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Muon-catalyzed fusion is a form of fusion where the electron in a hydrogen atom is replaced by a much heavier muon, which orbits 196 times closer than an electron and allows it to approach other hydrogen atoms at a distance 196 times closer than normal, a distance at which the probability of spontaneous fusion through quantum tunneling becomes a virtual certainty even at temperatures at or significantly below room temperature. It is also the only form of "cold fusion" with any form of scientific rigor behind it. Unfortunately on average each muon is not able to catalyze enough fusion reactions to recoup the energy needed to create it, meaning that the process consumes more energy than it produces. As such it remains a laboratory curiosity and technological dead-end barring some unexpected and highly unlikely breakthrough that allows for vastly more efficient muon generation.

Pycnonuclear is a compound word derived from "pycno-" meaning "dense" or "thick" and "nuclear" meaning, well, nuclear and I probably don't have to elaborate given the this question's intended audience. It is used to describe nuclear reactions which strongly depend on density with a weak to zero dependence on temperature. It is mainly used when describing theoretical fusion reactions in the core of white dwarfs and the crust of neutron stars where nuclei are so tightly packed that the probability of them spontaneously fusing through quantum tunneling becomes non-negligible even for relatively heavy nuclei which normally require extremely high temperatures to fuse.

But... doesn't that describe muon-catalyzed fusion too? In the sense of a nuclear reaction not initiated by enough heat to overcome the atoms Coulomb barriers but by decreasing the distance between (and thus the volume occupied by) two atoms until fusion results due to quantum tunneling.

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I would say not. My understanding of the term is that a "pycnonuclear" reaction is one where the reaction is driven by having a large (typically very large) macroscopic density of reactants. This is present in degenerate matter in compact objects and other exotic situations, but not really in muon-catalyzed fusion. A reaction gets labeled as "pycnonuclear" by having a large absolute density of matter (measured in kg/m$^{3}$, say).

In muon-catalyzed fusion, what is going on is that the muonic H$_{2}$ molecules are much smaller than usual ones (while not being that different in total mass). That means that near encounters by the two hydrogen nuclei in a single molecule (be they protons, deuterons, or tritons—as appropriate for the stage of the fusion process involved) are much more likely that in a normal H$_{2}$ molecule. However, increasing the total density, by cramming more muonic H$_{2}$ molecules into your reactor vessel, is not going to increase the reaction rate appreciably over what it would have been with the same mass of more rarefied gas, because interactions between nuclei in different molecules are still vanishingly rare. And that is what makes the reaction process not "pycnonuclear."

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