View source for Josiah Willard Gibbs

{{Short description|American scientist (1839–1903)}}
{{Use mdy dates|date=March 2022}}
{{for-multi|the American linguist and theologian|Josiah Willard Gibbs, Sr.|the United States Navy ship|USNS Josiah Willard Gibbs (T-AGOR-1)}}
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{{Infobox scientist
| image             = Josiah Willard Gibbs -from MMS-.jpg
| alt               = Portrait of Josiah Willard Gibbs
| birth_date        = {{birth date|1839|2|11}}
| birth_place       = [[New Haven, Connecticut]], U.S.
| death_date        = {{death date and age|1903|4|28|1839|2|11}}
| death_place       = New Haven, Connecticut, U.S.
| nationality       = American
| fields            = {{hlist|[[Chemistry]]|[[Mathematics]]|[[Physics]]}}
| workplaces        = [[Yale College]]
| alma_mater        = [[Yale College]] ([[Bachelor of Arts|BA]], [[PhD]])
| doctoral_advisor  = [[Hubert Anson Newton]]
| thesis_title      = On the form of the teeth of wheels in spur gearing
| thesis_url        = http://brbl-dl.library.yale.edu/vufind/Record/3439123
| thesis_year       = 1863
| doctoral_students = [[Edwin Bidwell Wilson]]<br />[[Irving Fisher]]<br />[[Henry Andrews Bumstead]]<br /> [[Lynde Wheeler]]<br />[[Lee De Forest]]
| notable_students  =
| known_for         = {{Collapsible list|[[Statistical mechanics]]|[[Chemical thermodynamics]]|[[Chemical potential]]|[[Cross product]]|[[Dyadics]]|[[Exergy]]|[[Principle of maximum work]]|[[Phase rule]]|[[Phase space]]|[[Physical optics]]|[[Phase separation|Physics of phase separation]]|[[Statistical ensemble (mathematical physics)|Statistical ensemble]]|[[Surface stress]]|[[Vector calculus]]|[[Gibbs entropy]]|[[Gibbs algorithm]]|[[Gibbs distribution]]|[[Surface rheology|Gibbs' elasticity]]|[[Massieu function|Gibbs function]]|[[Gibbs free energy]]|[[Gibbs' inequality]]|[[Gibbs isotherm]]|[[H-theorem#Gibbs' H-theorem|Gibbs' H theorem]]|[[Gibbs lemma]]|[[Gibbs measure]]|[[Gibbs paradox]]|[[Gibbs phenomenon]]|[[Gibbs state]]|[[Maxwell's thermodynamic surface|Gibbs' thermodynamic surface]]|[[Rotation formalisms in three dimensions#Rodrigues vector|Gibbs vector]]|[[Appell's equation of motion|Gibbs–Appell equation of motion]]|[[Gibbs–Donnan effect]]|[[Gibbs–Duhem equation]]|[[Gibbs–Helmholtz equation]]|[[Gibbs–Marangoni effect]]|[[Gibbs–Thomson equation]]|[[Gibbs–Thomson effect]]|[[Wulff construction|Gibbs–Wulff theorem]]}}
| author_abbrev_bot = 
| author_abbrev_zoo = 
| awards            = {{Plainlist|
* [[Rumford Prize]] (1880)
* [[Fellow of the Royal Society|ForMemRS]] (1897)<ref name=formemrs />
*  [[Copley Medal]] (1901)}}
| signature         = JWgibbs-signature.jpg
| signature_alt     = Gibbs's signature
| footnotes         =  
}}

'''Josiah Willard Gibbs''' ({{IPAc-en|ɡ|ɪ|b|z}};<ref>{{cite web |url=https://www.oxfordreference.com/view/10.1093/acref/9780199571123.001.0001/m_en_gb0334600 |title=Gibbs, Josiah Willard |work=Oxford Dictionary of English (3 ed.) |publisher=Oxford Reference |url-access=subscription}}</ref> February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to [[physics]], [[chemistry]], and [[mathematics]]. His work on the applications of [[thermodynamics]] was instrumental in transforming [[physical chemistry]] into a rigorous deductive science. Together with [[James Clerk Maxwell]] and [[Ludwig Boltzmann]], he created [[statistical mechanics]] (a term that he coined), explaining the [[laws of thermodynamics]] as consequences of the statistical properties of [[Statistical ensemble (mathematical physics)|ensembles]] of the possible states of a physical system composed of many particles. Gibbs also worked on the application of [[Maxwell's equations]] to problems in [[physical optics]].  As a mathematician, he created modern [[vector calculus]] (independently of the British scientist [[Oliver Heaviside]], who carried out similar work during the same period) and described the Gibbs phenomenon in the theory of Fourier analysis.

In 1863, [[Yale University]] awarded Gibbs the first American [[Doctor of Philosophy|doctorate]] in [[engineering]]. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was a professor of [[mathematical physics]] from 1871 until his death in 1903. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised by [[Albert Einstein]] as "the greatest mind in American history".<ref name="APS">{{cite web |url=http://www.aps.org/programs/outreach/history/historicsites/gibbs.cfm |title=J. Willard Gibbs |work=Physics History |publisher=American Physical Society |access-date=June 16, 2012|archive-url=https://web.archive.org/web/20080705151510/http://www.aps.org/programs/outreach/history/historicsites/gibbs.cfm |archive-date=July 5, 2008}}</ref> In 1901, Gibbs received what was then considered the highest honor awarded by the international scientific community, the [[Copley Medal]] of the [[Royal Society]] of London,<ref name="APS" /> "for his contributions to mathematical physics".<ref name="Copley">{{cite web |url=http://royalsociety.org/awards/copley-medal/ |title=Copley Medal |work=Premier Awards |publisher=Royal Society |access-date=June 16, 2012}}</ref>

Commentators and biographers have remarked on the contrast between Gibbs's quiet, solitary life in turn of the century [[New England]] and the great international impact of his ideas. Though his work was almost entirely theoretical, the practical value of Gibbs's contributions became evident with the development of industrial chemistry during the first half of the 20th century. According to [[Robert Andrews Millikan|Robert A.&nbsp;Millikan]], in pure science, Gibbs "did for statistical mechanics and thermodynamics what [[Pierre-Simon Laplace|Laplace]] did for celestial mechanics and Maxwell did for electrodynamics, namely, made his field a well-nigh finished theoretical structure".<ref name="Millikan">{{cite journal |last=Millikan |first=Robert A. |author-link=Robert Andrews Millikan |year=1938 |title=Biographical Memoir of Albert Abraham Michelson, 1852–1931 |journal=Biographical Memoirs of the National Academy of Sciences  |volume=19 |issue=4 |pages=121–146 |url=http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/michelson-a-a.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/michelson-a-a.pdf |archive-date=2022-10-09 |url-status=live}}</ref>

== Biography ==

=== Family background ===
[[File:A young Willard Gibbs.jpg|thumb|upright|left|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.|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 University|President of Harvard College]] from 1701 to 1707. On his mother's side, one of his ancestors was the Rev. [[Jonathan Dickinson (New Jersey minister)|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.<ref name="Bumstead">Bumstead 1928</ref> 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 [[La Amistad|Amistad]] case below.

The elder Gibbs was generally known to his family and colleagues as "Josiah", while the son was called "Willard".<ref>Cropper 2001, p. 121</ref> 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 [[Abolitionism in the United States|abolitionist]] who found an interpreter for the African passengers of the ship ''[[La Amistad|Amistad]]'', allowing them to testify during [[United States v. The Amistad|the trial]] that followed their rebellion against being sold as slaves.<ref>{{cite web |url=http://law2.umkc.edu/faculty/projects/ftrials/amistad/AMI_BGIB.HTM |title=Biography of Prof. Josiah Gibbs |last = Linder | first = Douglas |work=Famous American Trials: Amistad Trial |publisher=University of Missouri–Kansas City School of Law |access-date=June 16, 2012}}</ref>

=== Education ===

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.<ref name="MacTutor">{{cite web | url = http://www-groups.dcs.st-and.ac.uk/history/Biographies/Gibbs.html | title = Josiah Willard Gibbs | last1 = O'Connor | first1 = John J. | last2 = Robertson | first2 = Edmund F. | year = 1997 | work = The MacTutor History of Mathematics archive | publisher = University of St Andrews, Scotland. School of Mathematics and Statistics | access-date = June 16, 2012 | archive-date = October 30, 2014 | archive-url = https://web.archive.org/web/20141030174241/http://www-groups.dcs.st-and.ac.uk/history/Biographies/Gibbs.html | url-status = dead }}</ref> 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.<ref name="Rukeyser-CTAcademy">Rukeyser 1988, p. 104</ref>

Relatively few documents from the period survive and it is difficult to reconstruct the details of Gibbs's early career with precision.<ref name="Wheeler-college">Wheeler 1998, pp.&nbsp;23–24</ref> 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 [[Meteoroid|meteors]], who remained Gibbs's lifelong friend and confidant.<ref name="Rukeyser-CTAcademy" /><ref name="Wheeler-college" /> After the death of his father in 1861, Gibbs inherited enough money to make him financially independent.<ref name="Rukeyser-inheritance">Rukeyser 1998, pp.&nbsp;120, 142</ref>

Recurrent [[Lung|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 (eye)|astigmatism]], whose treatment was then still largely unfamiliar to [[Ophthalmology|oculists]], so that Gibbs had to diagnose himself and grind his own lenses.<ref name="Wheeler-astigmatism">Wheeler 1998, pp.&nbsp;29–31</ref><ref name="Rukeyser-astigmatism">Rukeyser 1988, p. 143</ref> Though in later years he used [[glasses]] only for reading or other close work,<ref name="Wheeler-astigmatism" /> Gibbs's delicate health and imperfect eyesight probably explain why he did not volunteer to fight in the [[American Civil War|Civil War]] of 1861–65.<ref name="Wheeler-war">Wheeler 1998, p. 30</ref>  He was not [[Conscription in the United States#Civil War|conscripted]] and he remained at Yale for the duration of the war.<ref name="Rukeyser-war">Rukeyser 1998, p. 134</ref>

[[File:JWGibbs-tutor.jpg|thumb|upright|alt=Portrait of Willard Gibbs as a Yale College tutor|Gibbs during his time as a tutor at Yale<ref>Wheeler 1998, p. 44</ref>]]

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 [[gear]]s.<ref name="MacTutor" /><ref name="Wheeler-PhD">Wheeler 1998, p. 32</ref><ref>{{cite book | url =http://brbl-dl.library.yale.edu/vufind/Record/3439123 | title =On the form of the teeth of wheels in spur gearing | last =Gibbs | first =Josiah W. | date =1863 | access-date =March 27, 2016| bibcode =1863PhDT.........1G }}</ref> In 1861, Yale had become the first US university to offer a PhD degree<ref name="ElmarsafyBernard2013">{{cite book|author1=Ziad Elmarsafy|author2=Anna Bernard|title=Debating Orientalism|url=https://books.google.com/books?id=VP6ARP2m-D0C&pg=PA85|date=June 13, 2013|publisher=Palgrave Macmillan|isbn=978-1-137-34111-2|page=85}}</ref> and Gibbs's was only the fifth PhD granted in the US in any subject.<ref name="Wheeler-PhD" />

=== Career, 1863–1873 ===

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).<ref name="Bumstead" /> In 1866, he patented a design for a [[railway brake]]<ref>US Patent No. 53,971, "Car Brake", April 17, 1866.  See ''The Early Work of Willard Gibbs in Applied Mechanics'', (New York: Henry Schuman, 1947), pp.&nbsp;51–62.</ref> 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.<ref name="Units">Wheeler 1998, appendix II.</ref>

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 [[University of Paris|Sorbonne]] and the {{Lang|fr|[[Collège de France]]|italic=no}}, given by such distinguished mathematical scientists as [[Joseph Liouville]] and [[Michel Chasles]].<ref>Wheeler 1998, p. 40.</ref> Having undertaken a punishing regimen of study, Gibbs caught a serious cold and a doctor, fearing tuberculosis, advised him to rest on the [[French Riviera|Riviera]], where he and his sisters spent several months and where he made a full recovery.<ref>Wheeler 1998, p. 41.</ref>

Moving to [[Berlin]], Gibbs attended the lectures taught by mathematicians [[Karl Weierstrass]] and [[Leopold Kronecker]], as well as by chemist [[Heinrich Gustav Magnus]].<ref>Wheeler 1998, p. 42.</ref> 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.<ref>Rukeyser 1988, p. 151.</ref> In [[Heidelberg University|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]].<ref>Rukeyser 1988, pp.&nbsp;158–161.</ref>

Gibbs returned to Yale in June 1869 and briefly taught French to engineering students.<ref name="Klein-proceedings">{{cite book | chapter = The Physics of J. Willard Gibbs in His Time | last = Klein | first = Martin J. | title = Proceedings of the Gibbs Symposium | year = 1990 |pages = 3, 7}}</ref> It was probably also around this time that he worked on a new design for a steam-engine [[Governor (device)|governor]], his last significant investigation in mechanical engineering.<ref name="Mayr">{{cite journal |doi=10.1098/rsnr.1971.0019 |title=Victorian physicists and speed regulation: An encounter between science and technology |journal=Notes and Records of the Royal Society of London |volume=26 |issue=2 |pages=205–228 |year=1971 |last1=Mayr |first1=Otto |s2cid=144525735 }}</ref><ref>Wheeler 1998, pp.&nbsp;54–55.</ref> 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.<ref name="Rukeyser-professor">Rukeyser 1988, pp.&nbsp;181–182.</ref>

=== Career, 1873–1880 ===
[[File:Maxwell's letters plate IV.jpg|thumb|upright|alt=Maxwell's handmade sketch of the thermodynamic surface for water|right|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.<ref name="MacTutor" /> 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 [[James Clerk Maxwell|Maxwell]] used in constructing his electromagnetic theory, which might not completely represent their corresponding phenomena.<ref>{{cite web | url =http://www.ub.uni-heidelberg.de/helios/fachinfo/www/math/htmg/gibbs-bumstead.htm | title =Josiah Willard Gibbs [Reprinted with some additions from the American Journal of Science, ser. 4, vol. xvi., September, 1903.] | last =Bumstead | first =Henry A. | publisher =Universitätsbibliothek Heidelberg | access-date =September 30, 2015 | archive-url =https://web.archive.org/web/20140427182622/http://www.ub.uni-heidelberg.de/helios/fachinfo/www/math/htmg/gibbs-bumstead.htm | archive-date =April 27, 2014 | url-status =dead }}</ref> 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 [[University of Cambridge|Cambridge]]. Maxwell even made, with his own hands, a [[Maxwell's thermodynamic surface|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.<ref>{{cite journal |doi=10.1103/physrevseriesi.10.228 |title=Gibbs' Thermodynamical Model |journal=Physical Review |series=Series I |volume=10 |issue=4 |pages=228–233 |year=1900 |last1=Boynton |first1=W. P. |bibcode=1900PhRvI..10..228B |url=https://zenodo.org/record/1996925 }}</ref><ref>{{cite web |url=http://esm.rkriz.net//classes/ESM4714/methods/Gibbs.html |title=Thermodynamic Case Study: Gibbs' Thermodynamic Graphical Method |last = Kriz | first = Ronald D. |year=2007 |publisher=Virginia Tech, Dept. of Engineering Science and Mechanics|archive-url=https://web.archive.org/web/20140201163858/http://www.sv.vt.edu/classes/ESM4714/methods/Gibbs.html|archive-date=February 1, 2014|url-status=live | access-date = September 30, 2015}}</ref>

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]]''.<ref>Rukeyser 1988, p. 201.</ref><ref>{{Cite EB1911|wstitle= Diagram |volume= 8 |last= Maxwell |first= James Clerk |author-link= James Clerk Maxwell | pages = 146–149 }}</ref> 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."<ref>Rukeyser 1988, p. 251.</ref>

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,<ref name="Cropper-monograph">Cropper 2001, p. 109.</ref> 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."<ref>Quoted in Rukeyser 1988, p. 233.</ref>

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.<ref name="Wheeler-thermodynamics">Wheeler 1998, ch. V.</ref> The work has been described as "the ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' of thermodynamics" and as a work of "practically unlimited scope".<ref name="Cropper-monograph" /> It solidly laid the foundation for physical Chemistry.<ref name="Jordan1910">{{cite book |author=David Starr Jordan |title=Leading American Men of Science |year=1910 |publisher=H. Holt |url=https://archive.org/details/leadingamerican00jordgoog |pages=[https://archive.org/details/leadingamerican00jordgoog/page/n389 350] |quote=for it laid the foundation of the new science of physical science}}</ref> [[Wilhelm Ostwald]], who translated Gibbs's monograph into German, referred to Gibbs as the "founder of chemical energetics".<ref name="Britannica-1911">{{Cite EB1911|wstitle=Gibbs, Josiah Willard}}</ref> According to modern commentators,
{{blockquote|
It is universally recognised that its publication was an event of the first importance in the history of chemistry&nbsp;... 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<ref name="MacTutor" />}}

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.<ref>Wheeler 1998, p. 91.</ref>

In 1879, Gibbs derived the [[Appell's equation of motion|Gibbs–Appell equation of motion]],<ref name="gibbs1879">{{cite journal |last=Gibbs |first=J. W. |year=1879 |title=On the Fundamental Formulae of Dynamics |journal=American Journal of Mathematics |volume=2 |issue=1 |pages=49–64 |doi=10.2307/2369196 |jstor=2369196}}</ref> rediscovered in 1900 by [[Paul Émile Appell]].<ref name="appell_1900a">{{cite journal |last=Appell |first=P |year=1900 |title=Sur une forme générale des équations de la dynamique |lang=fr |journal=Journal für die reine und angewandte Mathematik |volume=121 |pages=310}}</ref>

=== Career, 1880–1903 ===
[[File:SloaneLab.jpg|thumb|upright|alt=Yale University's old Sloane Physical Laboratory|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.<ref>Wheeler 1998, p. 86</ref>]]

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 product|dot]] and [[cross product]]s 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 [[quaternion]]ic 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 (journal)|Nature]]''.<ref name="Bumstead" />

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.<ref name="Bumstead" /> That book helped to popularize the "[[del]]" notation that is widely used today in [[Classical electromagnetism|electrodynamics]] and [[fluid mechanics]]. In other mathematical work, he re-discovered the "[[Gibbs phenomenon]]" in the theory of [[Fourier series]]<ref>{{Cite journal |last=Gibbs |first=J. Willard |date=1899 |title=Fourier's Series |url=https://www.nature.com/articles/059606a0 |journal=Nature |volume=59 |issue=606 |page=606 |doi=10.1038/059606a0|bibcode=1899Natur..59..606G }}</ref> (which, unbeknownst to him and to later scholars, had been described fifty years before by an obscure English mathematician, [[Henry Wilbraham]]).<ref>{{Cite journal |doi=10.1007/BF00330404 |title=The Gibbs-Wilbraham phenomenon: An episode in fourier analysis |journal=Archive for History of Exact Sciences |volume=21 |issue=2 |pages=129–160 |year=1979 |last1=Hewitt |first1=Edwin |last2=Hewitt |first2=Robert E. |s2cid=119355426 }}</ref>

[[File:Sine integral.svg|thumb|left|alt=Plot of the sine integral|The [[sine integral]] function, which gives the overshoot associated with the [[Gibbs phenomenon]] for the Fourier series of a [[Heaviside step function|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 [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] and others.<ref name="Bumstead" /> In his work on optics, just as much as in his work on thermodynamics,<ref name=Klein1990>{{cite journal |doi=10.1063/1.881258 |title=The Physics of J. Willard Gibbs in his Time |journal=Physics Today |volume=43 |issue=9 |pages=40–48 |year=1990 |last1=Klein |first1=Martin J. |bibcode=1990PhT....43i..40K }}</ref> 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.<ref>Wheeler 1998, p. 121, 124–125</ref>

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),<ref name="Klein-proceedings" /> and [[Statistical ensemble (mathematical physics)|statistical ensemble]] (1902).<ref name="Caldi">{{cite book | editor1= Caldi, D. G. | editor2 = Mostow, G. D.  | title = Proceedings of the Gibbs Symposium | year = 1990 |pages = 143–144}}</ref> 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.<ref name=Klein1990 />

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."<ref name="Wilson-reminiscences ">Wilson 1931</ref> Gibbs did supervise the doctoral thesis on mathematical economics written by [[Irving Fisher]] in 1891.<ref name="Fisher">{{cite journal |doi=10.1090/S0002-9904-1930-04919-8 |title=The application of mathematics to the social sciences |journal=Bulletin of the American Mathematical Society |volume=36 |issue=4 |pages=225–244 |year=1930 |last1=Fisher |first1=Irving |doi-access=free }}</ref> After Gibbs's death, Fisher financed the publication of his ''Collected Works''.<ref name="Celebrating-Fisher">{{cite book|last=Fisher |first=George W. |chapter=Foreword |title=Celebrating Irving Fisher: The Legacy of a Great Economist |publisher=Wiley-Blackwell |year=2005 |chapter-url=http://cowles.econ.yale.edu/books/gean/fisher.htm |url-status=dead |archive-url=https://web.archive.org/web/20060616203239/http://cowles.econ.yale.edu/books/gean/fisher.htm |archive-date=June 16, 2006 }}</ref> Another distinguished student was [[Lee De Forest]], later a pioneer of radio technology.<ref name="DeForest">{{cite journal |last=Schiff |first=Judith |title=The man who invented radio |url=http://www.yalealumnimagazine.com/articles/2285 |journal=Yale Alumni Magazine |volume=72 |issue=2 |date=November 2008 |access-date = December 28, 2013}}</ref>

Gibbs died in New Haven on April 28, 1903, at the age of 64, the victim of an acute intestinal obstruction.<ref name="Wilson-reminiscences " /> A funeral was conducted two days later at his home on 121&nbsp;High Street,<ref>Wheeler 1998, p. 197</ref> 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.<ref>Wheeler 1998, pp.&nbsp;197–199</ref>

=== Personal life and character ===
[[File:JWGibbs.jpg|thumb|upright|alt=Portrait of Willard Gibbs, circa 1895|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.<ref name="Wheeler-portrait">Wheeler 1998, pp.&nbsp;179–180.</ref>]]

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 [[Adirondack Mountains|Adirondacks]] (at [[Keene Valley, New York]]) and later at the [[White Mountains (New Hampshire)|White Mountains]] (in [[Intervale, New Hampshire]]),<ref name="Seeger-gentleman">Seeger 1974, pp.&nbsp;15–16.</ref> his sojourn in Europe in 1866–1869 was almost the only time that Gibbs spent outside New Haven.<ref name="Bumstead" /> He joined Yale's College Church (a [[Congregational church]]) at the end of his freshman year<ref name="Seeger-gentleman" /><ref name="obituary-church">{{cite book | title = Obituary Record of Graduates of Yale University, 1901–1910 | publisher = Tuttle, Morehouse & Taylor | year = 1910 | location = New Haven | url = https://books.google.com/books?id=rVkdAQAAIAAJ&pg=PA238 | page = 238}}</ref> and remained a regular attendant for the rest of his life.<ref name="Wheeler-views">Wheeler, 1998, p. 16.</ref> Gibbs generally voted for the [[Republican Party (United States)|Republican]] candidate in presidential elections but, like other "[[Mugwump]]s", his concern over the growing corruption associated with [[Political machine|machine politics]] led him to support [[Grover Cleveland]], a conservative [[Democratic Party (United States)|Democrat]], in the [[1884 United States presidential election|election of 1884]].<ref name="Samuelson-politics">{{cite book | last = Samuelson | first = Paul A. | author-link=Paul Samuelson | chapter = Gibbs in Economics | title = Proceedings of the Gibbs Symposium | year = 1990 | page = 255}}</ref> Little else is known of his religious or political views, which he mostly kept to himself.<ref name="Wheeler-views" />

Gibbs did not produce a substantial personal correspondence, and many of his letters were later lost or destroyed.<ref>Rukeyser 1988, pp.&nbsp;254, 345, 430.</ref> 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.&nbsp;A.&nbsp;Newton.<ref>Wheeler 1998, p. 95. See also the ''Collected Works'', vol.&nbsp;II.</ref> In Edward Bidwell Wilson's view,
{{blockquote|
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 ''ré''search&nbsp;... Gibbs was not a freak, he had no striking ways, he was a kindly dignified gentleman.
|E. B. Wilson, 1931<ref name="Wilson-reminiscences " />}}

According to [[Lynde Wheeler]], who had been Gibbs's student at Yale, in his later years Gibbs
{{blockquote|
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&nbsp;... His manner was cordial without being effusive and conveyed clearly the innate simplicity and sincerity of his nature.
|Lynde Wheeler, 1951<ref name="Wheeler-portrait" />}}

He was a careful investor and financial manager, and at his death in 1903 his estate was valued at $100,000<ref name="Seeger-gentleman" /> (roughly ${{Formatprice|{{Inflation|US|100000|1903|r=-1}}}} today{{Inflation-fn|US}}). For many years, he served as trustee, secretary, and treasurer of his alma mater, the Hopkins School.<ref name="Wheeler-HopkinsSchool">Wheeler, 1998, p. 144.</ref> US President [[Chester A.&nbsp;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.<ref name="Seeger-gentleman" /> A keen and skilled horseman,<ref>Rukeyser 1988, p. 191.</ref> Gibbs was seen habitually in New Haven driving his sister's [[carriage]].<ref>Rukeyser 1988, p. 224.</ref> In an obituary published in the ''[[American Journal of Science]]'', Gibbs's former student [[Henry Andrews Bumstead|Henry A. Bumstead]] referred to Gibbs's personal character:
{{blockquote|
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.
|[[Henry A. Bumstead|H. A. Bumstead]], 1903<ref name="Bumstead" />}}

== Major scientific contributions ==

=== Chemical and electrochemical thermodynamics ===
[[File:Wykres Gibbsa.svg|thumb|250px|alt=Diagram representing the free energy of a substance|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&nbsp;''U'' of a system in terms of the [[entropy]]&nbsp;''S'', in addition to the usual [[state variable]]s of volume&nbsp;''V'', pressure&nbsp;''p'', and temperature&nbsp;''T''. He also introduced the concept of the [[chemical potential]]&nbsp;<math>\mu</math> 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, d''U'', of a [[closed system]] in the form<ref name=Klein1990 />
: <math>\mathrm{d}U = T\mathrm{d}S - p \,\mathrm{d}V + \sum_i \mu_i \,\mathrm{d} N_i,</math>
where ''T'' is the [[absolute temperature]], ''p'' is the pressure, d''S'' is an infinitesimal change in entropy and d''V'' 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''<sub>i</sub>, of the ''i''-th species, multiplied by the infinitesimal change in the number of moles, d''N''<sub>i</sub> of that species. By taking the [[Legendre transformation|Legendre transform]] of this expression, he defined the concepts of [[enthalpy]] ''H'' and [[Gibbs free energy]] ''G'':
: <math>G_{(p,T)} = H - TS.</math>
This compares to the expression for [[Helmholtz free energy]] ''A'':
: <math>A_{(v,T)} = U - TS.</math>

When the Gibbs free energy for a chemical reaction is negative, the reaction will proceed spontaneously. When a chemical system is at [[equilibrium chemistry|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 state]]s:
: <math>\Delta G^\ominus = -RT \ln K^\ominus.</math>
[[Chemical potential]] is usually defined as partial molar Gibbs free energy:
: <math>\mu_i = \left(\frac{\partial G}{\partial N_i}\right)_{T,P,N_{j\neq i}}.</math>

Gibbs also obtained what later came to be known as the "[[Gibbs–Duhem equation]]".<ref name="Ott" />

In an [[Electrochemistry|electrochemical reaction]] characterized by an [[electromotive force]] ℰ and an amount of transferred charge ''Q'', Gibbs's starting equation becomes 
: <math>\mathrm{d}U = T\mathrm{d}S - p \,\mathrm{d}V + \mathcal{E}\mathrm{d}Q.</math>

[[File:Apparatus for investigating the Phase Rule of an iron-nitrogen system 9p290969x.tif|thumb|right|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]]" (1874–1878) is now regarded as a landmark in the development of [[chemistry]].<ref name="MacTutor" /> In it, Gibbs developed a rigorous mathematical theory for various [[transport phenomena]], including [[adsorption]], [[electrochemistry]], and the [[Marangoni effect]] in fluid mixtures.<ref name="Wheeler-thermodynamics" /> He also formulated the [[phase rule]]
: <math>F = C - P + 2</math>
for the number ''F'' of [[Intensive and extensive properties|variables]] that may be independently controlled in an equilibrium mixture of ''C'' components existing in ''P'' [[Phase (matter)|phases]]. The 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.<ref>Wheeler 1998, p.&nbsp;79.</ref>

=== Statistical mechanics ===

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 [[Statistical ensemble (mathematical physics)|ensembles]] of all possible physical states of a system composed of many particles. He introduced the concept of "[[phase space|phase of a mechanical system]]".<ref name=Nolte2010>{{cite journal |doi=10.1063/1.3397041 |title=The tangled tale of phase space |journal=Physics Today |volume=63 |issue=4 |pages=33–38 |year=2010 |last1=Nolte |first1=David D. |bibcode=2010PhT....63d..33N |s2cid=17205307 }}</ref>{{refn|For a mechanical system composed of ''n'' particles, the phase is represented by a point in a 2''n''-dimensional space, which he called "extension-in-phase" and is equivalent to our modern notion of phase space. However, the phrase "phase space" was not invented by him.<ref name=Nolte2010 />}} He used the concept to define the [[microcanonical ensemble|microcanonical]], [[canonical ensemble|canonical]], and [[grand canonical ensemble]]s; 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.<ref>Wheeler 1998, pp. 155–159.</ref>

Gibbs generalized Boltzmann's statistical interpretation of [[entropy]] <math>S</math> by defining the entropy of an arbitrary ensemble as
: <math>S = -k_\text{B}\,\sum_i p_i \ln p_i,</math>
where <math>k_\text{B}</math> is the [[Boltzmann constant]], while the sum is over all possible [[Microstate (statistical mechanics)|microstates]] <math>i</math>, with <math>p_i</math> the corresponding probability of the microstate (see [[Entropy (statistical thermodynamics)#Gibbs entropy formula|Gibbs entropy formula]]).<ref name="jaynes1965">{{cite journal |doi=10.1119/1.1971557 |title=Gibbs vs Boltzmann Entropies |journal=American Journal of Physics |volume=33 |issue=5 |pages=391–398 |year=1965 |last=Jaynes |first=E. T. |bibcode=1965AmJPh..33..391J |author-link= Edwin Thompson Jaynes}}</ref> 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.<ref name="Reif">{{cite book | last = Brillouin | first = Léon | title = Science and information theory | year = 1962 | author-link = Léon Brillouin | publisher = Academic Press |pages = 119–124}}</ref>

According to [[Henri Poincaré]], writing in 1904, even though Maxwell and Boltzmann had previously explained the [[Irreversible process|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''".<ref name="Poincare">{{cite book |last=Poincaré |first=Henri |author-link=Henri Poincaré |year=1904 |chapter=[[s:The Principles of Mathematical Physics|The Principles of Mathematical Physics]] |title=The Foundations of Science (The Value of Science) |pages=297–320 |publisher=Science Press |place=New York}}</ref> 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.<ref name="Wightman">{{cite book | chapter = On the Prescience of J. Willard Gibbs | last = Wightman | first = Arthur S. | author-link = Arthur Wightman | title = Proceedings of the Gibbs Symposium | year = 1990 |pages = 23–38}}</ref><ref name="Wiener">{{cite book | last = Wiener | first = Norbert | title =Cybernetics: or Control and Communication in the Animal and the Machine | chapter = II: Groups and Statistical Mechanics | edition = 2 | publisher = MIT Press | year = 1961 | isbn = 978-0-262-23007-0}}</ref>

Gibbs was well aware that the application of the [[equipartition theorem]] to large systems of classical particles failed to explain the measurements of the [[Heat capacity|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".<ref name=Klein1990 /> Gibbs's own framework for statistical mechanics, based on ensembles of macroscopically indistinguishable [[Microstate (statistical mechanics)|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.<ref name="MacTutor" /><ref>Wheeler 1998, pp. 160–161.</ref> 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 [[Identical particles|indistinguishability of particles]] required by quantum physics.<ref>See, e.g., {{cite book | last = Huang | first = Kerson | author-link = Kerson Huang | title = Statistical Mechanics | publisher = John Wiley & Sons | year = 1987 | edition = 2 | pages = 140–143 | isbn = 978-0-471-81518-1}}</ref>

=== Vector analysis ===
[[File:Cross product parallelogram.svg|alt=Diagram representing the cross product of two vectors|right|thumb|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 [[quaternion]]s 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 [[Euclidean vector|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 [[Edwin Bidwell Wilson|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.<ref>Wheeler 1998, pp.&nbsp;107-108, 110</ref>

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".<ref name="Grassmann-algebra">Letter by Gibbs to [[Victor Schlegel]], quoted in Wheeler 1998, pp.&nbsp;107–109</ref> 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 [[tide]]s 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]]).<ref name="Grassmann-tides">Wheeler 1998, pp.&nbsp;113–116</ref><ref name="MacTutor_Grassmann">{{cite web |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Grassmann.html |title=Hermann Günter Grassmann | last1 = O'Connor | first1 = John J. | last2 = Robertson | first2 = Edmund F. | year = 2005 | work=The MacTutor History of Mathematics archive |publisher=University of St Andrews, Scotland. School of Mathematics and Statistics}}</ref>

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 [[orbit]]s.<ref name="Crowe1967">{{cite book|author=[[Michael J. Crowe]]|title=A History of Vector Analysis: The Evolution of the Idea of a Vectorial System|year=1967|publisher=Courier Corporation|isbn=978-0-486-67910-5}}</ref>{{rp|160}} He also developed the concept of mutually reciprocal triads of vectors that later proved to be of importance in [[crystallography]].<ref>{{cite book | chapter = Reciprocal Space in Crystallography | last = Shmueli | first = Uri | title = International Tables for Crystallography | volume = B | year = 2006 | pages = 2–9 | chapter-url = http://www.mendeley.com/research/11-reciprocal-space-crystallography/#page-1 | access-date = February 25, 2012 | archive-date = November 4, 2013 | archive-url = https://web.archive.org/web/20131104024615/http://www.mendeley.com/research/11-reciprocal-space-crystallography/#page-1 | url-status = dead }}</ref>

=== Physical optics ===
[[File:Calcite.jpg|right|thumb|alt=Photograph showing birefrigence by a calcite crystal|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 (optics)|dispersion]], and [[Optical rotation|optical activity]].<ref name="Bumstead" /><ref name="Optics">Wheeler 1998, ch. VIII</ref>  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|luminiferous ether]]).  Gibbs also stressed that the absence of a [[longitudinal wave|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 fixing|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.<ref name="Optics" />

In his last paper on physical optics, Gibbs concluded that

{{blockquote|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<ref name="Bumstead" />}}

Shortly afterwards, the electromagnetic nature of light was demonstrated by the experiments of [[Heinrich Hertz]] in Germany.<ref name="Hertz">{{cite book | last = Buchwald | first = Jed Z. | author-link = Jed Buchwald |title = The Creation of Scientific Effects: Heinrich Hertz and Electric Waves |publisher = University of Chicago Press |year = 1994 |isbn = 978-0-226-07887-8}}</ref>

== Scientific recognition ==

Gibbs worked at a time when there was little tradition of rigorous theoretical science in the United States. His research was not easily understandable to his students or his colleagues, and he made no effort to popularize his ideas or to simplify their exposition to make them more accessible.<ref name="MacTutor" /> His seminal work on thermodynamics was published mostly in the ''Transactions of the Connecticut Academy'', a journal edited by his librarian brother-in-law, which was little read in the US and even less so in Europe. When Gibbs submitted his long paper on the equilibrium of heterogeneous substances to the academy, both [[Elias Loomis]] and H. A. Newton protested that they did not understand Gibbs's work at all, but they helped to raise the money needed to pay for the typesetting of the many mathematical symbols in the paper. Several Yale faculty members, as well as business and professional men in New Haven, contributed funds for that purpose.<ref name="Rukeyser-printing">Rukeyser 1998, pp.&nbsp;225–226</ref>

Even though it had been immediately embraced by Maxwell, Gibbs's graphical formulation of the laws of thermodynamics came into widespread use only in the mid 20th century, with the work of [[László Tisza]] and [[Herbert Callen]].<ref>Wightman 1979, pp. xiii, lxxx</ref> According to James Gerald Crowther,

{{blockquote|in his later years [Gibbs] was a tall, dignified gentleman, with a healthy stride and ruddy complexion, performing his share of household chores, approachable and kind (if unintelligible) to students. Gibbs was highly esteemed by his friends, but American science was too preoccupied with practical questions to make much use of his profound theoretical work during his lifetime. He lived out his quiet life at Yale, deeply admired by a few able students but making no immediate impress on American science commensurate with his genius.| J. G. Crowther, 1937<ref name="MacTutor" />}}

[[File:Burlington House ILN 1873.jpg|thumb|alt=Image of Burlington House, London, in 1873|[[Burlington House]], site of the Royal Society of London, in 1873]]

On the other hand, Gibbs did receive the major honors then possible for an academic scientist in the US. He was elected to the [[National Academy of Sciences]] in 1879 and received the 1880 [[Rumford Prize]] from the [[American Academy of Arts and Sciences]] for his work on chemical thermodynamics.<ref>{{cite book | last = Müller | first = Ingo | title = A History of Thermodynamics – the Doctrine of Energy and Entropy | publisher = Springer | year = 2007 | isbn = 978-3-540-46226-2}}</ref> In 1895, he was elected to the [[American Philosophical Society]] in 1895.<ref>{{Cite web |title=APS Member History |url=https://search.amphilsoc.org/memhist/search?creator=Josiah+Willard+Gibbs&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced |access-date=2024-03-19 |website=search.amphilsoc.org}}</ref> He was also awarded honorary doctorates by Princeton University and [[Williams College]].<ref name="Bumstead" />

In Europe, Gibbs was inducted as honorary member of the [[London Mathematical Society]] in 1892 and elected [[List of Fellows of the Royal Society elected in 1897|Foreign Member of the Royal Society in 1897]].<ref name=formemrs>{{cite web|archive-url=https://web.archive.org/web/20150316060617/https://royalsociety.org/about-us/fellowship/fellows/|archive-date=March 16, 2015|url=https://royalsociety.org/about-us/fellowship/fellows/|publisher=Royal Society|location=London|title=Fellows of the Royal Society}}</ref> He was elected as corresponding member of the [[Prussian Academy of Sciences|Prussian]] and [[French Academy of Sciences|French]] Academies of Science and received honorary doctorates from the universities of [[University of Dublin|Dublin]],<ref>{{Cite newspaper The Times |title=University intelligence |date=June 2, 1902 |page=9 |issue=36783}}</ref> [[University of Erlangen-Nuremberg|Erlangen]], and [[University of Oslo|Christiania]]<ref name="Bumstead" /> (now Oslo). The Royal Society further honored Gibbs in 1901 with the [[Copley Medal]], then regarded as the highest international award in the natural sciences,<ref name="APS" /> noting that he had been "the first to apply the second law of thermodynamics to the exhaustive discussion of the relation between chemical, electrical and thermal energy and capacity for external work."<ref name="Britannica-1911" /> Gibbs, who remained in New Haven, was represented at the award ceremony by Commander [[Richardson Clover]], the US naval attaché in London.<ref>Rukeyser 1998, p. 345</ref>

In his autobiography, mathematician [[Gian-Carlo Rota]] tells of casually browsing the mathematical stacks of [[Sterling Library]] and stumbling on a handwritten mailing list, attached to some of Gibbs's course notes, which listed over two hundred notable scientists of his day, including Poincaré, Boltzmann, [[David Hilbert]], and [[Ernst Mach]]. From this, Rota concluded that Gibbs's work was better known among the scientific elite of his day than the published material suggests.<ref name="Rota">{{cite book | last = Rota | first = Gian-Carlo | author-link=Gian-Carlo Rota | title = Indiscrete Thoughts | url = https://archive.org/details/indiscretethough0000rota | url-access = registration | publisher = Birkhäuser | year = 1996 | page = [https://archive.org/details/indiscretethough0000rota/page/25 25] |isbn = 978-0-8176-3866-5}}</ref> Lynde Wheeler reproduces that mailing list in an appendix to his biography of Gibbs.<ref name="Wheeler-mailing">Wheeler 1998, appendix IV</ref> That Gibbs succeeded in interesting his European correspondents in his work is demonstrated by the fact that his monograph "On the Equilibrium of Heterogeneous Substances" was translated into German (then the leading language for chemistry) by [[Wilhelm Ostwald]] in 1892 and into French by [[Henri Louis Le Châtelier]] in 1899.<ref>Wheeler 1998, pp.&nbsp;102–104</ref>

== Influence ==
Gibbs's most immediate and obvious influence was on physical chemistry and statistical mechanics, two disciplines which he greatly helped to found. During Gibbs's lifetime, his phase rule was experimentally validated by Dutch chemist [[Hendrik Willem Bakhuis Roozeboom|H. W. Bakhuis Roozeboom]], who showed how to apply it in a variety of situations, thereby assuring it of widespread use.<ref name="Roozeboom">{{cite book | last=Crowther | first=James Gerald | chapter=Josiah Willard Gibbs, 1839–1903 | title=Famous American Men of Science | chapter-url=https://archive.org/details/famousamericanme0000crow | chapter-url-access=registration | publisher=Books for Libraries | location=Freeport, NY | year=1969 |orig-year=1937 | pages=[https://archive.org/details/famousamericanme0000crow/page/277 277–278]| isbn=9780836900408 }}</ref> In industrial chemistry, Gibbs's thermodynamics found many applications during the early 20th century, from electrochemistry to the development of the [[Haber process]] for the synthesis of [[ammonia]].<ref name="Haber">{{cite journal |doi=10.1016/S0016-0032(25)90344-4 |title=Practical results of the theoretical development of chemistry |journal=Journal of the Franklin Institute |volume=199 |issue=4 |pages=437–456 |year=1925 |last1=Haber |first1=F. }}</ref>

When Dutch physicist [[Johannes Diderik van der Waals|J. D. van der Waals]] received the 1910 [[Nobel Prize in Physics|Nobel Prize]] "for his work on the [[equation of state]] for gases and liquids" he acknowledged the great influence of Gibbs's work on that subject.<ref name="vanderWaals">{{cite web |url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1910/waals-lecture.html | title = Nobel Lecture: The Equation of State for Gases and Liquids | last =van der Waals | first = J. D. | author-link = Johannes Diderik van der Waals | year= 1910 |work= Nobel Prize in Physics |publisher = Nobel Foundation}}</ref>  [[Max Planck]] received the 1918 Nobel Prize for his work on quantum mechanics, particularly his 1900 paper on [[Planck's law]] for quantized [[black-body radiation]]. That work was based largely on the thermodynamics of Kirchhoff, Boltzmann, and Gibbs. Planck declared that Gibbs's name "not only in America but in the whole world will ever be reckoned among the most renowned theoretical physicists of all times."<ref name="Planck">{{cite book | last=Planck | first=Max  |author-link=Max Planck | title=Eight Lectures on Theoretical Physics | chapter=Second Lecture: Thermodynamic States of Equilibrium in Dilute Solutions | publisher=Columbia University Press | location=New York | year=1915 | page=21 | isbn=978-1-4655-2188-0 | chapter-url=https://books.google.com/books?id=53DnAAAAMAAJ&pg=PA21}}</ref>

[[File:Gibbs-Elementary principles in statistical mechanics.png|thumb|upright|alt=Title page of Gibbs's ''Statistical Mechanics''|Title page of Gibbs's ''[[Elementary Principles in Statistical Mechanics]]'', one of the founding documents of that discipline, published in 1902]]

The first half of the 20th century saw the publication of two influential textbooks that soon came to be regarded as founding documents of [[chemical thermodynamics]], both of which used and extended Gibbs's work in that field: these were ''Thermodynamics and the Free Energy of Chemical Processes'' (1923), by [[Gilbert N. Lewis]] and [[Merle Randall]], and ''Modern Thermodynamics by the Methods of Willard Gibbs'' (1933), by [[Edward A. Guggenheim]].<ref name="Ott">{{cite book |last1 = Ott |first1 = Bevan J.  |last2 = Boerio-Goates |first2 = Juliana |title = Chemical Thermodynamics – Principles and Applications |publisher = Academic Press |year = 2000 |pages = 1, 213–214|isbn = 978-0-12-530990-5}}</ref>

Gibbs's work on statistical ensembles, as presented in his 1902 textbook, has had a great impact in both theoretical physics and in pure mathematics.<ref name="Wightman" /><ref name="Wiener" /> According to mathematical physicist [[Arthur Wightman]],

{{blockquote|It is one of the striking features of the work of Gibbs, noticed by every student of thermodynamics and statistical mechanics, that his formulations of physical concepts were so felicitously chosen that they have survived 100 years of turbulent development in theoretical physics and mathematics.| A. S. Wightman, 1990<ref name="Wightman" />}}

Initially unaware of Gibbs's contributions in that field, [[Albert Einstein]] wrote three papers on statistical mechanics, published between 1902 and 1904.  After reading Gibbs's textbook (which was translated into German by [[Ernst Zermelo]] in 1905), Einstein declared that Gibbs's treatment was superior to his own and explained that he would not have written those papers if he had known Gibbs's work.<ref name="Navarro-Einstein">{{cite journal |doi=10.1007/s004070050025 |title=Gibbs, Einstein and the Foundations of Statistical Mechanics |journal=Archive for History of Exact Sciences |volume=53 |issue=2 |pages=147–180 |year=1998 |last1=Navarro |first1=Luis |s2cid=26481725 }}</ref>
[[File:Wilson-2.jpg|alt=Title page to a 1907 copy of Vector Analysis|thumb|212x212px|Title page to a 1907 copy of ''Vector Analysis'']]
Gibbs's early papers on the use of graphical methods in thermodynamics reflect a powerfully original understanding of what mathematicians would later call "[[convex analysis]]",<ref>Wightman 1979, pp.&nbsp;x–xxxiv</ref> including ideas that, according to [[Barry Simon]], "lay dormant for about seventy-five years".<ref name="Simon">{{cite book | last = Simon | first = Barry | author-link = Barry Simon | title = Convexity: An Analytic Viewpoint | publisher = Cambridge University Press | year = 2011 | page = 287 | isbn = 978-1-107-00731-4}}</ref>  Important mathematical concepts based on Gibbs's work on thermodynamics and statistical mechanics include the [[Gibbs lemma]] in [[game theory]], the [[Gibbs' inequality|Gibbs inequality]] in [[information theory]], as well as [[Gibbs sampling]] in [[computational statistics]].

The development of vector calculus was Gibbs's other great contribution to mathematics. The publication in 1901 of E. B. Wilson's textbook ''[[Vector Analysis]]'', based on Gibbs's lectures at Yale, did much to propagate the use of vectorial methods and notation in both mathematics and theoretical physics, definitively displacing the quaternions that had until then been dominant in the scientific literature.<ref name="Marsden">{{cite book | last1 = Marsden | first1 = Jerrold E. | author-link1= Jerrold E. Marsden | last2 = Tromba | first2 = Anthony J. | title = Vector Calculus | edition = 3 | publisher = W. H. Freeman | year = 1988 | pages = 60–61 | isbn = 978-0-7167-1856-7}}</ref>

At Yale, Gibbs was also mentor to Lee De Forest, who went on to invent the [[triode]] amplifier and has been called the "father of radio".<ref name="Seeger-DeForest">Seeger 1974, p. 18</ref> De Forest credited Gibbs's influence for the realization "that the leaders in electrical development would be those who pursued the [[Electromagnetic radiation|higher theory of waves]] and oscillations and the transmission by these means of intelligence and power."<ref name="DeForest" /> Another student of Gibbs who played a significant role in the development of radio technology was Lynde Wheeler.<ref>{{cite journal |doi=10.1038/1831364b0 |title=Dr. Lynde P. Wheeler |journal=Nature |volume=183 |issue=4672 |pages=1364 |year=1959 |bibcode=1959Natur.183.1364. |doi-access=free }}</ref>

Gibbs also had an indirect influence on mathematical economics. He supervised the thesis of [[Irving Fisher]], who received the first PhD 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 theory|general equilibrium]] of markets, and he used Gibbs's vectorial notation.<ref name="Fisher" /><ref name="Celebrating-Fisher" />  Gibbs's protégé Edwin Bidwell Wilson became, in turn, a mentor to leading American economist and Nobel Laureate [[Paul Samuelson]].<ref name="Samuelson-lecture">{{cite book | chapter-url = https://www.nobelprize.org/uploads/2018/06/samuelson-lecture.pdf | chapter= Maximum Principles in Analytical Economics | last = Samuelson | first = Paul A. |title=Nobel Lectures, Economics 1969–1980 |editor=Assar Lindbeck |publisher=World Scientific Publishing |location=Singapore |year=1992 |orig-year = 1970 | citeseerx = 10.1.1.323.8705 }}</ref> In 1947, Samuelson published ''[[Foundations of Economic Analysis]]'', based on his doctoral dissertation, in which he used as [[Epigraph (literature)|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 [[Vilfredo Pareto|Pareto]] or [[Eugen Slutsky|Slutsky]], but to the great thermodynamicist, Willard Gibbs of Yale."<ref>{{cite book | last = Samuelson | first = Paul A. | title = The collected scientific papers of Paul A. Samuelson | volume = 5 | publisher = MIT Press | editor = Kate Crowley | year = 1986 | page = 863 | isbn = 978-0-262-19251-4}}</ref>

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."<ref>{{cite book | last = Wiener | first = Norbert | title =The Human Use of Human Beings: Cybernetics and Society | url = https://archive.org/details/humanuseofhumanb00wien | url-access = registration | publisher = Houghton Mifflin| year = 1950 | pages = [https://archive.org/details/humanuseofhumanb00wien/page/10 10–11]}}</ref>

== Commemoration ==
[[File:JWGibbs-bronze.jpg|thumb|upright|alt=Photograph of bronze memorial tablet of Willard Gibbs|Bronze memorial tablet, originally installed in 1912 at the Sloane Physics Laboratory, now at the entrance to the Josiah Willard Gibbs Laboratories, Yale University]]

When the German physical chemist [[Walther Nernst]] visited Yale in 1906 to give the [[Silliman Memorial Lectures|Silliman lecture]], he was surprised to find no tangible memorial for Gibbs. Nernst donated his $500 lecture fee to the university to help pay for a suitable monument. This was finally unveiled in 1912, in the form of a bronze bas-relief by sculptor [[Lee Lawrie]], installed in the Sloane Physics Laboratory.<ref name="Seeger-memorial">Seeger 1974, p. 21</ref> In 1910, the [[American Chemical Society]] established the [[Willard Gibbs Award]] for eminent work in pure or applied chemistry.<ref>{{cite web|title=Willard Gibbs Award|url=http://chicagoacs.org/content.php?page=Willard_Gibbs_Award|website=Chicago Section of the American Chemical Society|access-date=February 8, 2016}}</ref> In 1923, the [[American Mathematical Society]] endowed the [[Josiah Willard Gibbs Lectureship]], "to show the public some idea of the aspects of mathematics and its applications".<ref>{{cite web |url=https://www.ams.org/meetings/gibbs-lect.html |title=Josiah Willard Gibbs Lectures |work=Special Lectures |publisher=American Mathematical Society |access-date=June 16, 2012}}</ref>

[[File:JWGibbsLabs.jpg|thumb|left|alt=Photograph of the J. W. Gibbs Laboratories, Yale University|Building housing the Josiah Willard Gibbs Laboratories, at Yale University's [[Science Hill (Yale University)|Science Hill]]]]

In 1945, Yale University created the J. Willard Gibbs Professorship in Theoretical Chemistry, held until 1973 by [[Lars Onsager]]. Onsager, who much like Gibbs, focused on applying new mathematical ideas to problems in physical chemistry, won the 1968 Nobel Prize in chemistry.<ref>{{cite journal |doi=10.1063/1.3037438 |title=Lars Onsager |journal=Physics Today |volume=30 |issue=2 |pages=77 |year=1977 |last1=Montroll |first1=Elliott W. |bibcode=1977PhT....30b..77M }}</ref> In addition to establishing the Josiah Willard Gibbs Laboratories and the J. Willard Gibbs Assistant Professorship in Mathematics, Yale has also hosted two symposia dedicated to Gibbs's life and work, one in 1989 and another on the centenary of his death, in 2003.<ref>{{cite journal |year=2003 |title=Forum News |journal=History of Physics Newsletter |volume=8 |issue=6 |page=3| url=http://www.aps.org/units/fhp/newsletters/upload/february03.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.aps.org/units/fhp/newsletters/upload/february03.pdf |archive-date=2022-10-09 |url-status=live}}</ref> [[Rutgers University]] endowed a J. Willard Gibbs Professorship of Thermomechanics, held as of 2014 by Bernard Coleman.<ref>{{cite web|url=http://www.mechanics.rutgers.edu/BDC.html |title=Faculty webpage |last=Coleman |first=Bernard D |publisher=Rutgers University, Dept. of Mechanics and Materials Science |access-date=January 24, 2014 |url-status=dead |archive-url=https://web.archive.org/web/20150415022401/http://www.mechanics.rutgers.edu/BDC.html |archive-date=April 15, 2015 }}</ref>

Gibbs was elected in 1950 to the [[Hall of Fame for Great Americans]].<ref name="HallofFame">{{cite web |url=http://www.medalcollectors.org/Guides/HFGA/HFGA.html |title=The Hall of Fame for Great Americans at New York University |last=Johnson |first=D. Wayne |publisher=Medal Collectors of America |access-date=June 16, 2012 |archive-date=November 15, 2014 |archive-url=https://web.archive.org/web/20141115095902/http://www.medalcollectors.org/Guides/HFGA/HFGA.html |url-status=dead }}</ref> The [[oceanographic research ship]] [[USNS Josiah Willard Gibbs (T-AGOR-1)|USNS ''Josiah Willard Gibbs'' (T-AGOR-1)]] was in service with the [[United States Navy]] from 1958 to 1971.<ref name="ship">{{cite web|url=http://www.history.navy.mil/danfs/s4/san_carlos.htm |archive-url=http://webarchive.loc.gov/all/20110712144712/http://www.history.navy.mil/danfs/s4/san_carlos.htm |archive-date=July 12, 2011 |title=San Carlos |work=Dictionary of American Naval Fighting Ships |publisher=Naval History and Heritage Command |access-date=June 16, 2012 |url-status=dead }}</ref> [[Gibbs (crater)|Gibbs crater]], near the eastern [[Lunar limb|limb]] of the [[Moon]], was named in the scientist's honor in 1964.<ref>{{cite web |url=http://planetarynames.wr.usgs.gov/Feature/2155 |title=Gibbs |work=Gazetteer of Planetary Nomenclature |publisher=International Astronomical Union |access-date=December 11, 2012 |archive-url=https://web.archive.org/web/20171222162448/https://planetarynames.wr.usgs.gov/Feature/2155 |archive-date=December 22, 2017 |url-status=dead }}</ref>

[[Edward Guggenheim]] introduced the symbol ''G'' for the Gibbs free energy in 1933, and this was used also by [[Dirk ter Haar]] in 1966.<ref name="G">Seeger 1974, p. 96</ref> This notation is now universal and is recommended by the [[International Union of Pure and Applied Chemistry|IUPAC]].<ref name="IUPAC-G">{{Cite book| doi = 10.1351/goldbook.G02629 | chapter = Gibbs energy (function), G | title = IUPAC Compendium of Chemical Terminology | year = 2009 | isbn = 978-0-9678550-9-7 }}</ref> In 1960, William Giauque and others suggested the name "gibbs" (abbreviated gbs.) for the unit of entropy [[calorie]] per [[kelvin]],<ref name="Giauque-unit">{{cite journal |doi=10.1021/ja01486a014 |title=The Thermodynamic Properties of Aqueous Sulfuric Acid Solutions and Hydrates from 15 to 300K.1 |journal=Journal of the American Chemical Society |volume=82 |pages=62–70 |year=1960 |last1=Giauque |first1=W. F. |last2=Hornung |first2=E. W. |last3=Kunzler |first3=J. E. |last4=Rubin |first4=T. R. |issue=1 |bibcode=1960JAChS..82...62G }}</ref> but this usage did not become common, and the corresponding [[International System of Units|SI]] unit [[joule]] per kelvin carries no special name.

In 1954, a year before his death, Albert Einstein was asked by an interviewer who were the greatest thinkers that he had known. Einstein replied: "[[Hendrik Lorentz|Lorentz]]", adding "I never met Willard Gibbs; perhaps, had I done so, I might have placed him beside Lorentz."<ref name="Pais">{{cite book |title=Subtle is the Lord |url=https://archive.org/details/subtleislordscie00pais |url-access=registration |last=Pais |first=Abraham |author-link=Abraham Pais |year=1982 |publisher=Oxford University Press |location=Oxford |page = [https://archive.org/details/subtleislordscie00pais/page/73 73]|isbn= 978-0-19-280672-7}}</ref> Author [[Bill Bryson]] in his bestselling [[popular science]] book ''[[A Short History of Nearly Everything]]'' ranks Gibbs as "perhaps the most brilliant person that most people have never heard of".<ref>{{cite book |last1=Bryson |first1=Bill |title=A Short History of Nearly Everything |date=2003 |publisher=Broadway Books, Random House, Inc. |location=New York City |isbn=0-7679-0818-X |page=116 |edition=1st paperback |quote=Gibbs is perhaps the most brilliant person that most people have never heard of. Modest to the point of near invisibility, he passed virtually the whole of his life, apart from three years spent studying in Europe, within a three-block area bounded by his house and the Yale campus in New Haven, Connecticut. For his first ten years at Yale he didn’t even bother to draw a salary. (He had independent means.) From 1871, when he joined the university as a professor, to his death in 1903, his courses attracted an average of slightly over one student a semester. His written work was difficult to follow and employed a private form of notation that many found incomprehensible. But buried among his arcane formulations were insights of the loftiest brilliance.}}</ref>

In 1958, USS ''San Carlos'' was renamed [[USS San Carlos (AVP-51)#Oceanographic research ship|USNS ''Josiah Willard Gibbs'']] and re-designated as an oceanographic research ship.

=== In literature ===
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.<ref name="Mindel-Adams">{{cite journal |doi=10.2307/2708401 |jstor=2708401 |title=The Uses of Metaphor: Henry Adams and the Symbols of Science |journal=Journal of the History of Ideas |volume=26 |issue=1 |pages=89–102 |year=1965 |last1=Mindel |first1=Joseph }}</ref>  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 Adams|Brooks]].<ref name="Degradation">{{cite book |title=The Degradation of the Democratic Dogma |last=Adams |first=Henry |author-link=Henry Adams |editor-last = Adams |editor-first = Brooks |year=1919 |publisher=Macmillan |location=New York |url=https://archive.org/details/thedegradationof00adamuoft |access-date=May 5, 2012}}</ref>

[[File:Fortune June 1946.jpg|thumb|upright|alt=Cover of June 1946 issue of ''Fortune'' magazine, showing an artist's rendition of Gibbs's thermodynamic surface for water|Cover of the June 1946 issue of ''[[Fortune (magazine)|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).<ref name="Gander-Lives">{{cite book |last=Gander |first=Catherine |chapter-url=https://archive.org/details/murielrukeyserdo0000unse/page/73/mode/2up |title=Muriel Rukeyser and Documentary: The Poetics of Connection |publisher=Edinburgh University Press |year=2013 |isbn=978-0-7486-7053-6 |location=Edinburgh |pages=73–120 |chapter=The Lives |chapter-url-access=registration}}</ref>  According to Rukeyser:

{{blockquote|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<ref name="Rukeyser-PT">{{cite journal |doi=10.1063/1.3066422 |title=Josiah Willard Gibbs |journal=Physics Today |volume=2 |issue=2 |pages=6–27 |year=1949 |last1=Rukeyser |first1=Muriel |bibcode=1949PhT.....2b...6R }}</ref>}}

In 1946, ''[[Fortune (magazine)|Fortune]]'' magazine illustrated a cover story on "Fundamental Science" with a representation of the [[Maxwell's thermodynamic surface|thermodynamic surface]] that Maxwell had built based on Gibbs's proposal. Rukeyser called this surface a "statue of water"<ref>Rukeyser 1988, p. 203</ref> and the magazine saw in it "the abstract creation of a great American scientist that lends itself to the symbolism of contemporary art forms."<ref name="Fortune">{{cite journal |year=1946 |title=The Great Science Debate |journal=[[Fortune (magazine)|Fortune]] |volume=33 |issue=6 |page=117}}</ref> 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, [[Isaac Newton|Newton's]] apple, and a small rendition of a three-dimensional phase diagram.<ref name="Fortune" />

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.<ref name="family-papers">{{cite web |url=http://drs.library.yale.edu/HLTransformer/HLTransServlet?stylename=yul.ead2002.xhtml.xsl&pid=beinecke:gibbs&query=&clear-stylesheet-cache=yes&hlon=yes&big=&adv=&filter=&hitPageStart=&sortFields=&view=all |title=Guide to the Gibbs-Van Name Papers |last = Holeman | first = Heather L. |year = 1986 | publisher=Yale University Library |access-date = January 18, 2013}}</ref> Rukeyser's approach to Gibbs was also sharply criticized by Gibbs's former student and protégé Edwin Wilson.<ref name="Wilson-rev1">{{cite journal |doi=10.1126/science.99.2576.386 |pmid=17844056 |title=Starring Subjects in "american Men of Science" |journal=Science |volume=99 |issue=2576 |pages=386 |year=1944 |last1=Miller |first1=G. A. |bibcode=1944Sci....99..386M }}</ref>  With Van Name's and Wilson's encouragement, physicist Lynde Wheeler published a new biography of Gibbs in 1951.<ref name="Wheeler-preface">Wheeler 1998, pp.&nbsp;ix–xiii</ref><ref name="Wilson-rev2">{{Cite journal | last1 = Wilson | first1 = Edwin B. | year = 1951 | title = Josiah Willard Gibbs | journal = American Scientist | volume = 39 | issue = 2 | pages = 287–289 | jstor = 27826371 }}</ref>

Both Gibbs and Rukeyser's biography of him figure prominently in the poetry collection ''True North'' (1997) by [[Stephanie Strickland]].<ref name="Strickland">{{cite book |title=True North |last=Strickland |first=Stephanie |author-link=Stephanie Strickland |year=1997 |publisher=University of Notre Dame Press |location=Notre Dame, IN |isbn= 978-0-268-01899-3}}</ref>  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.<ref name="Pynchon">{{cite book |title=Against the Day |url=https://archive.org/details/againstday00pync |url-access=registration |last=Pynchon |first=Thomas |author-link=Thomas Pynchon |year=2006 |publisher=Penguin |location=New York |isbn= 978-1-59420-120-2}}</ref>

=== Gibbs stamp (2005) ===

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.<ref name="stamp-issue">{{cite news |title=Yale scientist featured in new stamp series |url=http://www.yale.edu/opa/arc-ybc/v33.n29/story3.html |newspaper=Yale Bulletin & Calendar |date=May 20, 2005 |volume=33 |number=28 |access-date=November 30, 2012 |url-status=dead |archive-url=https://web.archive.org/web/20141030122112/http://www.yale.edu/opa/arc-ybc/v33.n29/story3.html |archive-date=October 30, 2014 }}</ref>

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.<ref name="Jolls-stamp">{{cite news |title=Iowa State Chemical Engineer Drives Issue of New Stamp Honoring Father of Thermodynamics |url=http://www.eng.iastate.edu/coe/feature/jolls.asp |archive-url=https://web.archive.org/web/20121030193200/http://www.eng.iastate.edu/coe/feature/jolls.asp |archive-date=October 30, 2012 |newspaper=College Feature, Iowa State University, College of Engineering |year=2004 |access-date=November 17, 2012}}</ref><ref name="ISU-stamp">{{cite news |title=ISU professor helps develop postage stamp honoring noted scientist |first= Annette |last= Hacker |url=http://www.public.iastate.edu/~nscentral/news/04/nov/stamp.shtml |newspaper=News Service, Iowa State University |date=November 11, 2004 |access-date=November 17, 2012}}</ref><ref name="chem-eng-stamp">{{cite journal |year=2005 |title=Postal Service Pays Homage to Josiah Willard Gibbs |journal=Chemical Engineering Progress |volume=101 |issue=7 |page=57}}</ref>  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.<ref name="ISU-stamp" /><ref name="chem-eng-stamp" />  [[Microprinting]] on the [[Collar (clothing)|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.<ref name="Spakovszky-stamp">{{cite journal |last=Spakovszky |first= Zoltan |year=2005 |title=Stamp of Authenticity |journal=Mechanical Engineering |publisher=ASME |volume=128 |issue=4 |page=7 |url=http://web.mit.edu/16.unified/www/FALL/thermodynamics/figures/MechEng-4-2006-Gibbs.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://web.mit.edu/16.unified/www/FALL/thermodynamics/figures/MechEng-4-2006-Gibbs.pdf |archive-date=2022-10-09 |url-status=live}}</ref>

== Outline of principal work ==
* '''[[Physical chemistry]]''': [[Thermodynamic free energy|free energy]], [[phase diagram]], [[phase rule]], [[transport phenomena]]
* '''[[Statistical mechanics]]''': [[Statistical ensemble (mathematical physics)|statistical ensemble]], [[phase space]], [[chemical potential]], [[Gibbs entropy]], [[Gibbs paradox]]
* '''[[Mathematics]]''': ''[[Vector Analysis (Gibbs/Wilson)|Vector Analysis]]'', [[convex analysis]], [[Gibbs phenomenon]]
* '''[[Electromagnetism]]''': [[Maxwell's equations]], [[birefringence]]

== See also ==
{{Portal|Biography|Mathematics|Physics}}
* [[Concentration of measure#Concentration of measure in physics|Concentration of measure in physics]]
* [[Crystal growth#Morphology|Thermodynamics of crystal growth]]
* [[Governor (device)]]
* [[List of notable textbooks in statistical mechanics]]
* [[List of theoretical physicists]]
* [[List of things named after Josiah W. Gibbs]]
* [[Timeline of United States discoveries]]
* [[Timeline of thermodynamics]]

== References ==
{{Reflist|30em}}

=== Bibliography ===

=== Primary ===
* 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), [https://archive.org/stream/scientificpapers01gibbuoft#page/54/mode/2up pp.&nbsp;55–353] and ''The Collected Works of J. Willard Gibbs'' (1928), pp.&nbsp;55–353.
* [[Edwin Bidwell Wilson|E. B. Wilson]], ''[[Vector Analysis|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|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 [https://archive.org/stream/scientificpapers01gibbuoft#page/n5/mode/2up vol. I] and [https://archive.org/stream/scientificpapers02gibbuoft#page/n3/mode/2up 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 [http://gallica.bnf.fr/ark:/12148/bpt6k95192s vol. I] and [https://archive.org/details/collectedworksj00longgoog vol. II].

=== Secondary ===
* {{cite journal | doi = 10.2475/ajs.s4-16.93.187 | volume=s4-16 | issue=93 | title=Josiah Willard Gibbs | year=1903 | journal=American Journal of Science | pages=187–202 | last= Bumstead | first = H. A. | bibcode=1903AmJS...16..187A | url=https://zenodo.org/record/1450126 | author-link = Henry Andrews Bumstead }}.  Reprinted with some additions in both ''The Scientific Papers'', vol. I, [https://archive.org/stream/scientificpapers01gibbuoft#page/n15/mode/2up pp.&nbsp;xiii–xxviiii] (1906) and ''The Collected Works of J. Willard Gibbs'', vol. I, pp.&nbsp;xiii–xxviiii (1928).  Also available here [http://histmath-heidelberg.de/htmg/gibbs-bumstead.htm].
* 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.&nbsp;106–123.  {{isbn|0-19-517324-4}}
* M. J. Crowe, [[A History of Vector Analysis|''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 [https://archive.org/details/commentaryonscie01donn vol I.] is currently available online.
* [[Pierre Duhem|P. Duhem]], [https://archive.org/details/josiahwillardgib00duheuoft ''Josiah-Willard Gibbs à propos de la publication de ses Mémoires scientifiques''], (Paris: A. Herman, 1908).
* C. S. Hastings, [https://archive.org/stream/biographicalmemo011929mbp#page/n417/mode/2up "Josiah Willard Gibbs"], ''Biographical Memoirs of the National Academy of Sciences'', '''6''', 373–393 (1909).
* [[Martin J. Klein|M. J. Klein]], "Gibbs, Josiah Willard", in ''Complete Dictionary of Scientific Biography'', vol. 5, (Detroit: Charles Scribner's Sons, 2008), pp.&nbsp;386–393.
* [[Muriel Rukeyser|M. Rukeyser]], ''Willard Gibbs: American Genius'', (Woodbridge, CT: Ox Bow Press, 1988 [1942]).  {{isbn|0-918024-57-9}}
* [[Raymond Seeger|R. J. Seeger]], ''J. Willard Gibbs, American mathematical physicist'' par excellence, (Oxford and New York: Pergamon Press, 1974).  {{isbn|0-08-018013-2}}
* [[Lynde Wheeler|L. P. Wheeler]], ''Josiah Willard Gibbs, The History of a Great Mind'', (Woodbridge, CT: Ox Bow Press, 1998 [1951]). {{isbn|1-881987-11-6}}
* [[Arthur Wightman|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.&nbsp;ix–lxxxv.  {{isbn|0-691-08209-X}}
* [[Edwin Bidwell Wilson|E. B. Wilson]],  [http://projecteuclid.org/euclid.bams/1183494779 "Reminiscences of Gibbs by a student and colleague"], ''Bulletin of the American Mathematical Society'', '''37''', 401–416 (1931).

== External links ==
{{Sister project links | wikt=no | commons=Josiah Willard Gibbs | b=no | n=no | q=Josiah Willard Gibbs | s=Author:Josiah Willard Gibbs | v=no | voy=no | species=no | d=no | display=Josiah Willard Gibbs}}
* {{MacTutor Biography|id=Gibbs}}
* "[http://www.aip.org/history/gap/Gibbs/Gibbs.html Josiah Willard Gibbs] {{Webarchive|url=https://web.archive.org/web/20150501105303/http://www.aip.org/history/gap/Gibbs/Gibbs.html |date=May 1, 2015 }}", in ''Selected Papers of Great American Scientists'', American Institute of Physics, (2003 [1976])
* {{MathGenealogy|id=132578}}
* [https://web.archive.org/web/20131228162217/http://murielrukeyser.emuenglish.org/writing/gibbs/ "Gibbs"] by [[Muriel Rukeyser]]
* [https://arxiv.org/abs/1403.2460 Reflections on Gibbs: From Statistical Physics to the Amistad] by [[Leo Kadanoff]], Prof.
* [http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/gibbs-josiah.pdf National Academy of Sciences, Biography, Josiah Willard Gibbs]
*[[hdl:10079/fa/beinecke.jwgibbsbein|Josiah Willard Gibbs Papers]]. General Collection, Beinecke Rare Book and Manuscript Library, Yale University.

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