## Gibbs, Josiah (1839–1903)

## Gibbs, Josiah (1839–1903)

# GIBBS, JOSIAH

*(1839–1903)*

Josiah Gibbs, a theoretical physicist, was born and died in New Haven, Connecticut, and, aside from a few years studying physics in Europe, spent his academic career at Yale. He is one of the few distinguished American theoretical physicists prior to the twentieth century. Gibbs made advances in vector analysis, and he made major contributions to thermodynamics including an insightful diagrammatic method, work on equilibrium and stability, the definition of free energy, and his famous phase rule regarding coexistent phases of a substance. In a vital contribution to thermodynamics Gibbs extended this theory to deal with the rules that describe how chemical interactions are to be integrated with the other thermodynamic processes. He is the inventor of the notion of chemical potential, the key concept of chemical thermodynamics.

For philosophers it is Gibbs's work in statistical mechanics that is of great interest. This work is contained in his elegant *Elementary Principles in Statistical Mechanics* (1902). James Clerk Maxwell and Ludwig Boltzmann had previously developed a method for calculating equilibrium values by taking them to be averages over functions of the microscopic phase of the system using a time invariant probability distribution over a constant energy subspace of the phase space of the system. This technique reappears in Gibbs in the form of his "microcanonical ensembles."

But Gibbs introduced other ensembles as well. Most important of these is the canonical ensemble that introduces a time invariant probability distribution over the phase space that allows for different energies. For systems with a large number of degrees of freedom (a large number of molecules in a gas, for example), this probability distribution is highly concentrated about one specific energy. In these cases, averages calculated using the canonical distribution and those calculated using the microcanonical distribution will converge in the limit of an infinite number of degrees of freedom. Because calculations done using the canonical ensemble are much easier than those using the microcanonical, most practical statistical mechanics is done in the framework of Gibbs's canonical ensembles. Gibbs also developed the grand canonical ensemble whose use becomes necessary when chemical changes are part of the thermodynamic processes.

By showing how these ensembles and the features of them vary as constraints on the system are varied, Gibbs was able to show "analogies" between features of the ensembles and averages of features calculated with their probability distributions and standard thermodynamic quantities such as temperature and entropy. He is cautious in making any explicit "identification" of the latter with the former, possibly in part because of the known difficulties faced by standard statistical-kinetic reasoning at that time in correctly predicting such quantities as specific heats.

With the association of thermodynamic and statistical mechanical quantities, it is easy to understand the microcanonical ensemble as appropriate for a system energetically isolated from the rest of the world, and the canonical as appropriate for a system in perfect thermal contact with an infinite heat bath of constant temperature.

Gibbs's treatment of nonequilibrium is the source of one standard approach to that problem, but remains controversial to this day. Gibbs's ensembles can be thought of as a vast collection of systems identically prepared at the macroscopic level. Find the ensemble for such a collection of systems; now, change a constraint on the system (say by removing a partition in a box of gas): How will the ensemble, appropriate for equilibrium before the change of constraint, evolve? Will it evolve to the ensemble appropriate for equilibrium in the new constraint condition? This is what is most desirable because people want to show that, in some appropriate sense, the systems in the ensemble at a later time will be found, in general, to be ever closer to the equilibrium condition. But provably the Gibbs's ensemble cannot so evolve (Liouville's Theorem).

But, Gibbs argues, the ensemble may evolve in such a way as to approach the new equilibrium ensemble in a "coarse grained" sense. He uses the analogy of a glass mostly filled with water but partly filled with insoluble black ink. Stir the fluid. If one looks closely enough, the fluid always consists of pure water or pure ink, because the ink is insoluble. But looked at "coarsely," the fluid approaches a uniformly gray color. Gibbs was not able to show that such "mixing" would actually occur, but modern extensions of ergodic theory have been able to prove mixing theorems that hold under certain physical conditions. And idealized systems (such as molecules as "hard sphere in a box") have been shown to be mixing. It remains controversial, however, as to whether this model of an ensemble evolving in a coarse-grained sense is the appropriate one for characterizing the actual approach to equilibrium of nonequilibrium systems.

Gibbs is aware that "mixing" ought to be a time symmetric feature of his ensembles, given that it is driven by a time symmetric underlying dynamics of the molecules. But applying mixing in the past time direction would lead, incorrectly, to predict antithermodynamic behavior for systems. His solution is to argue that one ought to apply statistical inferences only into the unknown future, and that applying them to infer the a past that is already known is illegitimate. Paul and Tatiana Ehrenfest, in their important 1910 survey of statistical mechanics, called Gibbs's argument "incomprehensible." But it was later taken up and developed by Satosi Watanabe and Erwin Schrödinger. It also remains a subject of contemporary controversy in discussions of the relationship between the intuitive asymmetry of time and entropic features of the world.

** See also ** Boltzmann, Ludwig; Philosophy of Statistical Mechanics.

## Bibliography

Ehrenfest, Paul, and Tatiania Ehrenfest. *The Conceptual Foundations of the Statistical Approach in Mechanics*. Ithaca, NY: Cornell University Press, 1959.

Gibbs, Josiah Willard. *Elementary Principles in Statistical Mechanics*. New Haven, CT: Yale University Press, 1902.

Sklar, Lawrence. *Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics*. Cambridge, U.K.: Cambridge University Press, 1993.

Wheeler, Lynde Phelps. *Josiah Willard Gibbs: The History of a Great Mind*. Woodbridge, CT: Ox Bow Press, 1998.

*Lawrence Sklar (2005)*