Perturbation expansions and series acceleration procedures. Part I. ε-convergence and critical parameters
TL;DR: In this paper, a simple acceleration of convergence technique known as the e-convergence algorithm was applied to determine the critical temperatures and exponents of the virial equation of state.
Abstract: A simple acceleration of convergence technique known as the ‘e-convergence algorithm’ (ea) is applied to determine the critical temperatures and exponents. Several illustrations involving well-known series expansions appropriate to two- and three-dimensional Ising models, three-dimensional Heisenberg models, etc., are given. Apart from this, a few recently studied ferrimagnetic systems have also been analysed to emphasise the generality of the approach. Where exact solutions are available, our estimates obtained from this procedure are in excellent agreement. In the case of other models, the critical parameters we have obtained are consistent with other estimates such as those of the Pade approximants and group theoretic methods. The same procedure is applied to the partial virial series for hard spheres and hard discs and it is demonstrated that the divergence of pressure occurs when the close-packing density is reached. The asymptotic form for the virial equation of state is found to beP/ρkT ∼ (1 −ρ/ρ
c
−1 for hard spheres and hard discs. Apart from the estimation of ‘critical parameters’, we have applied theea and the parametrised Euler transformation to sum the partial, truncated virial series for hard spheres and hard discs. The resulting values of pressure so obtained, compare favourably with the molecular dynamics results.
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TL;DR: In this paper, the authors demonstrate the applicability of two equivalent procedures, viz. (1) the <-convergence algorithm for the acceleration of convergence of the original series, and (2) Pade's approximation scheme, a rational function derived using the first few terms of the given series.
23 citations
TL;DR: In this paper, the potential transients are obtained by using Pade approximants for all amplitudes of concentration polarization and current densities for several mechanistic schemes under constant current conditions.
9 citations
TL;DR: In this article, a parametrized version of the Euler transformation, introduced fairly recently, is employed to study the behaviour of functions, given their formal power-series (alternating) expansions in λ with finite radii of convergence, in the limit λ»∞.
Abstract: A parametrized version of the Euler transformation, introduced fairly recently, is employed to study the behaviour of functions, given their formal power-series (alternating) expansions in λ with finite radii of convergence, in the limit λ»∞. The strategy requires only the first few low-order data. Results are tested with quite a few known cases and found remarkably satisfactory. The role of some other methods in this context are briefly discussed.
4 citations
TL;DR: In this article, the convergence radii of the Rayleigh-Schrodinger perturbation series for the groundstate energy levels of the plane and linear polar rigid rotators in uniform electric fields are calculated.
Abstract: A new method is developed for studying real singularities of functions from their power series expansions. It is based on an appropriate calculation of the least‐squares deviation of the Taylor coefficients from their asymptotic behavior. Under certain conditions the procedure is shown to lead to the widely used ratio method and it is applied to perturbation‐theory problems and virial series. The convergence radii of the Rayleigh–Schrodinger perturbation series for the ground‐state energy levels of the plane and linear polar rigid rotators in uniform electric fields are calculated. The occurrence of the singularities in the compressibility factor for hard disks and spheres predicted by the Percus–Yevick and scaled‐particle theory is verified. The critical parameters characterizing the Kirkwood ‘‘phase transition’’ near close‐packing density and the asymptotic form of the equation of state in the neighborhood of the singularities are verified. Present results elucidate a critical‐exponent paradox.
3 citations
TL;DR: In this article, an improved version of a recently developed method for summing strongly divergent pertubation series is presented, where perturbation series are changed into a sequence of polynomials by means of a properly chosen order-dependent mapping.
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Book•
01 Jan 1972
TL;DR: The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results as discussed by the authors, and the major aim of this serial is to provide review articles that can serve as standard references for research workers in the field.
Abstract: The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results. It has moved into a central place in condensed matter studies. Statistical physics, and more specifically, the theory of transitions between states of matter, more or less defines what we know about 'everyday' matter and its transformations. The major aim of this serial is to provide review articles that can serve as standard references for research workers in the field, and for graduate students and others wishing to obtain reliable information on important recent developments.
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TL;DR: In this article, an equilibrium theory of rigid sphere fluids is developed based on the properties of a new distribution function G(r) which measures the density of rigid spheres molecules in contact with a rigid sphere solute of arbitrary size.
Abstract: An equilibrium theory of rigid sphere fluids is developed based on the properties of a new distribution function G(r) which measures the density of rigid sphere molecules in contact with a rigid sphere solute of arbitrary size. A number of exact relations which describe rather fully the functional form of G(r) are derived. These are based on both geometrical considerations and the virial theorem. A knowledge of G(a) where a is the diameter of a rigid sphere enables one to arrive at the equation of state. The resulting analytical expression which is exact up to the third virial coefficient gives the fourth virial coefficient within 3% and the fifth, insofar as it is known, within 5%. Furthermore over the entire range of fluid density, the equation of state derived from theory agrees with that computed using machine methods. Theory also gives an expression for the surface tension of a hard sphere fluid in contact with a perfectly repelling wall. The dependence of surface tension on curvature is also given. ...
1,237 citations