# Josiah Willard Gibbs : biography

According to Henri Poincaré, writing in 1904, even though Maxwell and Boltzmann had previously explained the irreversibility of macroscopic physical processes in probabilistic terms, "the one who has seen it most clearly, in a book too little read because it is a little difficult to read, is Gibbs, in his *Elementary Principles of Statistical Mechanics*." Gibbs's analysis of irreversibility, and his formulation of Boltzmann's H-theorem and of the ergodic hypothesis, were major influences on the mathematical physics of the 20th century.

Gibbs was well aware that the application of the equipartition theorem to large systems of classical particles failed to explain the measurements of the specific heats of both solids and gases, and he argued that this was evidence of the danger of basing thermodynamics on "hypotheses about the constitution of matter". Gibbs's own framework for statistical mechanics was so carefully constructed that it could be carried over almost intact after the discovery that the microscopic laws of nature obey quantum rules, rather than the classical laws known to Gibbs and to his contemporaries. His resolution of the so-called "Gibbs paradox", about the entropy of the mixing of gases, is now often cited as a prefiguration of the indistinguishability of particles required by quantum physics.See, e.g.,

### Vector analysis

Diagram showing the magnitude and direction of the cross product of two vectors, in the notation introduced by Gibbs

British scientists, including Maxwell, had relied on Hamilton's quaternions in order to express the dynamics of physical quantities, like the electric and magnetic fields, having both a magnitude and a direction in three-dimensional space. Gibbs, however, noted that the product of quaternions always had to be separated into two parts: a one-dimensional (scalar) quantity and a three-dimensional vector, so that the use of quaternions introduced mathematical complications and redundancies that could be avoided in the interest of simplicity and to facilitate teaching. He therefore proposed defining distinct dot and cross products for pairs of vectors and introduced the now common notation for them. He was also largely responsible for the development of the vector calculus techniques still used today in electrodynamics and fluid mechanics.

While he was working on vector analysis in the late 1870s, Gibbs discovered that his approach was similar to the one that Grassmann had taken in his "multiple algebra".Letter by Gibbs to Victor Schlegel, quoted in Wheeler 1998, pp. 107–109 Gibbs then sought to publicize Grassmann's work, stressing that it was both more general and historically prior to Hamilton's quaternionic algebra. To establish Grassmann's priority, Gibbs convinced Grassmann's heirs to seek the publication in Germany of the essay on tides that Grassmann had submitted in 1840 to the faculty at the University of Berlin, in which he had first introduced the notion of what would later be called a vector space.Wheeler 1998, pp. 113–116

As Gibbs had advocated in the 1880s and 1890s, quaternions were eventually all but abandoned by physicists in favor of the vectorial approach developed by him and, independently, by Oliver Heaviside. Gibbs applied his vector methods to the determination of planetary and comet orbits. He also developed the concept of mutually reciprocal triads of vectors that later proved to be of importance in crystallography.

### Physical optics

A [[calcite crystal produces birefringence (or "double refraction") of light, a phenomenon which Gibbs explained using Maxwell's equations for electromagnetic phenomena.]]

Though Gibbs's research on physical optics is less well known today than his other work, it made a significant contribution to classical electromagnetism by applying Maxwell's equations to the theory of optical processes such as birefringence, dispersion, and optical activity. In that work, Gibbs showed that those processes could be accounted for by Maxwell's equations without any special assumptions about the microscopic structure of matter or about the nature of the medium in which electromagnetic waves were supposed to propagate (the so-called luminiferous aether). Gibbs also stressed that the absence of a longitudinal electromagnetic wave, which is needed to account for the observed properties of light, is automatically guaranteed by Maxwell's equations (by virtue of what is now called their "gauge invariance"), whereas in mechanical theories of light, such as Lord Kelvin's, it must be imposed as an *ad hoc* condition on the properties of the aether.

In his last paper on physical optics, Gibbs concluded that

Shortly afterwards, the electromagnetic nature of light was conclusively demonstrated by the experiments of Heinrich Hertz in Germany.