Periodically driven systems thermalize to infinite temperature. This is the expected outcome: the drive pumps energy into the system, and without conservation laws to prevent it, the system heats indefinitely. For quantum systems, many-body localization can prevent this heating, but generic classical systems have no such protection.
The expectation has been standard for decades. The proof, for an infinite class of classical Floquet systems, is new. The paper establishes rigorously that a Gibbs state of any sufficiently uniform local differentiable Hamiltonian, subjected to a periodic drive, heats to infinite temperature at long times. The proof covers algebraically structured systems and uses the smoothness of the Hamiltonian and the uniformity of the interactions as its hypotheses.
The interesting part is not the destination (infinite temperature was expected) but the obstruction analysis. What could prevent heating? The only identified barrier: local observables that repeat periodically in time — quantities that oscillate synchronously with the drive rather than randomizing. If such observables exist, they lock the system's evolution to the drive period and prevent energy from distributing uniformly. If they don't exist, heating proceeds to completion.
The result draws a sharp line: a classical periodically driven system either has periodic local observables or it thermalizes. There is no middle ground — no partial thermalization, no slowly heating regime that persists indefinitely. The system either resonates with the drive (periodic) or absorbs from it (thermalizes). The physical intuition was always right; the theorem makes it exact.