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More abstractly, the topic is: amplification of quantum uncertainty within dynamically unstable systems.

I'd like to have a calculable toy model, e.g. maybe a quantum version of the famous "abelian sandpile model". We would start in a pure state in which the "grains of sand" each have an equal and uncorrelated uncertainty in position, and then we try to understand the probability distribution over large avalanches.

But other ideas which capture something of the topic are also welcome!

edit: A quote from "Chaos and the semiclassical limit of quantum mechanics" by Michael Berry:

the claim sometimes made, that chaos amplifies quantum indeterminacy, is misleading. The situation is more subtle: chaos magnifies any uncertainty, but in the quantum case h has a smoothing effect, which would suppress chaos if this suppression were not itself suppressed by externally-induced decoherence, that restores classicality (including chaos if the classical orbits are unstable)

An ideal model of "quantum earthquakes", therefore, might be one capable of showing the quantum suppression of classical "earthquake chaos", but also the counter-suppression due to decoherence, in a calculable way; as can be done for the tumbling moon Hyperion.

Mitchell Porter
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I do not think the probability of an earthquake can be tied through calculation to events occurring on the quantum level; here is why.

The quantum realm is fully manifest at the length scale of atomic nuclei. The energies associated with those length scales are of order ~gamma rays. This means that to "communicate" with the nucleus requires us to use gamma rays or particles with ~gamma ray energies.

Now we jump up to the characteristic scale length of earthquakes which involve strain energy stored over ~thousands of cubic kilometers of rock and ask ourselves, how strongly are gamma rays coupled to strain fields on those length scales? The answer is, not strongly- if at all.

For the coupling to be strong requires that the strain fields in deformed rock possess some mechanism by which they can either generate gamma rays or respond to gamma irradiation. But what's really happening with strained rock is tensile and compressive stresses applied to the interatomic forces between adjacent atoms, and shear stresses applied to covalent (and directional) bonds between atoms. The energies involved are of order ~tens of eV.

This means you cannot "talk" to the quantum realm with mechanical stresses acting on the outermost electrons of a solid consisting of ~cubic kilometers of atoms and molecules.

niels nielsen
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Taken literally, earthquakes are macroscopic phenomena, where quantum effects are negligible. Note that the macroscopic size itself does not preclude quantum effects, but it usually implies involvement of many particles, irregularities, and high temperatures, which result in short coherence lengths and negligible quantum effects. The macroscopic phenomena where quantum effects are still known to be crucial for dynamic properties - like Josephson junctions mentioned in the comments to the other answer - are still occurring on the scales several magnitudes lower than the earthquakes (millimeters vs. hundreds or thousands of kilometers.)

If we do not take the "earthquakes" in the Q. literally, but rather focus on the behavior of large collections of particles, then some strongly correlated phenomena, like superconductivity, superfluidity, quantum phase transitions are of interest. However, the study of this phenomena usually involves little of dynamical features, that are of interest to the author of the Q. This is why I suggest looking more into the physics of lasers and quantum devices.

In lasers the onset of stimulated radiation is an avalanche phenomenon, and is often modeled using simple versions of dynamical systems, with a few non-linear equations. See e.g., Nonlinearity of semiconductor lasers. In terms of many-particle instability, Dicke model might be of interest - it is both quantum and dynamical. Finally, devices like avalanche diode might be of some interest, although the models involved in their practical description are rather simplee.

Roger V.
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When information is copied out of a quantum system undergoing interference that interference is suppressed: this effect is called decoherence

https://arxiv.org/abs/quant-ph/0306072

Since macroscopic objects like large chunks of rock are undergoing interactions that copy information out of them a lot faster than they move their motion is decoherent and quantum effects don't come up much directly. Rather systems follow classical equations of motion on a macroscopic scale to a very good approximation

https://arxiv.org/abs/0903.1802

Quantum theory is indirectly relevant because quantum models explain radioactive decay, which generates a lot of the heat that drives plate tectonics and it explains the forces between atoms that lead to friction, elasticity of rock and so on. Quantum theory can also be used to derive equations used in statistical mechanics in the decoherent regime

https://arxiv.org/abs/2104.11223

alanf
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