  The Liouville theorem is a fundamental concept in understanding the
properties of systems that adhere to Hamilton's equations. However, the
traditional notion of the theorem may not always apply. Specifically, when the
entropy gradient in phase space fails to reach equilibrium, the phase-space
density may not have a zero time derivative, i.e., $\frac{d\rho}{dt}$ may not
be zero. This leads to the concept of the set of attainable states of a system
forming a compressible "fluid" in phase space. This observation provides
additional insights into Hamiltonian dynamics and suggests further examination
in the fields of statistical physics and fluid dynamics. In fact, this finding
sheds light on the limitations of the Liouville theorem and has practical
applications in fields such as beam stacking, stochastic cooling, and Rabi
oscillations, among others.
