Gibbs state

In plain terms

Here is the problem. Put a system in contact with a big environment at some temperature and wait. It does not fall into its lowest-energy configuration and stay there, because the environment keeps jostling it -- lending energy, taking it back. But it does not wander uniformly over all configurations either, because low energy is still favoured. What, exactly, is the compromise?

Gibbs wrote down the answer: weight each configuration by a decaying exponential of its energy, with temperature setting the decay rate. Hot systems spread out, cold systems concentrate near the bottom, and the crossover is quantitative.

What makes the answer canonical rather than plausible is a variational principle. Among all states with the same average energy, the Gibbs state is the one with the most entropy -- equivalently, the one minimising free energy. It is not a good guess. It is the unique solution to an optimisation problem.

In CSD

The thermodynamics strand of the corpus instantiates the Gibbs state on the field Hamiltonian and derives temperature and free energy from the variational principle -- deriving rather than assuming, which is the difference between formalising thermodynamics and decorating it.

It is also where two independently built strands of the corpus meet. The thermal side (Gibbs states, entropy, Landauer) and the field side (modes, propagators, light cones) developed separately for months; the exact KMS relation proved at the cutoff joined them, with the Gibbs state as the object on which they agree. Because the field Hamiltonian is diagonal, everything has closed form -- the thermal results are computations, not existence claims.

Mathematically

The density operator proportional to exp(-beta H), normalised by the partition function. Unique minimiser of free energy at inverse temperature beta; equivalently the entropy maximiser at fixed mean energy.

In the corpus the Hamiltonian is diagonal in the configuration basis, so the state is an explicit diagonal matrix with a finite geometric normalisation, the partition function factorises over modes, and the thermal two-point function and its vacuum limit are exact finite computations.

The name

Josiah Willard Gibbs (1839-1903) spent essentially his whole life in New Haven, Connecticut. Yale gave him America's first engineering doctorate in 1863; after three years in Paris, Berlin and Heidelberg he came home to a Yale professorship he held unpaid for nine years, living on inheritance, because the university had no money for him.

He published his great work in the Transactions of the Connecticut Academy of Arts and Sciences, a journal with a circulation approaching zero. Europe learned of him mainly through Maxwell, who read him, championed him, and made a plaster cast of one of Gibbs's thermodynamic surfaces with his own hands -- a copy went to Yale. Gibbs coined the very phrase statistical mechanics, in 1902, a year before he died.