Junget al. reply

Abstract

Jung et al. Reply:In his Comment [1], Rao has raised an interesting question about the role of fluctuations in the observation of the scaling law for dynamical hysteresis. In our experiments, the width and the area of the hysteresis loop are seen to grow as ft 2 ' 1 A , starting from a nonzero value at zero frequency, where ft is the modulation frequency and A is the modulation amplitude. These laws have been derived by studying the dynamics of a delayed saddle-node bifurcation in the absence of fluctuations and have been verified experimentally in a semiconductor laser. In contrast, Rao, Krishnamurthy, and Pandit [2] have studied the area of the hysteresis loop in a spin system, when modulating an external magnetic field periodically. For adiabatically small modulation frequencies ft and an amplitude much less than the mean-field spinodal value, the magnetization switches instantaneously (on the time scale of the external modulation) into the direction of the magnetic field due to fluctuations (zeroarea hysteresis loop). Increasing the modulation frequency, the noise-induced switching between positive and negative magnetization is delayed on the average, since the escape time of the system is now nonzero on the time scale of the modulation. This process can be understood as statistical hysteresis and cannot be compared directly with dynamical hysteresis, studied in our paper (Ref.[3]).The question, provoked by the Comment of Rao, is to what extent are both scaling laws compatible, or in other words, under what conditions are they valid? We first note that there are two different time scales in the spin system studied by Rao. One time scale t m is given by the modulation, while the other one / e sc is given by the statistically averaged waiting time for a fluctuation-induced switching between positive and negative magnetization. The time scale / esc becomes exponentially large for temperatures T much below the critical temperature To of the phase transition from a paramagnetic to a ferromagnetic phase. Considering the limit of vanishing temperatures for a fixed time scale t m , noise-induced switching events become extremely rare and can be neglected. In this situation the hysteresis loop is purely dynamical and the mechanism proposed in our paper (Ref.[3]) can be applied, yielding the scaling law /J 2/3 ft 2/3 for the growth of the hysteresis loop. All our experiments have been done in this regime, including those for small frequencies of modulation. When the escape time / e sc becomes smaller than the modulation time scale t m , thermal hopping governs the dynamics of the magnetization and the scaling law of Ref.[2] (A 2/3 ft ,/3 ) may apply.Noise randomly modifies individual realizations of the delayed dynamical bifurcation and shifts the delay randomly but symmetrically to smaller and larger values of the control parameter. These fluctuations, which are between 10% and 20% of the critical value of the control parameter, have been averaged out in our experiment by measuring thousands of hysteresis loops.There...

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