Gibbs state

In plain terms

Put a system in contact with a large environment at some temperature and wait. It doesn't fall into its lowest-energy configuration and stay there, because the environment keeps jostling it, lending energy and taking it back, but it doesn't wander uniformly over all configurations either, because low energy is still favoured, so you can ask what exactly the compromise is.

Gibbs wrote down the answer, which is to weight each configuration by a decaying exponential of its energy, with the temperature setting how fast the weight decays. Hot systems spread out over many configurations, cold systems concentrate near the bottom, and the crossover between the two is quantitative.

What makes the answer canonical rather than merely plausible is a variational principle, meaning that the state is singled out as the best solution to an optimisation problem. Among all states with the same average energy, the Gibbs state is the one with the most entropy, the standard measure of disorder or of missing information, and equivalently it's the one that minimises free energy, the quantity that balances energy against entropy at the given temperature. It isn't a good guess, as it's the unique solution to an optimisation problem.

In CSD

The thermodynamics strand of the corpus instantiates the Gibbs state on the field Hamiltonian, the energy function of the corpus's model of a field, and derives temperature and free energy from the variational principle. Deriving rather than assuming is the difference between formalising thermodynamics and decorating it.

It's also where two independently built strands of the corpus meet. The thermal side covers Gibbs states, entropy and Landauer's principle, and the field side covers modes, propagators and light cones. The two were built separately, and what joins them is the exact KMS relation at the cutoff, the identity that characterises thermal equilibrium by how correlations behave in time, with the Gibbs state as the object on which both sides agree. Because the field Hamiltonian is diagonal, meaning the energy of each configuration is simply read off, everything has closed form, so the thermal results are computations and not existence claims.

Mathematically

The Gibbs state is the density operator proportional to exp(-beta H), normalised by the partition function. It's the unique minimiser of free energy at inverse temperature beta, and 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, and after three years in Paris, Berlin and Heidelberg he came home to a Yale professorship he held unpaid for nine years, living on an 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, and a copy went to Yale. Gibbs coined the very phrase statistical mechanics in 1902, a year before he died.

Related entries

• Canonical typicalityA small piece of a large quantum system looks thermal not because anything prepared it that way, but because almost every state of the whole makes it look that way.
• KMS conditionThe algebraic signature of thermal equilibrium holds exactly on CSD's finite field arena, and cooling the thermal state all the way down recovers the vacuum.
• Landauer's principleForgetting costs heat at a rate set by temperature alone, and that's the point at which information stops being an abstraction and acquires a physical price.
• Klein's inequalityOne quantum state is never negatively distinguishable from another, as the relative entropy of a state with respect to a second is at least zero, and that single inequality drives the second law, subadditivity, concavity and the Holevo bound in the corpus.
• von Neumann entropyA quantum whole can be perfectly known while its parts are as uncertain as they could possibly be, and entropy is where that stops being a metaphor and becomes arithmetic.
• Jarzynski equalityRush a system between two equilibrium conditions as violently as you like, and the average of a particular exponential of the work you did still returns the equilibrium free energy difference exactly. Irreversible experiments measure a reversible quantity.