Phase transition as we vary disorder. Three H(M) curves for different levels of disorder: above, below, and near the critical disorderR_chen the burst disappears. (Select it for an animation!) For R > R_c the hysteresis loop is macroscopically smooth, although of course microscopically it is a sequence of sizable avalanches. The jump in the magnetization scales as (R-R_c)^beta, and the magnetization at R_c has a power-law form M - M_c ~ (H-H_c)^{1/delta}.
This is a simulation of a two-dimensional somewhat above the critical point, where none of the avalanches are very large.
This is a simulation closer to the transition. Notice the big avalanche which goes from one end of the system to the other! We're not sure yet whether there is a real transition in two dimensions: it's conceivable that no matter how weak the disorder, in a big enough system all avalanches would come to a halt at finite sizes. (We're sure that it happens that way in one dimension: there a single stuck spin can block the avalanche!)
Avalanches come in various sizes: Log-Log Plot of the avalanche-size distribution D(s) vs. avalanche size s, integrated over one sweep of the magnetic field from -infinity to +infinity, averaged over 5 systems of size 120^3. Notice the power-law region D(s) ~ s^{-(tau + sigma beta delta)} and the cutoff at s_{max} ~ (R-R_c)^{-1/sigma}.
The critical point is where an infinite avalanche first occurs. Naturally, near the critical point, some of the avalanches get rather large - indeed, one gets avalanches of all scales. At least in our model, one gets rather large avalanches even pretty far from the critical point: we believe the noise in hysteresis loops is due to this critical point.
Entertaining Science done at
LASSP.
James P. Sethna, sethna@lassp.cornell.edu
Statistical Mechanics: Entropy, Order Parameters, and Complexity,
now available at
Oxford University Press
(USA,
Europe).