ising model in one and two dimensions) or by direct numerical simulations (as with the three-dimensional Ising model) or by extrapolation from approximate solutions which become invalid in the vicinity of a phase transition (high-temperature expansions, for example). However, in a now-famous paper published in 1966, Kadanoff presented arguments which, he maintained, would allow one to simplify calculations in the critical regime to the point at which critical exponents could be extracted, without ever working out the partition function for the problem. However, though Kadanoff’s ideas showed great physical intuition about the processes giving rise to critical phenomena, they lacked the mathematical precision which might have given one faith in his results.
In this study we focus on the non-equilibrium phase transition whose asymptotic critical behavior is governed by the dynamic isotropic percolation universality class. In order to quantitatively characterize universal properties associated with this class, we employ a field-theoretic formulation of the model augmented with perturbative renormalization group technique. Utilizing such approach enables us to systematically compute the critical exponents in higher orders of perturbation theory. In particular, we perform an analysis to the three-loop precision. To this order we determine the renormalization constants and present relations for critical exponents in terms of critical dimensions. Finally, we also present numerically estimated values of critical exponents obtained from perturbation theory.