Statistical Mechanics I: Ensembles

a microstate

A microstate is the fullest possible description of a system: a complete, exhaustive specification of every microscopic detail at once. Imagine a snapshot that lists the exact position and velocity of every single molecule in a gas, all 10^23 of them, with nothing left unsaid. That impossibly detailed roll-call is one microstate.

Classically, a microstate is a single point in the 6N-dimensional phase space of N particles: 3N position coordinates plus 3N momentum coordinates. Quantum mechanically, it is one particular eigenstate of the system's Hamiltonian, one element of an orthonormal basis of the Hilbert space, labelled by a full set of quantum numbers. Enormously many microstates typically share the same macroscopic appearance; the number that do is the multiplicity of that macrostate.

Statistical mechanics is built on counting microstates. A crucial honesty: entropy is not a property of a single microstate. A microstate has no entropy of its own; entropy counts how many microstates realize a given macrostate. In quantum mechanics, permuting identical particles does not produce a new microstate, an indistinguishability that leaves deep fingerprints on the counting.

For four coins, one microstate is a definite sequence such as HTTH; there are 2^4 = 16 microstates in all, each an exact statement of every coin.

Every definite outcome of all the coins at once is one microstate.

A single microstate carries no entropy; entropy is a property of the macrostate or ensemble, counting how many microstates share it.

Also called
microscopic state微態微觀狀態