Derive Boltzmann weights from a reservoir

Assume the coupling is weak enough to neglect interaction energy when assigning subsystem and reservoir energies. If a subsystem microstate has energy Ei, the reservoir has total energy minus Ei. All compatible composite microstates are equally likely, so the subsystem weight is the reservoir multiplicity at that remaining energy. Expand the logarithm of this multiplicity around a reference reservoir energy. Its first derivative is inverse temperature divided by Boltzmann's constant, producing the exponential suppression of higher subsystem energies.

The second derivative contains the reservoir heat capacity. For an ordinary stable bath it is negative, making a quadratic correction to the exponent. The canonical approximation neglects that term over the subsystem's relevant energy range. A bath need not have infinitely much total energy, but exchanges must change its temperature negligibly. Strong coupling also needs a different treatment because interaction energy and subsystem equilibrium cannot generally be represented by the bare subsystem Hamiltonian alone.

The index in the partition function labels microstates, not necessarily distinct energy values. If an energy level has degeneracy g, its total probability includes g equal Boltzmann weights. One can either sum each microstate or multiply each distinct level by its degeneracy. Doing both counts the same states twice. In a classical description the corresponding integral must include its measure and Gibbs factor; those are part of the definition of the state sum.

Shifting every subsystem energy by the same constant multiplies the partition function by a common exponential factor. Normalized probabilities remain unchanged, while free energy and mean energy shift consistently. This check separates physical energy differences from an arbitrary energy zero. For independent subsystems, partition functions multiply because joint energies add and state choices factorize. Their logarithms therefore add, a feature that makes extensive thermodynamic quantities accessible from many simple constituents.

Differentiate the logarithm of the partition function with respect to beta while holding the Hamiltonian, volume, and particle number fixed. The first derivative gives minus mean energy; the second gives its variance. Converting the temperature derivative of mean energy gives the heat capacity relation. A nonnegative variance implies nonnegative canonical heat capacity for a temperature-independent Hamiltonian. This statement does not rule out every finite isolated microcanonical model with a different energy constraint.

For many independent two-level sites, the total energy is the excitation gap times a binomial count. The lab can therefore show exact canonical energy-count fluctuations once the excitation probability has been calculated from the Boltzmann weight. The probability control is not a temperature dial and the animation is not a thermalization history. Changing the probability recomputes the distribution; independent comparisons must reset the instrument and specify both its probability and number of sites.