partition function

History

The idea was born with the statistical mechanics of Ludwig Boltzmann in the 1870s, but in its modern form, the canonical sum was introduced by Josiah Willard Gibbs at the turn of the 20th century. Later, the quantum version accounting for particle indistinguishability was developed by Bose, Einstein, Fermi, and Dirac.

How it works

The summation is done with an exponential factor: the higher the energy of a state, the smaller its contribution—just as a steep climb on a mountain is encountered less often than a gentle slope on a walk. At low temperatures, only the lowest steps are 'populated'.

History

The idea was born with the statistical mechanics of Ludwig Boltzmann in the 1870s, but in its modern form, the canonical sum was introduced by Josiah Willard Gibbs at the turn of the 20th century. Later, the quantum version accounting for particle indistinguishability was developed by Bose, Einstein, Fermi, and Dirac.

How it works

The summation is done with an exponential factor: the higher the energy of a state, the smaller its contribution—just as a steep climb on a mountain is encountered less often than a gentle slope on a walk. At low temperatures, only the lowest steps are 'populated'.

History

Boltzmann introduced the concept of statistical weight and entropy in the 1870s, but the explicit formula Z = Σ e^{-βE_i} for the canonical ensemble was introduced by Gibbs in 1902 in his book 'Elementary Principles in Statistical Mechanics'. Einstein and Bose in 1924 generalized it to quantum gases with integer spin, and Fermi and Dirac to half-integer spin, leading to Bose–Einstein and Fermi–Dirac statistics.

How it works

All thermodynamic quantities are expressed through Z: internal energy U = -∂ ln Z/∂β, heat capacity C_v = ∂U/∂T, pressure P = k_B T (∂ ln Z/∂V)_T. In the quantum case, Z = Tr(e^{-βĤ}), where Ĥ is the Hamiltonian. For non-interacting systems, it factorizes over single-particle states, but in general requires approximate methods.

Open problems & fun facts

Calculation of Z for a strongly interacting system: no general analytical methods exist, and numerical calculations are exponentially complex.; Thermalization of isolated quantum systems: the applicability of the canonical ensemble to closed systems via the eigenstate thermalization hypothesis (ETH) remains unproven in general.; The role of long-range correlations: in systems with long-range interactions (gravity, plasma), the extensivity of entropy breaks down, and the usual Z does not yield correct thermodynamics.

💡 The partition function of water when freezing undergoes a sharp kink, and from its jump one can compute the latent heat of ice formation—the amount of energy the frost extracts from a puddle.