Universality beyond power laws and the average avalanche shape

Abstract

The study of critical phenomena and universal power laws has been one of the central advances in statistical mechanics during the second half of the past century, explaining traditional thermodynamic critical points 1 , avalanche behaviour near depinning transitions 2,3 and a wide variety of other phenomena 4 . Scaling, universality and the renormalization group claim to predict all behaviour at long length and timescales asymptotically close to critical points. In most cases, the comparison between theory and experiments has been limited to the evaluation of the critical exponents of the power-law distributions predicted at criticality. An excellent area for investigating scaling phenomena is provided by systems exhibiting crackling noise, such as the Barkhausen effect in ferromagnetic materials 5 . Here we go beyond powerlaw scaling and focus on the average functional form of the noise emitted by avalanches-the average temporal avalanche shape 4 . By analysing thin permalloy films and improving the data analysis methods, our experiments become quantitatively consistent with our calculation for the multivariable scaling function in the presence of a demagnetizing field and finite field-ramp rate.The average temporal avalanche shape has been measured for earthquakes 6 and for dislocation avalanches in plastically deformed metals 7,8 , but the primary experimental and theoretical focus has always been Barkhausen avalanches in magnetic systems 5,6,[9][10][11] . Theory and experiment agreed well for avalanche sizes and durations, but the strikingly asymmetric shapes found experimentally in ribbons 11 disagreed sharply with the theoretical predictions, for which the asymmetry in the scaling shapes under time reversal was at most very small 4,6 . (We note that the relevant models are not microscopically time-reversal invariant; temporal symmetry is thus emergent.) Doubts about universality 4 were resolved when eddy currents were shown to be responsible for the asymmetry, at least on short timescales 12 , but the exact form of the asymptotic universal scaling function of the Barkhausen avalanche shape still remained elusive.Here, we report an experimental study of Barkhausen noise in permalloy thin films, where a careful study of the average avalanche shapes leads to symmetric shapes, undistorted by eddy currents (which are suppressed by the sample geometry). We provide a quantitative explanation of the experimental results by solving exactly the mean-field theories for two general models of magnetic reversal: a domain-wall dynamics model 13 Time-series data (jagged line) are traditionally separated into avalanches using a threshold V th set above the instrumental noise (dotted blue line)-here breaking one avalanche into a few pieces. We instead do an optimal Wiener deconvolution (smoothed red curve, see text), allowing the use of a zero threshold (solid black line), which avoids distortions of the average shape and also gives more decades of size and duration scaling. Averaging over all avalanches with this duratio...

Universal fluctuations and extreme statistics of avalanches near the depinning transition

“…(5) is interpreted in the Stratonovich sense, as derived in Ref. [53]. In this interpretation, the x = 2 √ v transformation of the equation of motion is…”

“…[52]. The model was initially derived to mimic the dynamics of a single domain wall in soft magnets exhibiting Barkhausen noise, but has become a 'standard model' for interface depinning in the presence of longranged forces [1,10,28,53]. The dynamics of a single interface propagating across a disordered material results from a competition between internal elasticity of the interface, interactions with the quenched disorder experienced by the moving front and an external driving field that is increased at a constant rate.…”

“…[53], an update rule like Eq. (8) derived in the shell model of the random field Ising model [57] is equivalent to the ABBM model in the continuum limit.…”

Avalanche statistics from data with low time resolution

“…This problem can be addressed by optimally filtering the input data [24,25], by demanding that the stress derivative drops below some negative threshold (rather than a threshold of zero), or by dismissing avalanches below a certain size as indistinguishable from noise. Deciding on the appropriate method requires care because noise cannot only cause spurious avalanches to be detected, but can also cause avalanches to be incorrectly broken into pieces.…”

Crackling noise in fractional percolation

“…Magnification of the hysteresis curve of a magnetic material in a changing external field, for instance, reveals that the magnetization curve is not smooth but exhibits small discontinuities. This series of correlated jumps is called the Barkhausen effect, which is a standard example for crackling noise in physics [6][7][8] . Despite its importance, crackling noise is far from being understood.…”