Every Rock Cracks the Same Way

Scaling and Universality in Rock Fracture

Jorn Davidsen

PRL 98, 125502 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e125502

When a rock is squeezed, tiny cracks form inside. Each crack makes a sound. Davidsen recorded the waiting time between cracking sounds in a variety of rock samples and performed a statistical analysis. When the waiting times were divided by the mean waiting time for a given experiment, all the probability distributions collapsed onto a single curve. Even earthquake data falls onto the same curve when scaled by the mean waiting time between aftershocks.

The probability distribution for the scaled waiting time is a gamma distribution: a power law multiplied with an exponential. I suppose the name comes from the normalization constant. Davidsen showed that the distribution is independent of the sample, the mechanism used to crush the rock, and a cutoff intensity. (You get the same distribution even if you ignore the cracks you can't hear.) All of this suggests that the probability distribution is a universal feature of rock fracture.

A universal mechanism for cracking suggests that a detailed analysis of the molecular properties and bonding is unnecessary. Davidsen doesn't address this directly, but PhysicsWeb stressed the point. It could provide a useful check for numerical and analytic models of crack formation. If you use your favorite model to generate a series of crack, then analyze the scaled waiting time, your data should generate the same gamma distribution as real rocks and earthquakes. If not, then your model has failed to capture whatever mechanism is responsible for this universality.

Is it possible to work backwards with the renormalization group? I.e., knowing the universal scaling relation of the scaled waiting times, can one deduce something useful about the interactions on the microscopic level?

In concluding, Davidsen makes two interesting observations. First, although the waiting times for rock fracture and earthquakes have the same probability distribution, the correlation between waiting times do not. Second, the statistics of rare events often generates a Poisson distribution. The fact that the waiting times do not is important.