Find temperature by sharing a fixed energy

The microcanonical ensemble assigns equal probability to accessible microstates in a specified narrow energy interval at fixed volume and particle number. For discrete levels this means counting states in that interval; for classical phase space it means integrating a shell with the dimensionless measure. State the interval width before taking a logarithm. A density of states has inverse-energy units and cannot be used as a dimensionless multiplicity without a reference interval or an equivalent convention.

Here entropy means Boltzmann shell entropy. Differentiating its smooth large-system form at fixed volume and number defines inverse temperature. For a finite discrete system derivatives require smoothing or a finite-difference convention, so do not present a macroscopic derivative as an exact small-system measurement. Entropy defined from the cumulative volume below energy is a different finite-system convention. Its differences are particularly important for bounded spectra and must not be silently mixed with the shell convention used here.

Allow two weakly coupled systems to exchange energy without exchanging particles or changing volume. If the first has energy EA, the second has the fixed total minus EA. Their joint multiplicity is the product of their separate shell multiplicities. Taking its logarithm converts the product into an entropy sum. A stationary interior division has equal entropy slopes, hence equal temperatures. Weak coupling matters because assigning separate additive subsystem energies neglects interaction energy.

A stationary point must also be a maximum to describe stable thermal equilibrium. For ordinary positive heat capacities the entropy curvature is negative, giving a sharply peaked energy division in large systems. Temperature equality alone does not supply this stability check or the rate of equilibration. The statistical argument predicts an equilibrium distribution after accessibility and equilibration assumptions are met; it does not prove that an isolated integrable system explores every state on the relevant time scale.

Consider many fixed two-level sites with energies zero and epsilon. If K sites are excited, the energy is K times epsilon and the multiplicity is a binomial coefficient. The multiplicity rises to half excitation and then falls as the system approaches its fully excited state. In the large-site smooth approximation, the shell entropy therefore acquires a negative slope above half filling. This defines negative temperature within the specified bounded subsystem and shell-entropy convention.

A negative-temperature subsystem gives up energy when coupled to an ordinary positive-temperature reservoir. Its inverse temperature is below every positive inverse temperature, so it is hotter in the direction-of-energy-flow sense, not colder than zero kelvin. A free gas with unbounded kinetic energy cannot normalize a negative-beta Boltzmann weight. The binomial lab shows the bounded sites' degeneracy landscape at equal microstate weights; it is not a microcanonical dynamical evolution or a mechanism that prepares a population inversion.