Functional Integration Approach to Hysteresis

TL;DR

This paper introduces a general functional framework for scalar hysteresis based on generating functions and stochastic processes, linking it to the Preisach model and enabling analysis of magnetic and superconducting hysteresis.

Contribution

It develops a novel formulation of hysteresis using generating functions and stochastic measures, connecting ensemble responses to the Preisach model with explicit analytic expressions.

Findings

Hysteresis properties can be exactly described by the Preisach model under certain conditions.

The formulation reduces the problem to a level-crossing analysis of Markovian processes.

Analytic expressions for the Preisach distribution are derived from stochastic process parameters.

Abstract

A general formulation of scalar hysteresis is proposed. This formulation is based on two steps. First, a generating function g(x) is associated with an individual system, and a hysteresis evolution operator is defined by an appropriate envelope construction applied to g(x), inspired by the overdamped dynamics of systems evolving in multistable free energy landscapes. Second, the average hysteresis response of an ensemble of such systems is expressed as a functional integral over the space G of all admissible generating functions, under the assumption that an appropriate measure m has been introduced in G. The consequences of the formulation are analyzed in detail in the case where the measure m is generated by a continuous, Markovian stochastic process. The calculation of the hysteresis properties of the ensemble is reduced to the solution of the level-crossing problem for the stochastic process. In particular, it is shown that, when the process is translationally invariant (homogeneous), the ensuing hysteresis properties can be exactly described by the Preisach model of hysteresis, and the associated Preisach distribution is expressed in closed analytic form in terms of the drift and diffusion parameters of the Markovian process. Possible applications of the formulation are suggested, concerning the interpretation of magnetic hysteresis due to domain wall motion in quenched-in disorder, and the interpretation of critical state models of superconducting hysteresis.