Final answer:
Statistical mechanical equilibrium pertains to the static nature of the distribution function and is associated with energy distribution among degrees of freedom in a thermal system, as described by the equipartition theorem. It's distinct from dynamical equilibrium, where motions are dynamic but forces are balanced. This concept is foundational in understanding both classical and quantum states of matter.
Step-by-step explanation:
Equilibrium in the context of the distribution function f which lacks explicit time dependence, refers to statistical mechanical equilibrium rather than dynamical equilibrium. This form of equilibrium arises from the equipartition theorem which is applicable to classical systems in thermal equilibrium, and ensures that energy is equitably distributed amongst the system's degrees of freedom via collisions. In contrast, dynamical equilibrium describes a situation where forces are balanced, but objects may still be in motion, such as a car moving at constant velocity on a flat road.
Understanding the quantum mechanics aspect, Schrödinger's time-independent equation provides wave functions or 'stationary states' that are essential for determining the energy states of a quantum particle. These stationary states present a static probability density function, meaning the probability distribution of locating a particle at a certain position does not change with time, echoing the static nature of statistical mechanical equilibrium.
In conclusion, statistical mechanical equilibrium is concerned with the static aspects of distribution functions, without direct consideration of time, and is integral to accurately predicting states of matter in classical and quantum systems.