Task: Text Categorization
Input (English): The collisionless Boltzman equation is often referred to as this, and it can be used to extend the Runge-Gross theorem to time-dependent ensembles. A corollary of this result is that symplectic maps of Hamiltonian functions must preserve area. Applying this theorem to the expectation value of an observable in quantum mechanics gives a result that can also be derived by finding the expection value of the Heisenberg equation of motion, known as Ehrenfest's theorem. It can be expressed as the time derivative of a density equals the negative Poisson bracket of that density and the Hamiltonian. In the ergodic formulation, this theorem predicts smooth measure invariance under Hamiltonian flow, and it is equivalent to the Second Law of Thermodynamics. For 10 points, name this theorem in statistical mechanics which states the density of points is constant along a phase space trajectory with time.
Output (English): Science