How does the Quadrupole Accelerate?

Mathematically, we say that the basins of chaos are interconnected, not constrained by invariant tori. They are connected in an Arnol’d web that allows rapid stochastic acceleration.
More prosaically, compressions adiabatically increase the particle energy, while the center of the trap acts as a “mixmaster” for redistributing that energy into other invariants. Consequently particles diffuse in energy very rapidly.
A comparison to Fermi I,II is highly instructive for understanding the mechanism.

Mathematically, we say that the basins of chaos are interconnected, not constrained by invariant tori. They are connected in an Arnol’d web that allows rapid stochastic acceleration.