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Prof. Nolte: The tangled tale of phase space


Phase space has been called one of the most powerful inventions of modern science.  But its historical origins are clouded in a tangle of independent discovery and misattributions that persist today.  This Physics Today article unravels the twisted tale of the discovery and the naming of phase space that began with Liouville in 1838, but by no means ended there, culminating in an encyclopedia article of 1911 that had unintended and lasting etymological side effects never intended by its authors.

Though it was used originally to describe specific types of dynamical systems, today phase space has become synonymous with the idea of a large parameter set—whether they are stock prices in economics, the dust in Saturn’s rings, or high-energy particles in an accelerator—the degrees of freedom are loosely called the phase space of the respective systems.  The concept and its name are embedded in our scientific fluency and cultural literacy.  In his popular book Chaos on the history and science of chaos theory, James Gleich calls phase space “one of the most powerful inventions of modern science”.  But who invented it?  Who named it?  And why? 

The origins of both the concept of phase space, as well as its name, are historically obscure—which is surprising in view of the central role it plays in virtually every aspect of modern physics.  The historical origins have been further obscured by overly generous attribution.  In virtually every textbook on dynamics, classical or statistical, the first reference to phase space is placed firmly in the hands of the French mathematician Joseph Liouville, usually with a citation dated to his 1838 paper in which he supposedly derived the theorem on the conservation of volume in phase space.  In fact, Liouville makes no mention of phase space in his paper, let alone dynamical systems. Liouville’s paper is purely mathematical, on the behavior of a class of solutions to a specific kind of differential equation.  Though he lived to a ripe old age (he died in 1882), he was apparently unaware of its application to statistical mechanics by others even within his lifetime.  Therefore, Liouville’s famous paper, cited routinely as the origin of phase space by all the conventional textbooks, and even by noted chroniclers of the history of mathematics, surprisingly is not it!
How did this happen?  How did we lose track of the discovery and naming of one of our most important modern concepts in physics?  And if it was not discovered by Liouville, then by whom and when and why?  And where did it get its somewhat strange name of phase space?  Where’s the phase? 

In the search for the origins of phase space, there are two challenges.  The first is to identify those who contributed to the development of the concept of phase space. The second challenge is to discover who gave phase space its fully modern name.  This is the subject of the Physics Today article in the April 2010 issue.

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The path to phase space: Jacobi synthesized the unrelated work of Hamilton and Liouville into the first derivation of conservation of phase-space volume, but without concepts of space.  Boltzmann synthesized the unrelated work of Jacobi and Lissajou-Bowditch into a probabilistic theory of phase space, while Poincaré applied phase space concepts to systems of small number.  Notions of a trajectory in a 2n-dimensional space became common after Gibbs, and phase space (Phasenraum) was finally named by Ehrenfest.  The first explicit uses of the term “phase-space” in a paper were separately by Rosenthal and Plancheral, writing on the ergodic theory motivated by the work of Boltzmann and Poincaré.