A generalized approach to generate optimized approximations of the inverse Langevin function

Abstract

Proper approximation of the inverse Langevin function (ILF) is a well-recognized problem with significant relevance in many fields ranging from polymer physics to turbulence mechanics. While several estimations of the ILF has been developed recently, the accuracy/complexity trade-off has remained a major challenge in ILF estimation. Accurate estimations are computationally too expensive, and the low-cost estimations lack high accuracy. Here, a novel approach is developed that can provide a family of approximation functions for ILF with different degrees of accuracy. In the present paper, a simple procedure is presented, which can take current approximation functions with asymptotic behavior and enhance them by the addition of a power series of their induced error. The total error is thus correlated with a number of terms in the power series. We further propose different approaches to reduce the terms of the power series and increase the accuracy, the proposed approach is applied to four different classes of ILF approximations and shows significant improvement. The accuracy/complexity trade-off for the family of ILF approximations generated by the proposed approach is compared against those of other approaches to show the advantage of the proposed model. The level of error of this method can reach as low as 0.02%.

Improved approximations of non-Gaussian probability, force, and energy of a single polymer chain

“…Recently, many accurate ILF approximations with relative error less than 0.1% have been proposed in the literature (e.g. [26,27]) and thus using each of those, one can derive the strain energy of the chains from Eq. 23 as follows…”

“…For example, using L −1 (y) = x 1−x +2x− 8 9 x 2 (max relative error 1%) [26], the internal energy can be written as,…”

“…Recently, due to significant improvement of our computational power for simulating the entropic energy of the whole network, accurate approximation of the ILF has become a subject of interest. In the last decade, several high accuracy approximations with errors as low as 10 −4 % have been introduced [26][27][28][29]. While accurate approximations of the ILF can reduce the error of KG energy and force approximations, there still exists a significant error in those approximations due to the intrinsic error associated with KG PDF.…”

Inverse Langevin and Brillouin functions: mathematical properties and physical applications

“…With the development of very precise experimental techniques in the last 30 years, like single-molecule force spectroscopy (SMFS), the elastic strain energy and the force-displacement relationship, which can be written in terms of L −1 [13], [14], could be measured very precisely; to keep pace with SMFS, the theory had to produce more and more precise expressions for the inverse Langevin function [35], [36], [37].…”

“…the integral of the Brillouin function is simpler than the function itself. Consequently, S −1 J (x) can be obtained easier than B −1 J , and, if we really obtain it, and if we can write (37) in terms of inverse functions, this could provide us an alternative way of calculating B −1 J . This approach has not been explored yet, although several interesting relations between the integrals of direct and inverse functions were given in literature [17], [13].…”