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Classical microscopic states or microstates and ensembles

A microscopic state or microstate of a classical system is a specification of the complete set of positions and momenta of the system at any given time. In the language of phase space vectors, it is a specification of the complete phase space vector of a system at any instant in time. For a conservative system, any valid microstate must lie on the constant energy hypersurface, \(H (x) = E \). Hence, specifying a microstate of a classical system is equivalent to specifying a point on the constant energy hypersurface.

The concept of classical microstates now allows us to give a more formal definition of an ensemble. An ensemble is a collection of systems sharing one or more macroscopic characteristics but each being in a unique microstate. The complete ensemble is specified by giving all systems or microstates consistent with the common macroscopic characteristics of the ensemble.

The idea of ensemble averaging can also be expressed in terms of an average over all such microstates (which comprise the ensemble). A given macroscopic property, \(A\), and its microscopic function \(a = a (x) \), which is a function of the positions and momenta of a system, i.e. the phase space vector, are related by

\[ A = \langle a \rangle_{ensemble} = \frac {1}{N} \sum _{\lambda = 1}^N a(x_{\lambda})\]

where \(x_{\lambda}\) is the microstate of the \(\lambda \) th member of the ensemble.

Ergodic Hypothesis

However, recall the original problem of determining the microscopic detailed motion of each individual particle in a system. In reality, measurements are made only on a single system and all the microscopic detailed motion is present. However, what one observes is still an average, but it is an average over time of the detailed motion, an average that also washes out the microscopic details. Thus, the time average and the ensemble average should be equivalent, i.e.

\[ A = \langle a \rangle_{ensemble} = \lim _{T \to \infty } \frac {1}{T} \int _0^T dt   a(x (t)) \]

This statement is known as the ergodic hypothesis. A system that is ergodic is one for which, given an infinite amount of time, it will visit all possible microscopic states available to it (for Hamiltonian dynamics, this means it will visit all points on the constant energy hypersurface). No one has yet been able to prove that a particular system is truly ergodic, hence the above statement cannot be more than a supposition. However, it states that if a system is ergodic, then the ensemble average of a property \(A(x) \) can be equated to a time average of the property over an ergodic trajectory.