Editing Statistical mechanics (section)

== Principles: mechanics and ensembles ==
{{main|Mechanics|Statistical ensemble (mathematical physics)|l2=Statistical ensemble}}

In physics, two types of mechanics are usually examined: [[classical mechanics]] and [[quantum mechanics]]. For both types of mechanics, the standard mathematical approach is to consider two concepts:
*The complete state of the mechanical system at a given time, mathematically encoded as a [[phase space|phase point]] (classical mechanics) or a pure [[quantum state vector]] (quantum mechanics).
*An equation of motion which carries the state forward in time: [[Hamiltonian mechanics|Hamilton's equations]] (classical mechanics) or the [[Schrödinger equation]] (quantum mechanics)
Using these two concepts, the state at any other time, past or future, can in principle be calculated.
There is however a disconnect between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.

Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the [[Statistical ensemble (mathematical physics)|statistical ensemble]], which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a [[probability distribution]] over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a [[phase space]] with [[canonical coordinates|canonical coordinate]] axes. In quantum statistical mechanics, the ensemble is a probability distribution over pure states and can be compactly summarized as a [[density matrix]].

As is usual for probabilities, the ensemble can be interpreted in different ways:<ref name="gibbs" />
* an ensemble can be taken to represent the various possible states that a ''single system'' could be in ([[epistemic probability]], a form of knowledge), or
* the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner ([[empirical probability]]), in the limit of an infinite number of trials.
These two meanings are equivalent for many purposes, and will be used interchangeably in this article.

However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the [[Liouville's theorem (Hamiltonian)|Liouville equation]] (classical mechanics) or the [[von Neumann equation]] (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.

One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as ''equilibrium ensembles'' and their condition is known as ''statistical equilibrium''. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. (By contrast, ''[[mechanical equilibrium]]'' is a state with a balance of forces that has ceased to evolve.)  The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.