Josiah Willard Gibbs

How to use this course

Josiah Willard Gibbs spent almost his whole life in one house in New Haven and published mostly in the transactions of a small local academy. From there he reorganized three subjects. His memoir on heterogeneous equilibrium turned thermodynamics into a tool for chemists. His lecture notes on vectors replaced quaternions with the dot and cross products every engineer now uses. His last book founded statistical mechanics in the form physicists still teach. This course follows those three strands, plus a short excursion into Fourier series, alongside the story of his unusually quiet career.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The order follows Gibbs’s own: geometric thermodynamics (1873), heterogeneous equilibrium (1875–1878), vector analysis (1881–1884), and statistical mechanics (1902). Notation is modern, with Gibbs’s own symbols noted where they survive.

Timeline

New Haven and Yale, 1839–1866

Josiah Willard Gibbs was born in New Haven on 11 February 1839. His family had come from Warwickshire to Boston in the seventeenth century and included a long line of New England clergymen and scholars. His father, also Josiah Willard Gibbs, was a philologist and professor of sacred literature at Yale. In 1839 the elder Gibbs found an interpreter who could speak with the African captives of the slave ship Amistad, which allowed them to give testimony at their trial. The son grew up in the house his father built near the college and lived there for almost all his life.

He attended Hopkins Grammar School and entered Yale College in 1854, aged 15. He won prizes in Latin and mathematics and graduated in 1858. He stayed on for graduate study in engineering, guided by the mathematician and astronomer Hubert Anson Newton. His father died in 1861; the inheritance gave Gibbs a modest independent income, which later allowed him to work for years without pay.

The first engineering doctorate

In 1863 Yale awarded him a PhD for the thesis On the Form of the Teeth of Wheels in Spur Gearing. It was the first doctorate in engineering conferred in the United States, and one of the first American PhDs of any kind. The thesis treats the shape of gear teeth by the methods of geometry, a habit of mind that returns in his thermodynamics a decade later. He then served three years as a tutor at Yale, teaching Latin for two years and natural philosophy for the third. In 1866 he patented an improved railway car brake and read a paper to the Connecticut Academy on the proper size of units of length.

Paris, Berlin, Heidelberg, 1866–1869

In 1866 Gibbs left for three years in Europe, the only long period of his life spent away from New Haven. American science had no graduate schools of the European kind, and a year or two at a German university was the standard finishing school for ambitious scholars.

He spent the winter of 1866–67 in Paris, attending lectures by Joseph Liouville and Michel Chasles at the Sorbonne and the Collège de France. In Berlin he heard Karl Weierstrass, Leopold Kronecker and the physicist Gustav Magnus. In 1868–69 he was in Heidelberg, where Hermann von Helmholtz, Gustav Kirchhoff and Robert Bunsen were then working. Physics there was being rebuilt on the principle of energy conservation, and thermodynamics was a young subject.

He published nothing in Europe and seems to have read widely rather than worked with any one teacher. What he brought home was a command of the mathematics of continental physics, a familiarity with the thermodynamics of Rudolf Clausius, and the conviction that the subject needed a clearer foundation. He returned to New Haven in June 1869.

A professor without salary, 1869–1880

In 1871 Yale appointed Gibbs professor of mathematical physics, the first such chair in the United States. It carried no salary. Yale paid professors from endowed chairs and student fees, and there was neither an endowment nor many students for mathematical physics. Gibbs lived on his inheritance and in the household of his sister Julia and her husband Addison Van Name, the college librarian.

The unpaid decade was the most productive of his life. In 1873 he published two papers on geometrical methods in thermodynamics. Between 1875 and 1878 he published On the Equilibrium of Heterogeneous Substances, about three hundred pages with seven hundred numbered equations, in the Transactions of the Connecticut Academy of Arts and Sciences. Few chemists read that journal, and fewer still could follow the mathematics. Gibbs sent offprints to scientists across Europe, and the readers who mattered most found him.

The first was James Clerk Maxwell. He added a discussion of Gibbs’s thermodynamic surfaces to the 1875 edition of his Theory of Heat and built a clay model of the surface for water, sending Gibbs a plaster cast shortly before his death in 1879. In 1880 the new Johns Hopkins University offered Gibbs a chair at 3,000 dollars a year. Yale answered with 2,000 dollars, and Gibbs, attached to New Haven, accepted. He was elected to the National Academy of Sciences in 1879 and received the Rumford Prize of the American Academy of Arts and Sciences in 1880.

Graphical thermodynamics, 1873

For the general reader. Engineers drew steam engines on a pressure–volume chart. Gibbs asked which picture would make the laws of thermodynamics most transparent, and answered with a surface: energy plotted against entropy and volume. Temperature and pressure become slopes of that surface, and the stability of matter becomes its shape.

The fundamental equation

Clausius had shown that \(dS=\delta Q/T\) for reversible changes. Combined with the first law \(dU=\delta Q-P\,dV\), it gives the relation Gibbs took as his starting point:

Gibbs’s insight was that a single function \(U(S,V)\) contains all thermodynamic information about a simple fluid. He called such a function a fundamental equation. Its partial derivatives give the equations of state:

An equation of state such as \(PV=nRT\) alone is weaker: it does not determine \(U\) or \(S\).

Diagrams and surfaces

In his first 1873 paper Gibbs compared diagrams with coordinates chosen from \(P,V,T,S,U\). He favoured the entropy–temperature diagram, in which heat absorbed in a reversible process is the area \(\int T\,dS\) under the path, and the entropy–volume diagram. In the second paper he plotted \(U\) as a height over the \((S,V)\) plane. The tangent plane at a point has slopes \(T\) and \(-P\).

Why. Suppose the system, held in a medium at temperature \(T_0\) and pressure \(P_0\), could split or move to another state with the same total entropy and volume budget drawn from the medium. The available energy is \(U-T_0S+P_0V\), the height of the surface above the tangent plane. A state is stable when no other state has a lower value of this quantity. The plane touching at two points is then the geometric form of phase coexistence, as in the hero figure.

Chemical potential and equilibrium

For the general reader. Water evaporates, salt dissolves, a gas reacts. To predict which way matter will go, Gibbs gave each substance in each phase a number, its chemical potential. Matter flows from high potential to low, as heat flows from hot to cold, and equilibrium is reached when the potentials match.

The memoir opens with two lines from Clausius: the energy of the world is constant, and its entropy tends to a maximum. Gibbs turned them into a criterion.

The chemical potential

For a homogeneous phase containing amounts \(n_1,\dots,n_r\) of \(r\) independent components, Gibbs extended the fundamental equation to

He called \(\mu_i\) the potential of component \(i\); the symbol \(\mu\) is his.

Derivation. Apply \((\delta U)_S\ge0\) to two phases with \(\delta S^\alpha=-\delta S^\beta\), \(\delta V^\alpha=-\delta V^\beta\), \(\delta n_i^\alpha=-\delta n_i^\beta\). Then

for variations of either sign, so each bracket vanishes. If a component is absent from one phase, \(\delta n_i\) can only be positive there, and Gibbs obtained an inequality instead. He was careful about such boundary cases throughout.

Euler and Gibbs–Duhem

\(U\) is homogeneous of degree one in the extensive variables: \(U(\lambda S,\lambda V,\lambda n)=\lambda U(S,V,n)\). Differentiating at \(\lambda=1\) gives

Taking the differential and subtracting the fundamental equation gives the Gibbs–Duhem relation, which shows that the intensive variables of a phase are not independent:

Free energies and the phase rule

For the general reader. In a laboratory one controls temperature and pressure, not entropy and volume. Gibbs introduced energy-like functions adapted to each set of controls. The one for constant temperature and pressure now bears his name. He also found a simple rule that counts how many phases can coexist.

Gibbs’s three functions

Gibbs wrote \(\psi\), \(\chi\), \(\zeta\) for the functions now called the Helmholtz free energy, enthalpy and Gibbs free energy:

Each is a Legendre transform of \(U\) and is equally fundamental. By the Euler relation, \(G=\sum_i\mu_i n_i\): for a pure substance the chemical potential is the molar Gibbs energy.

The phase rule

Proof. Each phase’s intensive state is fixed by \(T\), \(P\) and \(C-1\) mole fractions: \(\Phi(C+1)\) variables. Equilibrium imposes equal \(T\), equal \(P\) and equal \(\mu_i\) across phases: \((\Phi-1)(C+2)\) equations. The difference is \(\Phi(C+1)-(\Phi-1)(C+2)=C-\Phi+2\). Gibbs himself phrased it as \(n+2-r\) variations of phase for \(n\) components in \(r\) phases.

Common tangents

For a binary mixture at fixed \(T,P\), let \(g(x)\) be the molar Gibbs energy at mole fraction \(x\). The chemical potentials are read off the tangent line: its intercepts at \(x=0\) and \(x=1\) are \(\mu_1\) and \(\mu_2\). Two phases with compositions \(x_\alpha,x_\beta\) coexist when they share a tangent line. Any mixture with \(x_\alpha\lt x\lt x_\beta\) lowers its free energy by splitting into those two phases, in proportions given by the lever rule.

Stability, critical points and surfaces

For the general reader. Some states of matter are fragile: superheated water can sit above its boiling point until a bubble forms. Gibbs separated stable, metastable and unstable states and described the critical point where two phases merge. He also showed how to do thermodynamics for the thin skin between two phases, which governs soap films, droplets and catalysis.

Both conditions follow from the convexity of \(U(S,V)\) in Lesson 04: the second derivative of \(F(T,V)\) in \(V\) is \(-(\partial P/\partial V)_T=1/(V\kappa_T)\), and convexity of \(U\) in \(V\) at fixed \(S\) passes to \(F\) at fixed \(T\) under the Legendre transform. Gibbs also analysed the work required to create a small nucleus of a new phase inside a metastable one. This is the starting point of nucleation theory.

The dividing surface

An interface between phases has a thickness of a few molecules, through which densities vary continuously. Gibbs replaced it with a mathematical dividing surface of area \(A\) and assigned to it surface excesses: for each component, the amount actually present minus the amount there would be if each bulk phase extended unchanged up to the surface. Writing \(\Gamma_i=n_i^{\mathrm s}/A\) and \(\gamma\) for the surface tension, the surface analogue of the Gibbs–Duhem relation is:

Vector analysis

For the general reader. William Rowan Hamilton’s quaternions multiply four-component numbers. Maxwell used them for electromagnetism, but most of the time physicists only needed two pieces of the quaternion product. Gibbs split them apart as the dot product and the cross product, and the resulting vector algebra is the one taught in every physics course today.

From quaternions to vectors

A pure quaternion \(a=a_1i+a_2j+a_3k\) can be identified with a vector \(\mathbf a\in\mathbb R^3\). With \(i^2=j^2=k^2=ijk=-1\), the product of two pure quaternions is

Gibbs’s Elements of Vector Analysis, privately printed for his students in 1881 and 1884, treated the scalar part and the vector part as separate products and dropped the quaternion. Oliver Heaviside reached the same system independently in his electromagnetic papers. Peter Guthrie Tait, the leading quaternionist, attacked the new notation in Nature in the early 1890s; Gibbs replied with a defence of vectors as the natural language of physics. Edwin Bidwell Wilson’s textbook Vector Analysis (1901), based on Gibbs’s lectures, carried the system into general use.

Dyadics

Gibbs’s most original contribution in this area was the dyadic, a formal sum \(\Phi=\sum_k\mathbf a_k\mathbf b_k\) of juxtaposed vector pairs, acting by \(\Phi\cdot\mathbf v=\sum_k\mathbf a_k(\mathbf b_k\cdot\mathbf v)\). A dyadic is a linear map of \(\mathbb R^3\), or equivalently a second-rank tensor. The stress in an elastic body and the inertia of a rigid body are dyadics, and Gibbs developed their algebra (conjugates, invariants, the idemfactor \(\mathbf{ii}+\mathbf{jj}+\mathbf{kk}\)) long before tensor notation was common in physics.

Statistical mechanics and ensembles

For the general reader. Thermodynamics describes heat without atoms. Ludwig Boltzmann and Maxwell explained it with molecules colliding in a gas. Gibbs took a more general and more abstract route: imagine a great many copies of a system, all prepared the same way, and ask how their states are distributed. Temperature, entropy and free energy then emerge as properties of that distribution.

Gibbs had worked on the subject since at least 1884, the year of a paper known only from its abstract. His book Elementary Principles in Statistical Mechanics appeared in March 1902. It considers a mechanical system with Hamiltonian \(H(q,p)\) on a \(2N\)-dimensional phase space and an ensemble of copies described by a density \(\rho(q,p,t)\).

The canonical ensemble

Gibbs singled out the stationary distribution he called canonically distributed:

He called \(\Theta\) the modulus and showed it behaves as temperature; today \(\Theta=k_BT\). In modern notation, with partition function \(Z=\int e^{-\beta H}\,dq\,dp/h^{3N}N!\) and \(\beta=1/k_BT\):

Gibbs called \(\eta=\ln\rho\) the index of probability; minus its ensemble average is the Gibbs entropy. He also defined the microcanonical ensemble (fixed energy) and the grand canonical ensemble, in which the number of particles varies and \(\rho\propto e^{\beta(\mu N-H)}\). He showed that for large systems the ensembles give the same thermodynamics, and he stated frankly that classical mechanics could not explain the specific heats of polyatomic gases, a failure resolved only by quantum theory.

The Gibbs paradox

Mixing two different ideal gases at equal \(T\) and \(P\) raises the entropy by \(\Delta S=-Nk_B\sum x_i\ln x_i\). If the gases are identical, removing the partition changes nothing, so \(\Delta S=0\). Yet the formula does not depend on how different the gases are. Gibbs resolved the discontinuity by treating permutations of identical particles as the same phase, which introduces the factor \(1/N!\) in \(Z\). Quantum mechanics later justified this as the indistinguishability of identical particles.

Fourier series and the Gibbs phenomenon

For the general reader. A Fourier series builds a jagged function out of smooth waves. Near a jump, the partial sums overshoot, and the overshoot never shrinks, no matter how many waves are added. Gibbs pointed this out in two short letters to Nature, and the effect bears his name, though an English mathematician had found it fifty years earlier.

History

In 1848 Henry Wilbraham analysed the overshoot in a paper that went largely unread. In 1898 Albert Michelson built a harmonic analyzer that summed Fourier series mechanically, and a correspondence in Nature between Michelson and Augustus Love discussed the series of a sawtooth wave. Gibbs joined late in 1898 with a note on the limit of the partial-sum curves; his first letter described that limit inaccurately. His correction of 27 April 1899 stated that the limiting curves extend beyond the jump by a definite amount. In 1906 Maxime Bôcher gave a full analysis and named the effect after Gibbs.

Derivation for the square wave

Let \(f(x)=\operatorname{sign}x\) on \((-\pi,\pi)\), so \(J=2\). Its series is \(f(x)=\frac4\pi\sum_{k\ge1}\frac{\sin(2k-1)x}{2k-1}\). Differentiate the partial sum with \(N\) terms:

The first critical point to the right of 0 is \(x=\pi/2N\). There

using \(t=2Nx\) and \(\sin x\approx x\). The overshoot above \(f=1\) is \(0.179\approx0.0895\times2\). The peak moves toward the jump as \(N\) grows, so the series converges at every fixed \(x\ne0\), but not uniformly.

Optics

Gibbs’s other contributions to mathematical physics in the 1880s were five papers on the electromagnetic theory of light (1882–1889). He showed that Maxwell’s equations account for double refraction, dispersion and optical rotation without special assumptions about the ether, and that they exclude longitudinal light waves automatically, where elastic-solid theories such as Kelvin’s had to impose that by hand.

Teacher and reluctant celebrity, 1880–1903

Gibbs taught a small number of advanced students for three decades. His courses covered thermodynamics, vector analysis, the electromagnetic theory of light and, later, statistical mechanics. Students described his lectures as clear but unhurried, and he was patient with anyone willing to work. His doctoral students included the economist Irving Fisher (1891), whose thesis on the mathematics of value and prices drew on Gibbs’s thermodynamics; the physicist Henry Andrews Bumstead (1897), later his editor and biographer; and Edwin Bidwell Wilson (1901), who wrote up the vector lectures and later worked in statistics and public health at Harvard. Lynde Wheeler and the radio pioneer Lee De Forest also took doctorates at Yale during Gibbs’s years and are often counted among his students.

Recognition came first from Europe. Wilhelm Ostwald translated the equilibrium memoir into German in 1892, and Henri Le Chatelier translated it into French in 1899. Dutch chemists, led by Hendrik Willem Bakhuis Roozeboom, used the phase rule to organize the study of alloys and salt solutions. Gibbs was elected an honorary member of the London Mathematical Society in 1892 and a Foreign Member of the Royal Society in 1897. In 1901 the Royal Society awarded him the Copley Medal, its highest honour.

He also contributed to astronomy, adapting vector methods to compute the orbit of a comet or planet from three observations and applying them to Swift’s comet of 1880. In 1879 he derived the equations of motion now called the Gibbs–Appell equations, a form of mechanics suited to systems with non-holonomic constraints.

Death and legacy

Gibbs died in New Haven on 28 April 1903, aged 64, after a short illness caused by an acute intestinal obstruction. He was buried in Grove Street Cemetery, a short walk from the house where he had lived. He never married. J. J. Thomson spoke at the memorial meeting Yale held in May 1903. Bumstead and Ralph Gibbs Van Name edited his Scientific Papers in 1906.

Albert Einstein, who had rederived much of statistical mechanics independently in 1902–1904, later said that Gibbs’s treatment was the better one and that he would not have written his own papers had he known it. Max Planck counted Gibbs among the most renowned theoretical physicists of all time.

Legacy

• Chemistry and materials: chemical potential, the Gibbs free energy, the phase rule and the common-tangent construction are the foundation of physical chemistry, metallurgy and chemical engineering.
• Physics: the ensemble formulation of statistical mechanics carried over directly into quantum statistical mechanics, where the density \(\rho\) becomes a density operator.
• Mathematics: vector algebra and dyadics, the Gibbs phenomenon, and the Gibbs measures of probability theory and ergodic theory, which extend the canonical distribution to infinite systems.
• Economics: through Irving Fisher and later Paul Samuelson, equilibrium analysis in economics borrowed Gibbs’s mathematical style.
• Honours: the American Mathematical Society has held the Josiah Willard Gibbs Lectureship since 1923. He was elected to the Hall of Fame for Great Americans in 1950 and appeared on a United States postage stamp in 2005. A lunar crater bears his name.

Connections

People linked to Josiah Willard Gibbs across this collection. A name without a link has no page here yet. See Josiah Willard Gibbs in the network of all 337 lives →

Teachers and advisors

• Hubert Anson Newton 1858-1863

Students

• Ronald Fisher 1891
• Edwin Bidwell Wilson 1901
• Henry Andrews Bumstead 1897
• Lee De Forest
• Lynde Phelps Wheeler

Influenced by

• Hermann Grassmann
• Hermann von Helmholtz 1868-1869
• Joseph Liouville 1866-1867
• Karl Weierstrass 1867-1868
• Leopold Kronecker 1867-1868
• Ludwig Boltzmann
• Rudolf Clausius
• William Rowan Hamilton

Influenced

• Albert Einstein
• Jacobus Henricus van 't Hoff
• Johannes Diderik van der Waals
• Max Planck
• Wilhelm Ostwald 1892
• Henri Le Chatelier 1899

Colleagues and collaborators

• Hermann von Helmholtz
• James Clerk Maxwell 1873-1879
• Ludwig Boltzmann
• Oliver Heaviside

Rivals

• Peter Guthrie Tait 1890-1893

Also mentioned in this life

• Robert Bunsen
• Michel Chasles
• Peter Gustav Lejeune Dirichlet
• Leonhard Euler
• Joseph Fourier
• J. J. Thomson
• William Thomson, Lord Kelvin
• Gustav Kirchhoff
• Joseph-Louis Lagrange
• Pierre-Simon Laplace
• Adrien-Marie Legendre
• Albert A. Michelson
• Isaac Newton
• Count Rumford

Fields

• thermodynamics 12 lives
• statistical mechanics 8 lives
• physical chemistry 9 lives
• mathematical physics 45 lives

On CoursesHub

• Maxwell & Vector Calculus · History of Mathematics & Physics
• Thermodynamics
• Statistical Mechanics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Josiah Willard Gibbs, MacTutor History of Mathematics.
2. Josiah Willard Gibbs, Wikipedia (with extensive references).
3. Josiah Willard Gibbs, Mathematics Genealogy Project.
4. H. A. Bumstead, Josiah Willard Gibbs, American Journal of Science 16 (1903), reprinted in the Scientific Papers.
5. L. P. Wheeler, Josiah Willard Gibbs: The History of a Great Mind, Yale University Press, 1951.
6. M. J. Crowe, A History of Vector Analysis, University of Notre Dame Press, 1967.

Physics and mathematics

1. J. W. Gibbs, Graphical methods in the thermodynamics of fluids; A method of geometrical representation of the thermodynamic properties of substances by means of surfaces, Trans. Connecticut Acad. 2 (1873).
2. J. W. Gibbs, On the equilibrium of heterogeneous substances, Trans. Connecticut Acad. 3 (1875–1878).
3. J. W. Gibbs, Elementary Principles in Statistical Mechanics, Scribner’s, 1902.
4. J. W. Gibbs, The Scientific Papers of J. Willard Gibbs, ed. H. A. Bumstead and R. G. Van Name, 2 vols., Longmans, 1906.
5. E. B. Wilson, Vector Analysis, founded upon the lectures of J. Willard Gibbs, Yale University Press, 1901.
6. J. W. Gibbs, Fourier’s series, Nature 59 (1898), 200, and 59 (1899), 606.
7. Gibbs phenomenon, Wikipedia, for the history from Wilbraham to Bôcher.
8. Elementary Principles in Statistical Mechanics, Wikipedia.
9. H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985: the modern textbook closest to Gibbs’s approach.