Inverse Langevin and Brillouin functions: mathematical properties and physical applications

Abstract

This paper gives a coherent and comprehensive review of the results concerning the inverse Langevin L (x) and Brillouin functions BJ (x) and the inverse of L (x) /x and BJ (x) /x. As these functions are used in several fields of physics, without evident interconnections -magnetism (ferromagnetism, superparamagnetism, nanomagnetism, hysteretic physics), rubber elasticity, rheology, solar energy conversion -the new results are not always efficiently transferred from a domain to another. The increasing accuracy of experimental investigations claims an increasing accuracy in the knowledge of these functions, so it is important to compare the accuracy of various approximants and even to obtain, in some cases, the exact form of the inverses of L (x), BJ (x) , L (x) /x and BJ (x) /x. This exact form can be obtained, in some cases, at least in principle, using the recently developed theory of generalized Lambert functions; in some particular -and also relevant -cases, explicit expressions for these new special functions are obtained. The paper contains also some new results, concerning both exact and approximate forms of the aforementioned inverse functions.

Multifrequency nonlinear model of magnetic material with artificial intelligence optimization

“…Langevin model for nonlinearity. We assume a simplest physical model for anhysteretic magnetization 6 and conversely assume derivative of Langevin (DL) function L ′ (x) 23,24 for nonlinear characteristics of inductance: with only two parameters needed: l 0 -a maximum inductance, and a -scaling factor. The DL derivative we implemented as follows 23 : L(x) = 1/3 − x 2 /15 + 2x 4 /189−x 6 /675 + 2x 8 /11686 for x < 1 , and L(x) = 1/x 2 − 1/ sinh 2 (x) elsewhere.…”