Entropy and the microcanonical ensemble

An isolated system has fixed particle number, volume, and energy. In quantum mechanics, count energy eigenstates inside a narrow shell around the specified energy, including degeneracies. The shell should be narrow on a macroscopic energy scale but contain many levels when a thermodynamic description is intended. We use Boltzmann's shell entropy, rather than the alternative Gibbs entropy counting all states below the energy; their finite-system derivatives can differ. The microcanonical postulate assigns equal probabilities to accessible shell states. It is an equilibrium ensemble assumption, not a claim that every isolated initial state immediately explores every eigenstate.

For $N$ distinguishable, noninteracting two-state units, let an excitation cost energy $\epsilon>0$. Exactly $q$ excitations give energy $q\epsilon$ and multiplicity $\binom Nq$. The labels could refer to localized spins, so no additional division by $N!$ is appropriate. Entropy is the logarithm of this multiplicity times Boltzmann's constant. Taking a logarithm turns products of independent state counts into sums, which explains why it is compatible with combining weakly interacting systems.

Put systems A and B in weak thermal contact while keeping their combined energy fixed. Negligible interaction energy lets us write the count for a chosen allocation as $\Omega_A(E_A)\Omega_B(E_{\mathrm{tot}}-E_A)$. The most probable allocation maximizes the sum of entropies. In a smooth large-system approximation, differentiating gives equality of their entropy slopes. Defining inverse temperature through that slope converts a counting statement into the familiar condition that bodies in thermal equilibrium have equal temperatures.

A maximum also requires suitable curvature: equality of slopes alone can describe an unstable stationary allocation. For ordinary short-range systems away from unusual constraints, concave entropy gives stable sharing. A bounded spectrum such as the two-state model can have decreasing multiplicity above half excitation, producing a negative entropy derivative and a negative absolute temperature in this definition. That population-inverted regime is possible only with an appropriate upper energy bound; it is not colder than a positive-temperature reservoir.

For macroscopic weakly correlated systems, the entropy maximum commonly overwhelms other allocations, giving reproducible bulk behavior despite microscopic motion. A six-spin example has no such overwhelming concentration: visibly different allocations retain substantial probability. The second law is therefore not a claim that every microscopic fluctuation raises entropy. Reversible microscopic equations and overwhelmingly probable macroscopic relaxation answer different questions, provided the initial preparation and coarse-graining have been specified.

Extensivity means that entropy grows proportionally to system size when energy, volume, and particle number are scaled together at fixed densities. It is a thermodynamic-limit property, not an exact identity for every finite count. Stirling corrections, boundaries, and constraints produce subleading terms. Long-range nonadditive interactions can require a different analysis. Always separate the exact finite combinatorics from a large-system approximation, and do not infer the time needed to equilibrate merely by counting equilibrium states.