# Josiah Willard Gibbs : biography

Gibbs also had an indirect influence on mathematical economics. He supervised the thesis of Irving Fisher, who received the first Ph.D. in economics from Yale in 1891. In that work, published in 1892 as *Mathematical Investigations in the Theory of Value and Prices*, Fisher drew a direct analogy between Gibbsian equilibrium in physical and chemical systems, and the general equilibrium of markets, and he used Gibbs’s vectorial notation. Gibbs’s protegé Edwin Bidwell Wilson became, in turn, a mentor to leading American economist and Nobel Laureate Paul Samuelson.

In 1947, Samuelson publishedFoundations of Economic Analysis, based on his doctoral dissertation, in which he used as epigraph a remark attributed to Gibbs: "Mathematics is a language." Samuelson later explained that in his understanding of prices his "debts were not primarily to Pareto or Slutsky, but to the great thermodynamicist, Willard Gibbs of Yale."

For his part, mathematician Norbert Wiener cited Gibbs’s use of probability in the formulation of statistical mechanics as "the first great revolution of twentieth century physics" and as a major influence on his conception of cybernetics. Wiener explained in the preface to his book *The Human Use of Human Beings* that it was "devoted to the impact of the Gibbsian point of view on modern life, both through the substantive changes it has made to working science, and through the changes it has made indirectly in our attitude to life in general."

## Major scientific contributions

### Chemical thermodynamics

Gibbs’s papers from the 1870s introduced the idea of expressing the internal energy *U* of a system in terms of the entropy *S*, in addition to the usual state-variables of volume *V*, pressure *p*, and temperature *T*. He also introduced the concept of the chemical potential mu of a given chemical species, defined to be the rate of the increase in *U* associated with the increase in the number *N* of molecules of that species (at constant entropy and volume). Thus, it was Gibbs who first combined the first and second laws of thermodynamics by expressing the infinitesimal change in the energy of a system in the form:

mathrmU = TmathrmS – p ,mathrmV + sum_i mu_i ,mathrm N_i,

where the sum in the last term is over the different chemical species. By taking the Legendre transform of this expression, he defined the concepts of enthalpy and "free energy", including what is now known as the "Gibbs free energy" (a thermodynamic potential which is especially useful to chemists since it determines whether a reaction will proceed spontaneously at a fixed temperature and pressure). In a similar way, he also obtained what later came to be known as the "Gibbs–Duhem equation".

The publication of the paper "On the Equilibrium of Heterogeneous Substances" (1874–78) is now regarded as a landmark in the development of physical chemistry. In it, Gibbs developed a rigorous mathematical theory for various transport phenomena, including adsorption, electrochemistry, and the Marangoni effect in fluid mixtures. He also formulated the phase rule

F;=;C;-;P;+;2

for the number *F* of variables that may be independently controlled in an equilibrium mixture of *C* components existing in *P* phases. Awareness of this rule led to the widespread use of phase diagrams by chemists.

### Statistical mechanics

Together with James Clerk Maxwell and Ludwig Boltzmann, Gibbs is considered one of the founders of statistical mechanics. It was Gibbs who coined the term "statistical mechanics" to identify the branch of theoretical physics that accounts for the observed thermodynamic properties of systems in terms of the statistics of large ensembles of particles. He introduced the concept of phase space and used it to define the microcanonical, canonical, and grand canonical ensembles, thus obtaining a more general formulation of the statistical properties of many-particle systems than what Maxwell and Boltzmann had achieved before.