A method is developed for determining the critical point and the critical exponent from terms of a series expansion for a restricted class of functions. The advantage of the method is that under certain reasonable assumptions it provides estimates for errors which are narrower than those usually given. As examples, we consider the high-temperature susceptibility series for the Ising model and the analogous chain-generating function for the excluded volume problem.
A method is developed for determining the critical point and the critical exponent from terms of a series expansion for a restricted class of functions. The advantage of the method is that under certain reasonable assumptions it provides estimates for errors which are narrower than those usually given. As examples, we consider the high-temperature susceptibility series for the Ising model and the analogous chain-generating function for the excluded volume problem.
A simple method is proposed for determining characteristic parameters of critical-point singularities from power-series expansions. It applies to a broad class of functions of physical interest and provides acceptable, simultaneous approximations to the critical point, exponent, and amplitude. Several examples of low- and high-temperature series expansions for thermodynamic properties of spin-(1/2) Ising models are studied, and quite accurate results are obtained for both close- and loose-packed arrays.
For pt. III, see abstr. A21890 of 1970. A new low-temperature expansion variable for the Ising model is given. Expansions in the new variable, x=1-tanh(J/kT), then converge right up to the critical point. Power series in the new variable are derived, and then analysed by the Pade method and the method of N-point fits. For the spontaneous magnetization of the Ising model in three dimensions. beta =0.312+or-0.002 for the simple cubic lattice, beta =0.3125+or-0.0015 for the body-centred cubic lattice, and beta =0.312+or-0.004 for the face-centred cubic lattice. For the low-temperature susceptibility gamma =1.30+or-0.02, gamma =1.28+or-0.04, and gamma '=1.29+or-0.04 for the simple cubic, body-centred cubic and face-centred cubic lattices respectively. For the low-temperature specific heat the authors estimate 1 / 16 <or= alpha =0.2 for all three lattices.
Closed-form approximations in crystal statistics have suffered from the defect that no steady series of approximations was available from which the rate of convergence could be assessed. The method of Yvon, based on a cluster-integral type of development, can furnish such a series of approximations. It is shown how to construct the partition functions for such approximations, the key being the use of the Mayer theory for multi-component assemblies. The method is applied to various properties of the Ising model of a ferromagnet and antiferromagnet, and results are obtained which are consistent with those based on series expansions. Previous investigations of Fournet which gave different results are shown to have made use of an insufficient number of terms in the approximation. The statistical mechanics of interacting systems on crystal lattices has attracted a good deal of attention in recent years. Exact results have been confined to a restricted class of problems in two dimensions, but for more general problems two independent approaches have been pursued. Closed-form approximations have been developed which ignore certain features of the statistical problem; and limited numbers of terms of exact series expansions for the partition function at high and low temperatures have been calculated. (See Domb (i 960) for a detailed review.) Series expansions are most useful when the terms are all consistent in sign. Convergence is then assured in the critical region, and they can yield reliable information on critical behaviour. An example of this kind is provided by the magnetic susceptibility of the Ising model above the Curie point (Domb & Sykes 1957; Fisher i959). When the terms are not consistent in sign some rearrangement must be sought, and the position is less satisfactory. For some problems, such as antiferromagnetism in a magnetic field, little information about critical behaviour can be derived from series expansions. Closed-form approximations could also be used for estimating critical behaviour if a series of successive approximations were available, so that the general trend of a particular property could be assessed. However, for most of the approximations discussed in the literature, the labour increases rapidly with the order of the approximation, and the process has not been followed for more than two or three stages. A method which does offer the possibility of a steady series of approximations,
Low temperature Ising model series in the usual variable z = exp(−4 J/kT ) are analysed. The positions of the singularities in the disk|z| <or=|z c | and the values of the corresponding critical exponents are determined for the simple cubic, body-centred cubic and face-centred cubic lattices. We obtain the following estimates of the physical critical exponents: β = 0·312 ± 0·002 for the simple cubic and face-centred cubic lattices, and β = 0·312 ± 0·004 for the body-centred cubic lattice. γprime = 1·31 ± 0·02 for the simple cubic lattice, γprime = 1·28 ± 0·04 for the body-centred cubic lattice, and γprime = 1·3 ± 0·1 for the face-centred cubic lattice. For the low temperature specific heat exponent we obtain fraction one-sixteens <or= αprime <or= ¼.
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
Low temperature Ising model series in the usual variable z = exp(−4J/kT) are analysed. The positions of the singularities in the disk|z| <or=|zc| and the values of the corresponding critical exponents are determined for the simple cubic, body-centred cubic and face-centred cubic lattices. We obtain the following estimates of the physical critical exponents: β = 0312 ± 0002 for the simple cubic and face-centred cubic lattices, and β = 0312 ± 0004 for the body-centred cubic lattice. γprime = 131 ± 002 for the simple cubic lattice, γprime = 128 ± 004 for the body-centred cubic lattice, and γprime = 13 ± 01 for the face-centred cubic lattice. For the low temperature specific heat exponent we obtain fraction one-sixteens <or= αprime <or= ¼.