Universal excursion and bridge shapes in ABBM/CIR/Bessel processes

Abstract

Several years ago, in the context of the physics of hysteresis in magnetic materials, a simple stochastic model has been introduced: the ABBM model. Later, the ABBM model has been advocated as a paradigm for the description of a broad class of diverse phenomena, baptized ‘crackling noise phenomena’. The model reproduces many statistical features of such intermittent signals, as for instance the statistics of burst (or avalanche) durations and sizes, in particular the power law exponents that would characterize the dynamics as critical. In order to go beyond such ‘critical exponents’, the measure of the average shape of the avalanche has also been proposed. Here, the exact calculation of the average (as well as the fluctuations) of the avalanche shape for the ABBM model is presented, showing that its normalised shape does not depend on the external drive. Moreover, the average (and the fluctuations) of the multi-avalanche shape, that is the average shape of a sequence of avalanches of fixed total duration, is also computed. Surprisingly, the two quantities (avalanche and multi-avalanche normalised shapes) are exactly the same. This result is obtained using the exact solution of the ABBM model, which is obtained leveraging the equivalence with the Cox–Ingersoll–Ross process (CIR), rigorously obtained with a so called ‘time change’. A simple presentation of this and other known relevant exact results is provided: notably the correspondence of the ABBM/CIR model with the Rayleigh model and, more importantly, with the generalised Bessel process, which describes the dynamics of the modulus of the multi dimensional Ornstein–Uhlenbeck process (exactly as the Bessel process does for the Brownian process). As a consequence, our main finding, that is the correspondence between the excursion (avalanche) and bridge (multi-avalanche) shape distributions, turns to apply to all the aforementioned stochastic processes. In simple words: if we consider the distance from the origin of such diffusive particles, the (normalised) average shape of its trajectory (and the fluctuations around that) until a return in a time T is the same, whether it has returned before T or not.