Josiah Willard Gibbs

Willard Gibbs as a young man

Gibbs was born in New Haven, Connecticut. He belonged to an old Yankee family that had produced distinguished American clergymen and academics since the 17th century. He was the fourth of five children and the only son of Josiah Willard Gibbs Sr., and his wife Mary Anna, née Van Cleve. On his father's side, he was descended from Samuel Willard, who served as acting President of Harvard College from 1701 to 1707. On his mother's side, one of his ancestors was the Rev. Jonathan Dickinson, the first president of the College of New Jersey (later Princeton University). Gibbs's given name, which he shared with his father and several other members of his extended family, derived from his ancestor Josiah Willard, who had been Secretary of the Province of Massachusetts Bay in the 18th century.[6] His paternal grandmother, Mercy (Prescott) Gibbs, was the sister of Rebecca Minot Prescott Sherman, the wife of American founding father Roger Sherman; and he was the second cousin of Roger Sherman Baldwin, see the Amistad case below.

The elder Gibbs was generally known to his family and colleagues as "Josiah", while the son was called "Willard".[7] Josiah Gibbs was a linguist and theologian who served as professor of sacred literature at Yale Divinity School from 1824 until his death in 1861. He is chiefly remembered today as the abolitionist who found an interpreter for the African passengers of the ship Amistad, allowing them to testify during the trial that followed their rebellion against being sold as slaves.[8]

Willard Gibbs was educated at the Hopkins School and entered Yale College in 1854 at the age of 15. At Yale, Gibbs received prizes for excellence in mathematics and Latin, and he graduated in 1858, near the top of his class.[9] He remained at Yale as a graduate student at the Sheffield Scientific School. At age 19, soon after his graduation from college, Gibbs was inducted into the Connecticut Academy of Arts and Sciences, a scholarly institution composed primarily of members of the Yale faculty.[10]

Relatively few documents from the period survive and it is difficult to reconstruct the details of Gibbs's early career with precision.[11] In the opinion of biographers, Gibbs's principal mentor and champion, both at Yale and in the Connecticut Academy, was probably the astronomer and mathematician Hubert Anson Newton, a leading authority on meteors, who remained Gibbs's lifelong friend and confidant.[10][11] After the death of his father in 1861, Gibbs inherited enough money to make him financially independent.[12]

Recurrent pulmonary trouble ailed the young Gibbs and his physicians were concerned that he might be susceptible to tuberculosis, which had killed his mother. He also suffered from astigmatism, whose treatment was then still largely unfamiliar to oculists, so that Gibbs had to diagnose himself and grind his own lenses.[13][14] Though in later years he used glasses only for reading or other close work,[13] Gibbs's delicate health and imperfect eyesight probably explain why he did not volunteer to fight in the Civil War of 1861–65.[15] He was not conscripted and he remained at Yale for the duration of the war.[16]

Gibbs during his time as a tutor at Yale[17]

In 1863, Gibbs received the first Doctorate of Philosophy (PhD) in engineering granted in the US, for a thesis entitled "On the Form of the Teeth of Wheels in Spur Gearing", in which he used geometrical techniques to investigate the optimum design for gears.[9][18][19] In 1861, Yale had become the first US university to offer a PhD degree[20] and Gibbs's was only the fifth PhD granted in the US in any subject.[18]

After graduation, Gibbs was appointed as tutor at the college for a term of three years. During the first two years, he taught Latin and during the third year, he taught "natural philosophy" (i.e., physics).[6] In 1866, he patented a design for a railway brake[21] and read a paper before the Connecticut Academy, entitled "The Proper Magnitude of the Units of Length", in which he proposed a scheme for rationalizing the system of units of measurement used in mechanics.[22]

After his term as tutor ended, Gibbs traveled to Europe with his sisters. They spent the winter of 1866–67 in Paris, where Gibbs attended lectures at the Sorbonne and the Collège de France, given by such distinguished mathematical scientists as Joseph Liouville and Michel Chasles.[23] Having undertaken a punishing regimen of study, Gibbs caught a serious cold and a doctor, fearing tuberculosis, advised him to rest on the Riviera, where he and his sisters spent several months and where he made a full recovery.[24]

Moving to Berlin, Gibbs attended the lectures taught by mathematicians Karl Weierstrass and Leopold Kronecker, as well as by chemist Heinrich Gustav Magnus.[25] In August 1867, Gibbs's sister Julia was married in Berlin to Addison Van Name, who had been Gibbs's classmate at Yale. The newly married couple returned to New Haven, leaving Gibbs and his sister Anna in Germany.[26] In Heidelberg, Gibbs was exposed to the work of physicists Gustav Kirchhoff and Hermann von Helmholtz, and chemist Robert Bunsen. At the time, German academics were the leading authorities in the natural sciences, especially chemistry and thermodynamics.[27]

Gibbs returned to Yale in June 1869 and briefly taught French to engineering students.[28] It was probably also around this time that he worked on a new design for a steam-engine governor, his last significant investigation in mechanical engineering.[29][30] In 1871, he was appointed Professor of Mathematical Physics at Yale, the first such professorship in the United States. Gibbs, who had independent means and had yet to publish anything, was assigned to teach graduate students exclusively and was hired without salary.[31]

Maxwell's sketch of the lines of constant temperature and pressure, made in preparation for his construction of a solid model based on Gibbs's definition of a thermodynamic surface for water (see Maxwell's thermodynamic surface)

Gibbs published his first work in 1873.[9] His papers on the geometric representation of thermodynamic quantities appeared in the Transactions of the Connecticut Academy. These papers introduced the use of different type phase diagrams, which were his favorite aids to the imagination process when doing research, rather than the mechanical models, such as the ones that Maxwell used in constructing his electromagnetic theory, which might not completely represent their corresponding phenomena.[32] Although the journal had few readers capable of understanding Gibbs's work, he shared reprints with correspondents in Europe and received an enthusiastic response from James Clerk Maxwell at Cambridge. Maxwell even made, with his own hands, a clay model illustrating Gibbs's construct. He then produced two plaster casts of his model and mailed one to Gibbs. That cast is on display at the Yale physics department.[33][34]

Maxwell included a chapter on Gibbs's work in the next edition of his Theory of Heat, published in 1875. He explained the usefulness of Gibbs's graphical methods in a lecture to the Chemical Society of London and even referred to it in the article on "Diagrams" that he wrote for the Encyclopædia Britannica.[35][36] Prospects of collaboration between him and Gibbs were cut short by Maxwell's early death in 1879, aged 48. The joke later circulated in New Haven that "only one man lived who could understand Gibbs's papers. That was Maxwell, and now he is dead."[37]

Gibbs then extended his thermodynamic analysis to multi-phase chemical systems (i.e., to systems composed of more than one form of matter) and considered a variety of concrete applications. He described that research in a monograph titled "On the Equilibrium of Heterogeneous Substances", published by the Connecticut Academy in two parts that appeared respectively in 1875 and 1878. That work, which covers about three hundred pages and contains exactly seven hundred numbered mathematical equations,[38] begins with a quotation from Rudolf Clausius that expresses what would later be called the first and second laws of thermodynamics: "The energy of the world is constant. The entropy of the world tends towards a maximum."[39]

Gibbs's monograph rigorously and ingeniously applied his thermodynamic techniques to the interpretation of physico-chemical phenomena, explaining and relating what had previously been a mass of isolated facts and observations.[40] The work has been described as "the Principia of thermodynamics" and as a work of "practically unlimited scope".[38] It solidly laid the foundation for physical Chemistry.[41] Wilhelm Ostwald, who translated Gibbs's monograph into German, referred to Gibbs as the "founder of chemical energetics".[42] According to modern commentators,

It is universally recognised that its publication was an event of the first importance in the history of chemistry ... Nevertheless it was a number of years before its value was generally known, this delay was due largely to the fact that its mathematical form and rigorous deductive processes make it difficult reading for anyone, and especially so for students of experimental chemistry whom it most concerns.

— J. J. O'Connor and E. F. Robertson, 1997[9]

Gibbs continued to work without pay until 1880, when the new Johns Hopkins University in Baltimore, Maryland offered him a position paying $3,000 per year. In response, Yale offered him an annual salary of $2,000, which he was content to accept.[43]

Yale's Sloane Physical Laboratory, as it stood between 1882 and 1931 at the current location of Jonathan Edwards College. Gibbs's office was on the second floor, to the right of the tower in the picture.[44]

From 1880 to 1884, Gibbs worked on developing the exterior algebra of Hermann Grassmann into a vector calculus well-suited to the needs of physicists. With this object in mind, Gibbs distinguished between the dot and cross products of two vectors and introduced the concept of dyadics. Similar work was carried out independently, and at around the same time, by the British mathematical physicist and engineer Oliver Heaviside. Gibbs sought to convince other physicists of the convenience of the vectorial approach over the quaternionic calculus of William Rowan Hamilton, which was then widely used by British scientists. This led him, in the early 1890s, to a controversy with Peter Guthrie Tait and others in the pages of Nature.[6]

Gibbs's lecture notes on vector calculus were privately printed in 1881 and 1884 for the use of his students, and were later adapted by Edwin Bidwell Wilson into a textbook, Vector Analysis, published in 1901.[6] That book helped to popularize the "del" notation that is widely used today in electrodynamics and fluid mechanics. In other mathematical work, he re-discovered the "Gibbs phenomenon" in the theory of Fourier series (which, unbeknownst to him and to later scholars, had been described fifty years before by an obscure English mathematician, Henry Wilbraham).[45]

The sine integral function, which gives the overshoot associated with the Gibbs phenomenon for the Fourier series of a step function on the real line

From 1882 to 1889, Gibbs wrote five papers on physical optics, in which he investigated birefringence and other optical phenomena and defended Maxwell's electromagnetic theory of light against the mechanical theories of Lord Kelvin and others.[6] In his work on optics, just as much as in his work on thermodynamics,[46] Gibbs deliberately avoided speculating about the microscopic structure of matter and purposefully confined his research problems to those that can be solved from broad general principles and experimentally confirmed facts. The methods that he used were highly original and the obtained results showed decisively the correctness of Maxwell's electromagnetic theory.[47]

Gibbs coined the term statistical mechanics and introduced key concepts in the corresponding mathematical description of physical systems, including the notions of chemical potential (1876),[28] and statistical ensemble (1902).[48] Gibbs's derivation of the laws of thermodynamics from the statistical properties of systems consisting of many particles was presented in his highly influential textbook Elementary Principles in Statistical Mechanics, published in 1902, a year before his death.[46]

Gibbs's retiring personality and intense focus on his work limited his accessibility to students. His principal protégé was Edwin Bidwell Wilson, who nonetheless explained that "except in the classroom I saw very little of Gibbs. He had a way, toward the end of the afternoon, of taking a stroll about the streets between his study in the old Sloane Laboratory and his home—a little exercise between work and dinner—and one might occasionally come across him at that time."[49] Gibbs did supervise the doctoral thesis on mathematical economics written by Irving Fisher in 1891.[50] After Gibbs's death, Fisher financed the publication of his Collected Works.[51] Another distinguished student was Lee De Forest, later a pioneer of radio technology.[52]

Gibbs died in New Haven on April 28, 1903, at the age of 64, the victim of an acute intestinal obstruction.[49] A funeral was conducted two days later at his home on 121 High Street,[53] and his body was buried in the nearby Grove Street Cemetery. In May, Yale organized a memorial meeting at the Sloane Laboratory. The eminent British physicist J. J. Thomson was in attendance and delivered a brief address.[54]

Photograph taken around 1895. According to his student Lynde Wheeler, of the existing portraits, this is the most faithful to Gibbs's kindly habitual expression.[55]

Gibbs never married, living all his life in his childhood home with his sister Julia and her husband Addison Van Name, who was the Yale librarian. Except for his customary summer vacations in the Adirondacks (at Keene Valley, New York) and later at the White Mountains (in Intervale, New Hampshire),[56] his sojourn in Europe in 1866–69 was almost the only time that Gibbs spent outside New Haven.[6] He joined Yale's College Church (a Congregational church) at the end of his freshman year[56][57] and remained a regular attendant for the rest of his life.[58] Gibbs generally voted for the Republican candidate in presidential elections but, like other "Mugwumps", his concern over the growing corruption associated with machine politics led him to support Grover Cleveland, a conservative Democrat, in the election of 1884.[59] Little else is known of his religious or political views, which he mostly kept to himself.[58]

Gibbs did not produce a substantial personal correspondence and many of his letters were later lost or destroyed.[60] Beyond the technical writings concerning his research, he published only two other pieces: a brief obituary for Rudolf Clausius, one of the founders of the mathematical theory of thermodynamics, and a longer biographical memoir of his mentor at Yale, H. A. Newton.[61] In Edward Bidwell Wilson's view,

Gibbs was not an advertiser for personal renown nor a propagandist for science; he was a scholar, scion of an old scholarly family, living before the days when research had become résearch ... Gibbs was not a freak, he had no striking ways, he was a kindly dignified gentleman.

— E. B. Wilson, 1931[49]

According to Lynde Wheeler, who had been Gibbs's student at Yale, in his later years Gibbs

was always neatly dressed, usually wore a felt hat on the street, and never exhibited any of the physical mannerisms or eccentricities sometimes thought to be inseparable from genius ... His manner was cordial without being effusive and conveyed clearly the innate simplicity and sincerity of his nature.

— Lynde Wheeler, 1951[55]

He was a careful investor and financial manager, and at his death in 1903 his estate was valued at $100,000[56] (roughly $3.26 million today[62]). For many years, he served as trustee, secretary, and treasurer of his alma mater, the Hopkins School.[63] US President Chester A. Arthur appointed him as one of the commissioners to the National Conference of Electricians, which convened in Philadelphia in September 1884, and Gibbs presided over one of its sessions.[56] A keen and skilled horseman,[64] Gibbs was seen habitually in New Haven driving his sister's carriage.[65] In an obituary published in the American Journal of Science, Gibbs's former student Henry A. Bumstead referred to Gibbs's personal character:

Unassuming in manner, genial and kindly in his intercourse with his fellow-men, never showing impatience or irritation, devoid of personal ambition of the baser sort or of the slightest desire to exalt himself, he went far toward realizing the ideal of the unselfish, Christian gentleman. In the minds of those who knew him, the greatness of his intellectual achievements will never overshadow the beauty and dignity of his life.

— H. A. Bumstead, 1903[6]

Graphical representation of the free energy of a body, from the latter of the papers published by Gibbs in 1873. This shows a plane of constant volume, passing through the point A that represents the body's initial state. The curve MN is the section of the "surface of dissipated energy". AD and AE are, respectively, the energy (ε) and entropy (η) of the initial state. AB is the "available energy" (now called the Helmholtz free energy) and AC the "capacity for entropy" (i.e., the amount by which the entropy can be increased without changing the energy or volume).

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 ${\displaystyle \mu }$ 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 internal energy, dU, of a closed system in the form:[46]

${\displaystyle \mathrm {d} U=T\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}\,}$

where T is the absolute temperature, p is the pressure, dS is an infinitesimal change in entropy and dV is an infinitesimal change of volume. The last term is the sum, over all the chemical species in a chemical reaction, of the chemical potential, μ_i, of the i^th species, multiplied by the infinitesimal change in the number of moles, dN_i of that species. By taking the Legendre transform of this expression, he defined the concepts of enthalpy, H and Gibbs free energy, G.

${\displaystyle G_{(p,T)}=H-TS}$

This compares to the expression for Helmholtz free energy, A.

${\displaystyle A_{(v,T)}=U-TS\,}$

When the Gibbs free energy for a chemical reaction is negative the reaction will proceed spontaneously. When a chemical system is at equilibrium, the change in Gibbs free energy is zero. An equilibrium constant is simply related to the free energy change when the reactants are in their standard states.

${\displaystyle \Delta G^{\ominus }=-RT\ln K^{\ominus }}$

Chemical potential is usually defined as partial molar Gibbs free energy.

${\displaystyle \mu _{i}=\left({\frac {\partial G}{\partial N_{i}}}\right)_{T,P,N_{j\neq i}}}$

Gibbs also obtained what later came to be known as the "Gibbs–Duhem equation".[66]

In an electrochemical reaction characterized by an electromotive force ℰ and an amount of transferred charge Q, Gibbs's starting equation becomes

${\displaystyle \mathrm {d} U=T\mathrm {d} S-p\,\mathrm {d} V+{\mathcal {E}}\mathrm {d} Q}$.

Apparatus for investigating the Phase Rule of an iron-nitrogen system, U.S. Fixed Nitrogen Research Laboratory, 1930

The publication of the paper "On the Equilibrium of Heterogeneous Substances" (1874–78) is now regarded as a landmark in the development of chemistry.[9] In it, Gibbs developed a rigorous mathematical theory for various transport phenomena, including adsorption, electrochemistry, and the Marangoni effect in fluid mixtures.[40] He also formulated the phase rule

${\displaystyle F\;=\;C\;-\;P\;+\;2}$

for the number F of variables that may be independently controlled in an equilibrium mixture of C components existing in P phases. Phase rule is very useful in diverse areas, such as metallurgy, mineralogy, and petrology. It can also be applied to various research problems in physical chemistry.[67]

Together with James Clerk Maxwell and Ludwig Boltzmann, Gibbs founded "statistical mechanics", a term that he coined to identify the branch of theoretical physics that accounts for the observed thermodynamic properties of systems in terms of the statistics of ensembles of all possible physical states of a system composed of many particles. He introduced the concept of "phase of a mechanical system".[68][69] He used the concept to define the microcanonical, canonical, and grand canonical ensembles; all related to the Gibbs measure, thus obtaining a more general formulation of the statistical properties of many-particle systems than Maxwell and Boltzmann had achieved before him.[70]

Gibbs generalized Boltzmann's statistical interpretation of entropy ${\displaystyle S}$ by defining the entropy of an arbitrary ensemble as

${\displaystyle S=-k_{\text{B}}\,\sum _{i}p_{i}\ln \,p_{i}}$,

where ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant, while the sum is over all possible microstates ${\displaystyle i}$, with ${\displaystyle p_{i}}$ the corresponding probability of the microstate (see Gibbs entropy formula).[71] This same formula would later play a central role in Claude Shannon's information theory and is therefore often seen as the basis of the modern information-theoretical interpretation of thermodynamics.[72]

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."[73] 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.[74][75]

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".[46] Gibbs's own framework for statistical mechanics, based on ensembles of macroscopically indistinguishable microstates, 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.[9][76] 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.[77]

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. Following W. K. Clifford in his Elements of Dynamic (1888), Gibbs noted that the product of quaternions could be separated into two parts: a one-dimensional (scalar) quantity and a three-dimensional vector, so that the use of quaternions involved mathematical complications and redundancies that could be avoided in the interest of simplicity and to facilitate teaching. In his Yale classroom notes he defined distinct dot and cross products for pairs of vectors and introduced the now common notation for them. Through the 1901 textbook Vector Analysis prepared by E. B. Wilson from Gibbs notes, he was largely responsible for the development of the vector calculus techniques still used today in electrodynamics and fluid mechanics.[78]

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".[79] 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 priority of Grassmann's ideas, Gibbs convinced Grassmann's heirs to seek the publication in Germany of the essay "Theorie der Ebbe und Flut" 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 (linear space).[80][81]

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.[82]^: 160 He also developed the concept of mutually reciprocal triads of vectors that later proved to be of importance in crystallography.[83]

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.[6][84] 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 ether). 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.[84]

In his last paper on physical optics, Gibbs concluded that

it may be said for the electrical theory [of light] that it is not obliged to invent hypotheses, but only to apply the laws furnished by the science of electricity, and that it is difficult to account for the coincidences between the electrical and optical properties of media unless we regard the motions of light as electrical.

— J. W. Gibbs, 1889[6]

Shortly afterwards, the electromagnetic nature of light was demonstrated by the experiments of Heinrich Hertz in Germany.[85]

In 1909, the American historian and novelist Henry Adams finished an essay entitled "The Rule of Phase Applied to History", in which he sought to apply Gibbs's phase rule and other thermodynamic concepts to a general theory of human history. William James, Henry Bumstead, and others criticized both Adams's tenuous grasp of the scientific concepts that he invoked, as well as the arbitrariness of his application of those concepts as metaphors for the evolution of human thought and society.[121] The essay remained unpublished until it appeared posthumously in 1919, in The Degradation of the Democratic Dogma, edited by Henry Adams's younger brother Brooks.[122]

Cover of the June 1946 issue of Fortune, by artist Arthur Lidov, showing Gibbs's thermodynamic surface of water and his formula for the phase rule

In the 1930s, feminist poet Muriel Rukeyser became fascinated by Willard Gibbs and wrote a long poem about his life and work ("Gibbs", included in the collection A Turning Wind, published in 1939), as well as a book-length biography (Willard Gibbs, 1942).[123] According to Rukeyser:

Willard Gibbs is the type of the imagination at work in the world. His story is that of an opening up which has had its effect on our lives and our thinking; and, it seems to me, it is the emblem of the naked imagination—which is called abstract and impractical, but whose discoveries can be used by anyone who is interested, in whatever "field"—an imagination which for me, more than that of any other figure in American thought, any poet, or political, or religious figure, stands for imagination at its essential points.

— Muriel Rukeyser, 1949[124]

In 1946, Fortune magazine illustrated a cover story on "Fundamental Science" with a representation of the thermodynamic surface that Maxwell had built based on Gibbs's proposal. Rukeyser called this surface a "statue of water"[125] and the magazine saw in it "the abstract creation of a great American scientist that lends itself to the symbolism of contemporary art forms."[126] The artwork by Arthur Lidov also included Gibbs's mathematical expression of the phase rule for heterogeneous mixtures, as well as a radar screen, an oscilloscope waveform, Newton's apple, and a small rendition of a three-dimensional phase diagram.[126]

Gibbs's nephew, Ralph Gibbs Van Name, a professor of physical chemistry at Yale, was unhappy with Rukeyser's biography, in part because of her lack of scientific training. Van Name had withheld the family papers from her and, after her book was published in 1942 to positive literary but mixed scientific reviews, he tried to encourage Gibbs's former students to produce a more technically oriented biography.[127] Rukeyser's approach to Gibbs was also sharply criticized by Gibbs's former student and protégé Edwin Wilson.[128] With Van Name's and Wilson's encouragement, physicist Lynde Wheeler published a new biography of Gibbs in 1951.[129][130]

Both Gibbs and Rukeyser's biography of him figure prominently in the poetry collection True North (1997) by Stephanie Strickland.[131] In fiction, Gibbs appears as the mentor to character Kit Traverse in Thomas Pynchon's novel Against the Day (2006). That novel also prominently discusses the birefringence of Iceland spar, an optical phenomenon that Gibbs investigated.[132]

In 2005, the United States Postal Service issued the American Scientists commemorative postage stamp series designed by artist Victor Stabin, depicting Gibbs, John von Neumann, Barbara McClintock, and Richard Feynman. The first day of issue ceremony for the series was held on May 4 at Yale University's Luce Hall and was attended by John Marburger, scientific advisor to the president of the United States, Rick Levin, president of Yale, and family members of the scientists honored, including physician John W. Gibbs, a distant cousin of Willard Gibbs.[133]

Kenneth R. Jolls, a professor of chemical engineering at Iowa State University and an expert on graphical methods in thermodynamics, consulted on the design of the stamp honoring Gibbs.[134][135][136] The stamp identifies Gibbs as a "thermodynamicist" and features a diagram from the 4th edition of Maxwell's Theory of Heat, published in 1875, which illustrates Gibbs's thermodynamic surface for water.[135][136] Microprinting on the collar of Gibbs's portrait depicts his original mathematical equation for the change in the energy of a substance in terms of its entropy and the other state variables.[137]

• L. P. Wheeler, E. O. Waters and S. W. Dudley (eds.),The Early Work of Willard Gibbs in Applied Mechanics, (New York: Henry Schuman, 1947). ISBN 1-881987-17-5. This contains previously unpublished work by Gibbs, from the period between 1863 and 1871.
• J. W. Gibbs, "On the Equilibrium of Heterogeneous Substances", Transactions of the Connecticut Academy of Arts and Sciences, 3, 108–248, 343–524, (1874–1878). Reproduced in both The Scientific Papers (1906), pp. 55–353 and The Collected Works of J. Willard Gibbs (1928), pp. 55–353.
• E. B. Wilson, Vector Analysis, a text-book for the use of students of Mathematics and Physics, founded upon the Lectures of J. Willard Gibbs, (New Haven: Yale University Press, 1929 [1901]).
• J. W. Gibbs, Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics, (New York: Dover Publications, 1960 [1902]).

Gibbs's other papers are included in both:

• The Scientific Papers of J. Willard Gibbs, in two volumes, eds. H. A. Bumstead and R. G. Van Name, (Woodbridge, CT: Ox Bow Press, 1993 [1906]). ISBN 0-918024-77-3, 1-881987-06-X. For scans of the 1906 printing, see vol. I and vol. II.
• The Collected Works of J. Willard Gibbs, in two volumes, eds. W. R. Longley and R. G. Van Name, (New Haven: Yale University Press, 1957 [1928]). For scans of the 1928 printing, see vol. I and vol. II.

• Bumstead, H. A. (1903). "Josiah Willard Gibbs". American Journal of Science. s4-16 (93): 187–202. Bibcode:1903AmJS...16..187A. doi:10.2475/ajs.s4-16.93.187.. Reprinted with some additions in both The Scientific Papers, vol. I, pp. xiii–xxviiii (1906) and The Collected Works of J. Willard Gibbs, vol. I, pp. xiii–xxviiii (1928). Also available here .
• D. G. Caldi and G. D. Mostow (eds.), Proceedings of the Gibbs Symposium, Yale University, May 15–17, 1989, (American Mathematical Society and American Institute of Physics, 1990).
• W. H. Cropper, "The Greatest Simplicity: Willard Gibbs", in Great Physicists, (Oxford: Oxford University Press, 2001), pp. 106–123. ISBN 0-19-517324-4
• M. J. Crowe, A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, (New York: Dover, 1994 [1967]). ISBN 0-486-67910-1
• J. G. Crowther, Famous American Men of Science, (Freeport, NY: Books for Libraries Press, 1969 [1937]). ISBN 0-8369-0040-5
• F. G. Donnan and A. E. Hass (eds.), A Commentary on the Scientific Writings of J. Willard Gibbs, in two volumes, (New York: Arno, 1980 [1936]). ISBN 0-405-12544-5. Only vol I. is currently available online.
• P. Duhem, Josiah-Willard Gibbs à propos de la publication de ses Mémoires scientifiques, (Paris: A. Herman, 1908).
• C. S. Hastings, "Josiah Willard Gibbs", Biographical Memoirs of the National Academy of Sciences, 6, 373–393 (1909).
• M. J. Klein, "Gibbs, Josiah Willard", in Complete Dictionary of Scientific Biography, vol. 5, (Detroit: Charles Scribner's Sons, 2008), pp. 386–393.
• M. Rukeyser, Willard Gibbs: American Genius, (Woodbridge, CT: Ox Bow Press, 1988 [1942]). ISBN 0-918024-57-9
• R. J. Seeger, J. Willard Gibbs, American mathematical physicist par excellence, (Oxford and New York: Pergamon Press, 1974). ISBN 0-08-018013-2
• L. P. Wheeler, Josiah Willard Gibbs, The History of a Great Mind, (Woodbridge, CT: Ox Bow Press, 1998 [1951]). ISBN 1-881987-11-6
• A. S. Wightman, "Convexity and the notion of equilibrium state in thermodynamics and statistical mechanics". Published as an introduction to R. B. Israel, Convexity in the Theory of Lattice Gases, (Princeton, NJ: Princeton University Press, 1979), pp. ix–lxxxv. ISBN 0-691-08209-X
• E. B. Wilson, "Reminiscences of Gibbs by a student and colleague", Bulletin of the American Mathematical Society, 37, 401–416 (1931).