  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.
