Stochastic model of hysteresis

The methods of the probability theory have been used in order to build up a model of hysteresis which is different from the well-known Preisach model. It is assumed that the system consists of large number of abstract particles in which the variation of an external control parameter (e.g., the magnetic field) may result in transitions between two states S((+)) and S((-)). The state of a particle is characterized by the value +1 or -1 of a random variable (e.g., the magnetization direction parallel or antiparallel to the magnetic field). The transitions are governed by two further random variables corresponding to the S((-))-->S((+)) and the S((+))-->S((-)) transitions (e.g., "up switching" and "down switching magnetic field"). The method presented here makes it possible to calculate the probability distribution and consequently the expectation value of the number of particles in the S((+)) (or S((-))) state for both increasing and decreasing parameter values, i. e., the hysteresis curves of the transitions can be determined. It turns out that the reversal points of the control parameter are Markov points which determine the stochastic evolution of the process. It has been shown that the branches of the hysteresis loop are converging to fixed limit curves when the number of cyclic back-and-forth variations of the control parameter between two consecutive reversal points is large enough. This convergence to limit curves gives a clear explanation of the accommodation process. The accommodated minor loops show the return-point memory property but this property is obviously absent in the case of nonaccommodated minor loops which are not congruent and generally not closed. In contrast to the traditional Preisach model the reversal point susceptibilities are nonzero finite values. The stochastic model can provide a surprisingly good approximation of the Raylaigh quadratic law when the external parameter varies between two sufficiently small values. The practical benefits of the model can be seen in the numerical analysis of the derived equations. On one hand the calculated curves are in good qualitative agreement with the experimental observations and on the other hand, the estimation of the joint distribution function of the up and down switching fields can be performed by using the measured hysteresis curves.