Josiah Willard Gibbs (; 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

_{i}, of the ''i''^{th} species, multiplied by the infinitesimal change in the number of moles, d''N''_{i} of that species. By taking the

Vector Analysis
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...

'',

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 901. * J. W. Gibbs, '' Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics'', (New York: Dover Publications, 1960

vol. I

an

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

vol. I

an

vol. II

pp. xiii–xxviiii

(1906) and ''The Collected Works of J. Willard Gibbs'', vol. I, pp. xiii–xxviiii (1928). Also available her

* 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. * M. J. Crowe, ''A History of Vector Analysis: The Evolution of the Idea of a Vectorial System'', (New York: Dover, 1994 967. * J. G. Crowther, ''Famous American Men of Science'', (Freeport, NY: Books for Libraries Press, 1969

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

"Reminiscences of Gibbs by a student and colleague"

''Bulletin of the American Mathematical Society'', 37, 401–416 (1931).

, in ''Selected Papers of Great American Scientists'', American Institute of Physics, (2003

"Gibbs"

by

Reflections on Gibbs: From Statistical Physics to the Amistad

by

National Academy of Sciences, Biography, Josiah Willard Gibbs

* Josiah Willard Gibbs Papers. General Collection, Beinecke Rare Book and Manuscript Library, Yale University. {{DEFAULTSORT:Gibbs, Josiah Willard 1839 births 1903 deaths Thermodynamicists Mathematical analysts American physical chemists 19th-century American mathematicians 20th-century American mathematicians Foreign Members of the Royal Society Yale School of Engineering & Applied Science alumni Heidelberg University alumni Yale University faculty Hopkins School alumni Scientists from New Haven, Connecticut Recipients of the Copley Medal Burials at Grove Street Cemetery Theoretical physicists Fluid dynamicists Hall of Fame for Great Americans inductees Connecticut Republicans Philosophers of science Yale College alumni Statistical physicists

thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...

was instrumental in transforming physical chemistry
Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical mecha ...

into a rigorous inductive science. Together with James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...

and Ludwig Boltzmann
Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...

, he created statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...

(a term that he coined), explaining the laws of thermodynamics
The laws of thermodynamics are a set of scientific laws which define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium. The laws also use various paramet ...

as consequences of the statistical properties of ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on the application of Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...

to problems in physical optics
In physics, physical optics, or wave optics, is the branch of optics that studies interference, diffraction, polarization, and other phenomena for which the ray approximation of geometric optics is not valid. This usage tends not to include effec ...

. As a mathematician, he invented modern vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...

(independently of the British scientist Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...

, who carried out similar work during the same period).
In 1863, Yale
Yale University is a private research university in New Haven, Connecticut. Established in 1701 as the Collegiate School, it is the third-oldest institution of higher education in the United States and among the most prestigious in the wor ...

awarded Gibbs the first American doctorate
A doctorate (from Latin ''docere'', "to teach"), doctor's degree (from Latin ''doctor'', "teacher"), or doctoral degree is an academic degree awarded by universities and some other educational institutions, derived from the ancient formalism ''l ...

in engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...

. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was a professor of mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...

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
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

as "the greatest mind in American history." In 1901, Gibbs received what was then considered the highest honor awarded by the international scientific community, the Copley Medal
The Copley Medal is an award given by the Royal Society, for "outstanding achievements in research in any branch of science". It alternates between the physical sciences or mathematics and the biological sciences. Given every year, the medal is t ...

of the Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...

of London, "for his contributions to mathematical physics."
Commentators and biographers have remarked on the contrast between Gibbs's quiet, solitary life in turn of the century New England
New England is a region comprising six states in the Northeastern United States: Connecticut, Maine, Massachusetts, New Hampshire, Rhode Island, and Vermont. It is bordered by the state of New York to the west and by the Canadian provinces ...

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 A. Millikan
Robert Andrews Millikan (March 22, 1868 – December 19, 1953) was an American experimental physicist honored with the Nobel Prize for Physics in 1923 for the measurement of the Elementary charge, elementary electric charge and for his work on ...

, in pure science, Gibbs "did for statistical mechanics and thermodynamics what Laplace did for celestial mechanics and Maxwell did for electrodynamics, namely, made his field a well-nigh finished theoretical structure."
Biography

Family background

Gibbs was born in New Haven, Connecticut. He belonged to an oldYankee
The term ''Yankee'' and its contracted form ''Yank'' have several interrelated meanings, all referring to people from the United States. Its various senses depend on the context, and may refer to New Englanders, residents of the Northern United St ...

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
Josiah Willard Gibbs (; 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 t ...

Sr., and his wife Mary Anna, ''née'' Van Cleve. On his father's side, he was descended from Samuel Willard
Samuel Willard (January 31, 1640 – September 12, 1707) was a New England Puritan clergyman. He was born in Concord, Massachusetts, graduated from Harvard College in 1659, and was minister at Groton from 1663 to 1676, before being driven out by ...

, 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
Jonathan Dickinson (1663–1722) was a merchant from Port Royal, Jamaica who was shipwrecked on the southeast coast of Florida in 1696, along with his family and the other passengers and crew members of the ship.
The party was held captive by Job ...

, the first president of the College of New Jersey (later Princeton University
Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...

). 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
The Province of Massachusetts Bay was a colony in British America which became one of the Thirteen Colonies, thirteen original states of the United States. It was chartered on October 7, 1691, by William III of England, William III and Mary II ...

in the 18th century.Bumstead 1928 His paternal grandmother, Mercy (Prescott) Gibbs, was the sister of Rebecca Minot Prescott
Rebecca Minot Prescott (1742–1813) was the second wife of United States Founding Father Roger Sherman.
She was the daughter of Benjamin and Rebecca Minot Prescott from Salem, Massachusetts; the niece of Roger Sherman’s brother Rev. Josiah Sher ...

Sherman, the wife of American founding father Roger Sherman
Roger Sherman (April 19, 1721 – July 23, 1793) was an American statesman, lawyer, and a Founding Father of the United States. He is the only person to sign four of the great state papers of the United States related to the founding: the Cont ...

; and he was the second cousin of Roger Sherman Baldwin
Roger Sherman Baldwin (January 4, 1793 – February 19, 1863) was an American politician who served as the 32nd Governor of Connecticut from 1844 to 1846 and a United States senator from 1847 to 1851. As a lawyer, his career was most notable ...

, see the Amistad case below.
The elder Gibbs was generally known to his family and colleagues as "Josiah", while the son was called "Willard". Josiah Gibbs was a linguist and theologian who served as professor of sacred literature at Yale Divinity School
Yale Divinity School (YDS) is one of the twelve graduate and professional schools of Yale University in New Haven, Connecticut.
Congregationalist theological education was the motivation at the founding of Yale, and the professional school has ...

from 1824 until his death in 1861. He is chiefly remembered today as the abolitionist
Abolitionism, or the abolitionist movement, is the movement to end slavery. In Western Europe and the Americas, abolitionism was a historic movement that sought to end the Atlantic slave trade and liberate the enslaved people.
The British ...

who found an interpreter for the African passengers of the ship '' Amistad'', allowing them to testify during the trial
''The Trial'' (german: Der Process, link=no, previously , and ) is a novel written by Franz Kafka in 1914 and 1915 and published posthumously on 26 April 1925. One of his best known works, it tells the story of Josef K., a man arrested and pr ...

that followed their rebellion against being sold as slaves.
Education

Willard Gibbs was educated at theHopkins School
Hopkins School is a private, college-preparatory, coeducational, day school for grades 7–12 located in New Haven, Connecticut.
In 1660, Edward Hopkins, seven-time governor of the Connecticut Colony, bequeathed a portion of his estate to found s ...

and entered Yale College
Yale College is the undergraduate college of Yale University. Founded in 1701, it is the original school of the university. Although other Yale schools were founded as early as 1810, all of Yale was officially known as Yale College until 1887, ...

in 1854 at the age of 15. At Yale, Gibbs received prizes for excellence in mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...

, and he graduated in 1858, near the top of his class. He remained at Yale as a graduate student at the Sheffield Scientific School
Sheffield Scientific School was founded in 1847 as a school of Yale College in New Haven, Connecticut, for instruction in science and engineering. Originally named the Yale Scientific School, it was renamed in 1861 in honor of Joseph E. Sheffield, ...

. At age 19, soon after his graduation from college, Gibbs was inducted into the Connecticut Academy of Arts and Sciences
The Connecticut Academy of Arts and Sciences is a learned society founded in 1799 in New Haven, Connecticut "to cultivate every art and science which may tend to advance the interest and happiness of a free and virtuous people." Its purpose is the ...

, a scholarly institution composed primarily of members of the Yale faculty.Rukeyser 1988, p. 104
Relatively few documents from the period survive and it is difficult to reconstruct the details of Gibbs's early career with precision.Wheeler 1998, pp. 23–24 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
Prof Hubert Anson Newton FRS HFRSE (19 March 1830 – 12 August 1896), usually cited as H. A. Newton, was an American astronomer and mathematician, noted for his research on meteors.
Biography
Newton was born at Sherburne, New York, and gradu ...

, a leading authority on meteors
A meteoroid () is a small rocky or metallic body in outer space.
Meteoroids are defined as objects significantly smaller than asteroids, ranging in size from grains to objects up to a meter wide. Objects smaller than this are classified as mic ...

, who remained Gibbs's lifelong friend and confidant. After the death of his father in 1861, Gibbs inherited enough money to make him financially independent.Rukeyser 1998, pp. 120, 142
Recurrent pulmonary
The lungs are the primary organs of the respiratory system in humans and most other animals, including some snails and a small number of fish. In mammals and most other vertebrates, two lungs are located near the backbone on either side of th ...

trouble ailed the young Gibbs and his physicians were concerned that he might be susceptible to tuberculosis
Tuberculosis (TB) is an infectious disease usually caused by '' Mycobacterium tuberculosis'' (MTB) bacteria. Tuberculosis generally affects the lungs, but it can also affect other parts of the body. Most infections show no symptoms, in ...

, which had killed his mother. He also suffered from astigmatism
Astigmatism is a type of refractive error due to rotational asymmetry in the eye's refractive power. This results in distorted or blurred vision at any distance. Other symptoms can include eyestrain, headaches, and trouble driving at nig ...

, whose treatment was then still largely unfamiliar to oculists
Ophthalmology ( ) is a surgical subspecialty within medicine that deals with the diagnosis and treatment of eye disorders.
An ophthalmologist is a physician who undergoes subspecialty training in medical and surgical eye care. Following a medic ...

, so that Gibbs had to diagnose himself and grind his own lenses.Wheeler 1998, pp. 29–31Rukeyser 1988, p. 143 Though in later years he used glasses
Glasses, also known as eyeglasses or spectacles, are vision eyewear, with lenses (clear or tinted) mounted in a frame that holds them in front of a person's eyes, typically utilizing a bridge over the nose and hinged arms (known as temples or ...

only for reading or other close work, Gibbs's delicate health and imperfect eyesight probably explain why he did not volunteer to fight in the Civil War
A civil war or intrastate war is a war between organized groups within the same state (or country).
The aim of one side may be to take control of the country or a region, to achieve independence for a region, or to change government policies ...

of 1861–65.Wheeler 1998, p. 30 He was not conscripted
Conscription (also called the draft in the United States) is the state-mandated enlistment of people in a national service, mainly a military service. Conscription dates back to antiquity and it continues in some countries to the present day und ...

and he remained at Yale for the duration of the war.Rukeyser 1998, p. 134
In 1863, Gibbs received the first Doctorate of Philosophy
A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...

(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
A gear is a rotating circular machine part having cut teeth or, in the case of a cogwheel or gearwheel, inserted teeth (called ''cogs''), which mesh with another (compatible) toothed part to transmit (convert) torque and speed. The basic pr ...

s.Wheeler 1998, p. 32 In 1861, Yale had become the first US university to offer a PhD degree and Gibbs's was only the fifth PhD granted in the US in any subject.
Career, 1863–73

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). In 1866, he patented a design for arailway brake
A railway brake is a type of brake used on the cars of railway trains to enable deceleration, control acceleration (downhill) or to keep them immobile when parked. While the basic principle is similar to that on road vehicle usage, operational f ...

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.Wheeler 1998, appendix II
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
Sorbonne may refer to:
* Sorbonne (building), historic building in Paris, which housed the University of Paris and is now shared among multiple universities.
*the University of Paris (c. 1150 – 1970)
*one of its components or linked institution, ...

and the Collège de France
The Collège de France (), formerly known as the ''Collège Royal'' or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment (''grand établissement'') in France. It is located in Paris ne ...

, given by such distinguished mathematical scientists as Joseph Liouville
Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work
He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...

and Michel Chasles
Michel Floréal Chasles (; 15 November 1793 – 18 December 1880) was a French mathematician.
Biography
He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth Coal ...

. Having undertaken a punishing regimen of study, Gibbs caught a serious cold and a doctor, fearing tuberculosis, advised him to rest on the Riviera
''Riviera'' () is an Italian word which means "coastline", ultimately derived from Latin , through Ligurian . It came to be applied as a proper name to the coast of Liguria, in the form ''Riviera ligure'', then shortened in English. The two areas ...

, where he and his sisters spent several months and where he made a full recovery.
Moving to Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...

, Gibbs attended the lectures taught by mathematicians Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...

and Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...

, as well as by chemist Heinrich Gustav Magnus
Heinrich Gustav Magnus (; 2 May 1802 – 4 April 1870) was a notable German experimental scientist. His training was mostly in chemistry but his later research was mostly in physics. He spent the great bulk of his career at the University of Ber ...

. 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. In Heidelberg
Heidelberg (; Palatine German language, Palatine German: ''Heidlberg'') is a city in the States of Germany, German state of Baden-Württemberg, situated on the river Neckar in south-west Germany. As of the 2016 census, its population was 159,914 ...

, Gibbs was exposed to the work of physicists Gustav Kirchhoff
Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects.
He coine ...

and Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...

, and chemist Robert Bunsen
Robert Wilhelm Eberhard Bunsen (;
30 March 1811
– 16 August 1899) was a German chemist. He investigated emission spectra of heated elements, and discovered caesium (in 1860) and rubidium (in 1861) with the physicist Gustav Kirchhoff. The Bu ...

. At the time, German academics were the leading authorities in the natural sciences, especially chemistry and thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...

.
Gibbs returned to Yale in June 1869 and briefly taught French to engineering students. It was probably also around this time that he worked on a new design for a steam-engine governor
A governor is an administrative leader and head of a polity or political region, ranking under the head of state and in some cases, such as governors-general, as the head of state's official representative. Depending on the type of political ...

, his last significant investigation in mechanical engineering. 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.Rukeyser 1988, pp. 181–182
Career, 1873–80

Gibbs published his first work in 1873. 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. Although the journal had few readers capable of understanding Gibbs's work, he shared reprints with correspondents in Europe and received an enthusiastic response fromJames Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...

at Cambridge
Cambridge ( ) is a university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge bec ...

. 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.
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
The Chemical Society was a scientific society formed in 1841 (then named the Chemical Society of London) by 77 scientists as a result of increased interest in scientific matters. Chemist Robert Warington was the driving force behind its creation.
...

of London and even referred to it in the article on "Diagrams" that he wrote for the ''Encyclopædia Britannica
The (Latin for "British Encyclopædia") is a general knowledge English-language encyclopaedia. It is published by Encyclopædia Britannica, Inc.; the company has existed since the 18th century, although it has changed ownership various time ...

''. 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."
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
In the history of thermodynamics, ''On the Equilibrium of Heterogeneous Substances'' is a 300-page paper written by American chemical physicist Willard Gibbs. It is one of the founding papers in thermodynamics, along with German physicist Hermann ...

", 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,Cropper 2001, p. 109 begins with a quotation from Rudolf Clausius
Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Sadi Carnot's principle ...

that expresses what would later be called the first and second laws of thermodynamics
The laws of thermodynamics are a set of scientific laws which define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium. The laws also use various paramet ...

: "The energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...

of the world is constant. The entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...

of the world tends towards a maximum."
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.Wheeler 1998, ch. V The work has been described as "the '' Principia'' of thermodynamics" and as a work of "practically unlimited scope". It solidly laid the foundation for physical Chemistry. Wilhelm Ostwald
Friedrich Wilhelm Ostwald (; 4 April 1932) was a Baltic German chemist and German philosophy, philosopher. Ostwald is credited with being one of the founders of the field of physical chemistry, with Jacobus Henricus van 't Hoff, Walther Nernst, ...

, who translated Gibbs's monograph into German, referred to Gibbs as the "founder of chemical energetics". According to modern commentators,
Gibbs continued to work without pay until 1880, when the new Johns Hopkins University
Johns Hopkins University (Johns Hopkins, Hopkins, or JHU) is a private university, private research university in Baltimore, Maryland. Founded in 1876, Johns Hopkins is the oldest research university in the United States and in the western hem ...

in Baltimore, Maryland
Baltimore ( , locally: or ) is the List of municipalities in Maryland, most populous city in the U.S. state of Maryland, fourth most populous city in the Mid-Atlantic (United States), Mid-Atlantic, and List of United States cities by popula ...

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.
Career, 1880–1903

From 1880 to 1884, Gibbs worked on developing theexterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...

of Hermann Grassmann
Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...

into a vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...

well-suited to the needs of physicists. With this object in mind, Gibbs distinguished between the dot and cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...

s of two vectors and introduced the concept of dyadics In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.
There are numerous ways to multiply two Euclidean vectors. The dot product takes in two v ...

. Similar work was carried out independently, and at around the same time, by the British mathematical physicist and engineer Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...

. Gibbs sought to convince other physicists of the convenience of the vectorial approach over the quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

ic calculus of William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...

, which was then widely used by British scientists. This led him, in the early 1890s, to a controversy with Peter Guthrie Tait
Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook '' Treatise on Natural Philosophy'', which he co-wrote wi ...

and others in the pages of ''Nature
Nature, in the broadest sense, is the physics, physical world or universe. "Nature" can refer to the phenomenon, phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. ...

''.
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
Edwin Bidwell Wilson (April 25, 1879 – December 28, 1964) was an American mathematician, statistician, physicist and general polymath. He was the sole protégé of Yale University physicist Josiah Willard Gibbs and was mentor to MIT economist ...

into a textbook, ''Vector Analysis
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...

'', published in 1901. That book helped to popularize the "del
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...

" notation that is widely used today in electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

and fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...

. In other mathematical work, he re-discovered the "Gibbs phenomenon
In mathematics, the Gibbs phenomenon, discovered by Available on-line at:National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. and rediscovered by , is the oscillatory behavior of the Fourier series of a piecewise continuousl ...

" in the theory of Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...

(which, unbeknownst to him and to later scholars, had been described fifty years before by an obscure English mathematician, Henry Wilbraham).
From 1882 to 1889, Gibbs wrote five papers on physical optics
In physics, physical optics, or wave optics, is the branch of optics that studies interference, diffraction, polarization, and other phenomena for which the ray approximation of geometric optics is not valid. This usage tends not to include effec ...

, in which he investigated birefringence
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefring ...

and other optical phenomena and defended Maxwell's electromagnetic theory of light against the mechanical theories of Lord Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...

and others. In his work on optics, just as much as in his work on thermodynamics, 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.
Gibbs coined the term ''statistical mechanics'' and introduced key concepts in the corresponding mathematical description of physical systems, including the notions of chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...

(1876), and statistical ensemble
In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...

(1902). 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.
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."Wilson 1931 Gibbs did supervise the doctoral thesis on mathematical economics written by Irving Fisher
Irving Fisher (February 27, 1867 – April 29, 1947) was an American economist, statistician, inventor, eugenicist and progressive social campaigner. He was one of the earliest American neoclassical economists, though his later work on debt def ...

in 1891. After Gibbs's death, Fisher financed the publication of his ''Collected Works''. Another distinguished student was Lee De Forest
Lee de Forest (August 26, 1873 – June 30, 1961) was an American inventor and a fundamentally important early pioneer in electronics. He invented the first electronic device for controlling current flow; the three-element "Audion" triode va ...

, later a pioneer of radio technology.
Gibbs died in New Haven on April 28, 1903, at the age of 64, the victim of an acute intestinal obstruction. A funeral was conducted two days later at his home on 121 High Street, and his body was buried in the nearby Grove Street Cemetery
Grove Street Cemetery or Grove Street Burial Ground is a cemetery in New Haven, Connecticut, that is surrounded by the Yale University campus. It was organized in 1796 as the New Haven Burying Ground and incorporated in October 1797 to replace the ...

. In May, Yale organized a memorial meeting at the Sloane Laboratory. The eminent British physicist J. J. Thomson
Sir Joseph John Thomson (18 December 1856 – 30 August 1940) was a British physicist and Nobel Laureate in Physics, credited with the discovery of the electron, the first subatomic particle to be discovered.
In 1897, Thomson showed that ...

was in attendance and delivered a brief address.
Personal life and character

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 theAdirondacks
The Adirondack Mountains (; a-də-RÄN-dak) form a massif in northeastern New York with boundaries that correspond roughly to those of Adirondack Park. They cover about 5,000 square miles (13,000 km2). The mountains form a roughly circular d ...

(at Keene Valley, New York
Keene is a town in central Essex County, New York, United States. It includes the hamlets of Keene, Keene Valley, and St. Huberts, with a total population of 1,144 as of the 2020 census
The town is part of the Adirondack Park, and includes 15 ...

) and later at the White Mountains (in Intervale, New Hampshire
Intervale is an unincorporated community in Carroll County, New Hampshire, located on the boundary between the towns of Bartlett and Conway in the White Mountains Region. The village is part of the Mount Washington Valley, a resort area that als ...

),Seeger 1974, pp. 15–16 his sojourn in Europe in 1866–69 was almost the only time that Gibbs spent outside New Haven. He joined Yale's College Church (a Congregational church
Congregational churches (also Congregationalist churches or Congregationalism) are Protestant churches in the Calvinist tradition practising congregationalist church governance, in which each congregation independently and autonomously runs its ...

) at the end of his freshman year and remained a regular attendant for the rest of his life.Wheeler, 1998, p. 16 Gibbs generally voted for the Republican
Republican can refer to:
Political ideology
* An advocate of a republic, a type of government that is not a monarchy or dictatorship, and is usually associated with the rule of law.
** Republicanism, the ideology in support of republics or agains ...

candidate in presidential elections but, like other "Mugwump
The Mugwumps were Republican political activists in the United States who were intensely opposed to political corruption. They were never formally organized. Typically they switched parties from the Republican Party by supporting Democratic ...

s", his concern over the growing corruption associated with machine politics
In the politics of representative democracies, a political machine is a party organization that recruits its members by the use of tangible incentives (such as money or political jobs) and that is characterized by a high degree of leadership co ...

led him to support Grover Cleveland
Stephen Grover Cleveland (March 18, 1837June 24, 1908) was an American lawyer and politician who served as the 22nd and 24th president of the United States from 1885 to 1889 and from 1893 to 1897. Cleveland is the only president in American ...

, a conservative Democrat
Democrat, Democrats, or Democratic may refer to:
Politics
*A proponent of democracy, or democratic government; a form of government involving rule by the people.
*A member of a Democratic Party:
**Democratic Party (United States) (D)
**Democratic ...

, in the election of 1884. Little else is known of his religious or political views, which he mostly kept to himself.
Gibbs did not produce a substantial personal correspondence and many of his letters were later lost or destroyed. Beyond the technical writings concerning his research, he published only two other pieces: a brief obituary for Rudolf Clausius
Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Sadi Carnot's principle ...

, one of the founders of the mathematical theory of thermodynamics, and a longer biographical memoir of his mentor at Yale, H. A. Newton. In Edward Bidwell Wilson's view,
According to Lynde Wheeler, who had been Gibbs's student at Yale, in his later years Gibbs
He was a careful investor and financial manager, and at his death in 1903 his estate was valued at $100,000 (roughly $ today). For many years, he served as trustee, secretary, and treasurer of his alma mater, the Hopkins School.Wheeler, 1998, p. 144 US President Chester A. Arthur
Chester Alan Arthur (October 5, 1829 – November 18, 1886) was an American lawyer and politician who served as the 21st president of the United States from 1881 to 1885. He previously served as the 20th vice president under President James ...

appointed him as one of the commissioners to the National Conference of Electricians, which convened in Philadelphia
Philadelphia, often called Philly, is the largest city in the Commonwealth of Pennsylvania, the sixth-largest city in the U.S., the second-largest city in both the Northeast megalopolis and Mid-Atlantic regions after New York City. Sinc ...

in September 1884, and Gibbs presided over one of its sessions. A keen and skilled horseman, Gibbs was seen habitually in New Haven driving his sister's carriage
A carriage is a private four-wheeled vehicle for people and is most commonly horse-drawn. Second-hand private carriages were common public transport, the equivalent of modern cars used as taxis. Carriage suspensions are by leather strapping an ...

. In an obituary published in the '' American Journal of Science'', Gibbs's former student Henry A. Bumstead referred to Gibbs's personal character:
Major scientific contributions

Chemical and electrochemical thermodynamics

Gibbs's papers from the 1870s introduced the idea of expressing the internal energy ''U'' of a system in terms of theentropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...

''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
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...

$\backslash mu$ 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
The laws of thermodynamics are a set of scientific laws which define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium. The laws also use various paramet ...

by expressing the infinitesimal change in the internal energy, d''U'', of a closed system
A closed system is a natural physical system that does not allow transfer of matter in or out of the system, although — in contexts such as physics, chemistry or engineering — the transfer of energy (''e.g.'' as work or heat) is allowed.
In ...

in the form:
:$\backslash mathrmU\; =\; T\backslash mathrmS\; -\; p\; \backslash ,\backslash mathrmV\; +\; \backslash sum\_i\; \backslash mu\_i\; \backslash ,\backslash mathrm\; N\_i\backslash ,$
where ''T'' is the absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...

, ''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''Legendre transform
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...

of this expression, he defined the concepts of enthalpy
Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...

, ''H'' and Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pr ...

, ''G''.
:$G\_\; =\; H\; -\; TS$
This compares to the expression for Helmholtz free energy
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal
In thermodynamics, an isotherma ...

, ''A''.
:$A\_\; =\; U-TS\backslash ,$
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
The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...

is simply related to the free energy change when the reactants are in their standard state
In chemistry, the standard state of a material (pure substance, mixture or solution) is a reference point used to calculate its properties under different conditions. A superscript circle ° (degree symbol) or a Plimsoll (⦵) character is use ...

s.
:$\backslash Delta\; G^\backslash ominus=-RT\; \backslash ln\; K^\backslash ominus$
Chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...

is usually defined as partial molar Gibbs free energy.
:$\backslash mu\_i=\backslash left(\backslash frac\backslash right)\_$
Gibbs also obtained what later came to be known as the "Gibbs–Duhem equation In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system:
:\sum_^I N_i \mathrm\mu_i = - S \mathrmT + V \mathrmp
where N_i is the number of moles of com ...

".
In an electrochemical reaction
Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential difference, as a measurable and quantitative phenomenon, and identifiable chemical change, with the potential difference as an outco ...

characterized by an electromotive force
In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical ''transd ...

ℰ and an amount of transferred charge Q, Gibbs's starting equation becomes
:$\backslash mathrmU\; =\; T\backslash mathrmS\; -\; p\; \backslash ,\backslash mathrmV\; +\; \backslash mathcal\backslash mathrmQ$.
The publication of the paper "On the Equilibrium of Heterogeneous Substances
In the history of thermodynamics, ''On the Equilibrium of Heterogeneous Substances'' is a 300-page paper written by American chemical physicist Willard Gibbs. It is one of the founding papers in thermodynamics, along with German physicist Hermann ...

" (1874–78) is now regarded as a landmark in the development of chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...

. In it, Gibbs developed a rigorous mathematical theory for various transport phenomena
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechan ...

, including adsorption
Adsorption is the adhesion of atoms, ions or molecules from a gas, liquid or dissolved solid to a surface. This process creates a film of the ''adsorbate'' on the surface of the ''adsorbent''. This process differs from absorption, in which a f ...

, electrochemistry
Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential difference, as a measurable and quantitative phenomenon, and identifiable chemical change, with the potential difference as an outco ...

, and the Marangoni effect
The Marangoni effect (also called the Gibbs–Marangoni effect) is the mass transfer along an interface between two phases due to a gradient of the surface tension. In the case of temperature dependence, this phenomenon may be called thermo-capill ...

in fluid mixtures. He also formulated the phase rule
In thermodynamics, the phase rule is a general principle governing "pVT" systems, whose thermodynamic states are completely described by the variables pressure (), volume () and temperature (), in thermodynamic equilibrium. If is the number of d ...

:$F\backslash ;=\backslash ;C\backslash ;-\backslash ;P\backslash ;+\backslash ;2$
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.
Statistical mechanics

Together with James Clerk Maxwell andLudwig Boltzmann
Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...

, 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". He used the concept to define the microcanonical, canonical
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical example ...

, and grand canonical ensemble
In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium ...

s; all related to the Gibbs measure In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems.
Th ...

, thus obtaining a more general formulation of the statistical properties of many-particle systems than Maxwell and Boltzmann had achieved before him.
Gibbs generalized Boltzmann's statistical interpretation of entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...

$S$ by defining the entropy of an arbitrary ensemble as
: $S\; =\; -k\_\backslash text\backslash ,\backslash sum\_i\; p\_i\; \backslash ln\; \backslash ,p\_i$,
where $k\_\backslash text$ is the Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...

, while the sum is over all possible microstates
A microstate or ministate is a sovereign state having a very small population or very small land area, usually both. However, the meanings of "state" and "very small" are not well-defined in international law.Warrington, E. (1994). "Lilliputs ...

$i$, with $p\_i$ the corresponding probability of the microstate (see Gibbs entropy formula
The concept entropy was first developed by German physicist Rudolf Clausius in the mid-nineteenth century as a thermodynamic property that predicts that certain spontaneous processes are irreversible or impossible. In statistical mechanics, entropy ...

). This same formula would later play a central role in Claude Shannon
Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American people, American mathematician, electrical engineering, electrical engineer, and cryptography, cryptographer known as a "father of information theory".
As a 21-year-o ...

's information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...

and is therefore often seen as the basis of the modern information-theoretical interpretation of thermodynamics.
According to Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

, writing in 1904, even though Maxwell and Boltzmann had previously explained the irreversibility
In science, a process that is not reversible is called irreversible. This concept arises frequently in thermodynamics. All complex natural processes are irreversible, although a phase transition at the coexistence temperature (e.g. melting of i ...

of macroscopic physical processes in probabilistic terms, "the one who has seen it most clearly, in a book too little read because it is a little difficult to read, is Gibbs, in his ''Elementary Principles of Statistical Mechanics''." Gibbs's analysis of irreversibility, and his formulation of Boltzmann's H-theorem
In classical statistical mechanics, the ''H''-theorem, introduced by Ludwig Boltzmann in 1872, describes the tendency to decrease in the quantity ''H'' (defined below) in a nearly-ideal gas of molecules.
L. Boltzmann,Weitere Studien über das Wä ...

and of the ergodic hypothesis, were major influences on the mathematical physics of the 20th century.
Gibbs was well aware that the application of the equipartition theorem
In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...

to large systems of classical particles failed to explain the measurements of the specific heats
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...

of both solids and gases, and he argued that this was evidence of the danger of basing thermodynamics on "hypotheses about the constitution of matter". Gibbs's own framework for statistical mechanics, based on ensembles of macroscopically indistinguishable microstates
A microstate or ministate is a sovereign state having a very small population or very small land area, usually both. However, the meanings of "state" and "very small" are not well-defined in international law.Warrington, E. (1994). "Lilliputs ...

, could be carried over almost intact after the discovery that the microscopic laws of nature obey quantum rules, rather than the classical laws known to Gibbs and to his contemporaries. His resolution of the so-called "Gibbs paradox
In statistical mechanics, a semi-classical derivation of entropy that does not take into account the indistinguishability of particles yields an expression for entropy which is not extensive (is not proportional to the amount of substance in qu ...

", about the entropy of the mixing of gases, is now often cited as a prefiguration of the indistinguishability of particles required by quantum physics.
Vector analysis

British scientists, including Maxwell, had relied on Hamilton'squaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

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
''Elements of Dynamic'' is a book published by William Kingdon Clifford in 1878. In 1887 it was supplemented by a fourth part and an appendix. The subtitle is "An introduction to motion and rest in solid and fluid bodies". It was reviewed positiv ...

'' (1888), Gibbs noted that the product of quaternions could be separated into two parts: a one-dimensional (scalar) quantity and a three-dimensional vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...

, 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
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...

'' prepared by E. B. Wilson from Gibbs notes, he was largely responsible for the development of the vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...

techniques still used today in electrodynamics and fluid mechanics.
While he was working on vector analysis in the late 1870s, Gibbs discovered that his approach was similar to the one that Grassmann had taken in his "multiple algebra".Letter by Gibbs to Victor Schlegel
Victor Schlegel (4 March 1843 – 22 November 1905) was a German mathematician. He is remembered for promoting the geometric algebra of Hermann Grassmann and for a method of visualizing polytopes called Schlegel diagrams.
In the nineteenth centur ...

, quoted in Wheeler 1998, pp. 107–109 Gibbs then sought to publicize Grassmann's work, stressing that it was both more general and historically prior to Hamilton's quaternionic algebra. To establish 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
Tides are the rise and fall of sea levels caused by the combined effects of the gravity, gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another.
Tide t ...

s that Grassmann had submitted in 1840 to the faculty at the University of Berlin
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative o ...

, in which he had first introduced the notion of what would later be called a vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

(linear space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

).Wheeler 1998, pp. 113–116
As Gibbs had advocated in the 1880s and 1890s, quaternions were eventually all but abandoned by physicists in favor of the vectorial approach developed by him and, independently, by Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...

. Gibbs applied his vector methods to the determination of planetary and comet orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...

s. He also developed the concept of mutually reciprocal triads of vectors that later proved to be of importance in crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...

.
Physical optics

Though Gibbs's research on physical optics is less well known today than his other work, it made a significant contribution to classicalelectromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

by applying Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...

to the theory of optical processes such as birefringence
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefring ...

, dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...

, and optical activity
Optical rotation, also known as polarization rotation or circular birefringence, is the rotation of the orientation of the plane of polarization about the optical axis of linearly polarized light as it travels through certain materials. Circul ...

.Wheeler 1998, ch. VIII 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
Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated Transmission medium, medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty ...

). Gibbs also stressed that the absence of a longitudinal
Longitudinal is a geometric term of location which may refer to:
* Longitude
** Line of longitude, also called a meridian
* Longitudinal engine, an internal combustion engine in which the crankshaft is oriented along the long axis of the vehicl ...

electromagnetic wave, which is needed to account for the observed properties of light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...

, is automatically guaranteed by Maxwell's equations (by virtue of what is now called their "gauge invariance
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie group ...

"), whereas in mechanical theories of light, such as Lord Kelvin's, it must be imposed as an ''ad hoc'' condition on the properties of the aether.
In his last paper on physical optics, Gibbs concluded that
Shortly afterwards, the electromagnetic nature of light was demonstrated by the experiments of Heinrich Hertz
Heinrich Rudolf Hertz ( ; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's Maxwell's equations, equations of electrom ...

in Germany.
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. 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, bothElias Loomis
Elias Loomis (August 7, 1811 – August 15, 1889) was an American mathematician. He served as a professor of mathematics and natural philosophy at Case Western Reserve University, Western Reserve College (now Case Western Reserve University), the ...

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.Rukeyser 1998, pp. 225–226
Even though it had been immediately embraced by Maxwell, Gibbs's graphical formulation of the laws of thermodynamics only came into widespread use in the mid 20th century, with the work of László Tisza
László Tisza (July 7, 1907 – April 15, 2009) was a Hungarian-born American physicist who was Professor of Physics Emeritus at MIT. He was a colleague of famed physicists Edward Teller, Lev Landau and Fritz London, and initiated the two-flui ...

and Herbert Callen
Herbert Bernard Callen (July 1, 1919 – May 22, 1993) was an American physicist specializing in thermodynamics and statistical mechanics. He is considered one of the founders of the modern theory of irreversible thermodynamics, and is the author ...

. According to James Gerald Crowther,
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
The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...

in 1879 and received the 1880 Rumford Prize
Founded in 1796, the Rumford Prize, awarded by the American Academy of Arts and Sciences, is one of the oldest scientific prizes in the United States. The prize recognizes contributions by scientists to the fields of heat and light. These terms ...

from the American Academy of Arts and Sciences
The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and ...

for his work on chemical thermodynamics. He was also awarded honorary doctorates by Princeton University and Williams College
Williams College is a Private college, private liberal arts colleges in the United States, liberal arts college in Williamstown, Massachusetts. It was established as a men's college in 1793 with funds from the estate of Ephraim Williams, a col ...

.
In Europe, Gibbs was inducted as honorary member of the London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...

in 1892 and elected Foreign Member of the Royal Society in 1897. He was elected as corresponding member of the Prussian
Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an e ...

and French Academies of Science and received honorary doctorates from the universities of Dublin
Dublin (; , or ) is the capital and largest city of Republic of Ireland, Ireland. On a bay at the mouth of the River Liffey, it is in the Provinces of Ireland, province of Leinster, bordered on the south by the Dublin Mountains, a part of th ...

, Erlangen
Erlangen (; East Franconian German, East Franconian: ''Erlang'', Bavarian language, Bavarian: ''Erlanga'') is a Middle Franconian city in Bavaria, Germany. It is the seat of the administrative district Erlangen-Höchstadt (former administrative d ...

, and Christiania (now Oslo). The Royal Society further honored Gibbs in 1901 with the Copley Medal
The Copley Medal is an award given by the Royal Society, for "outstanding achievements in research in any branch of science". It alternates between the physical sciences or mathematics and the biological sciences. Given every year, the medal is t ...

, then regarded as the highest international award in the natural sciences, 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." Gibbs, who remained in New Haven, was represented at the award ceremony by Commander Richardson Clover
Richardson Clover (July 11, 1846 – October 14, 1919) was an officer of the United States Navy. An 1867 graduate of the United States Naval Academy, he was a noted hydrographer, served as Director of Naval Intelligence, and commanded the gunb ...

, the US naval attaché in London.
In his autobiography, mathematician Gian-Carlo Rota
Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, pro ...

tells of casually browsing the mathematical stacks of Sterling Library
Sterling Memorial Library (SML) is the main library building of the Yale University Library system in New Haven, Connecticut, United States. Opened in 1931, the library was designed by James Gamble Rogers as the centerpiece of Yale's Gothic Re ...

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
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

, and Ernst Mach
Ernst Waldfried Josef Wenzel Mach ( , ; 18 February 1838 – 19 February 1916) was a Moravian-born Austrian physicist and philosopher, who contributed to the physics of shock waves. The ratio of one's speed to that of sound is named the Mach ...

. From this, Rota concluded that Gibbs's work was better known among the scientific elite of his day than the published material suggests. Lynde Wheeler reproduces that mailing list in an appendix to his biography of Gibbs.Wheeler 1998, appendix IV 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
Friedrich Wilhelm Ostwald (; 4 April 1932) was a Baltic German chemist and German philosophy, philosopher. Ostwald is credited with being one of the founders of the field of physical chemistry, with Jacobus Henricus van 't Hoff, Walther Nernst, ...

in 1892 and into French by Henri Louis Le Châtelier in 1899.
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 H. W. Bakhuis Roozeboom, who showed how to apply it in a variety of situations, thereby assuring it of widespread use. In industrial chemistry, Gibbs's thermodynamics found many applications during the early 20th century, from electrochemistry to the development of theHaber process
The Haber process, also called the Haber–Bosch process, is an artificial nitrogen fixation process and is the main industrial procedure for the production of ammonia today. It is named after its inventors, the German chemists Fritz Haber and C ...

for the synthesis of ammonia
Ammonia is an inorganic compound of nitrogen and hydrogen with the formula . A stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a distinct pungent smell. Biologically, it is a common nitrogenous was ...

.
When Dutch physicist J. D. van der Waals
Johannes Diderik van der Waals (; 23 November 1837 – 8 March 1923) was a Dutch theoretical physics, theoretical physicist and thermodynamicist famous for his pioneering work on the equation of state for gases and liquids. Van der Waals starte ...

received the 1910 Nobel Prize
The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...

"for his work on the equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal ...

for gases and liquids" he acknowledged the great influence of Gibbs's work on that subject. Max Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial contributions to theoretical p ...

received the 1918 Nobel Prize for his work on quantum mechanics, particularly his 1900 paper on Planck's law
In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the body and its environment.
At ...

for quantized black-body radiation
Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spect ...

. 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."
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
Chemical thermodynamics is the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. Chemical thermodynamics involves not only laboratory measurement ...

, 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
Merle Randall (January 29, 1888 – March 17, 1950) was an American physical chemist famous for his work with Gilbert N. Lewis, over a period of 25 years, in measuring reaction heat of chemical compounds and determining their corresponding free ...

, and ''Modern Thermodynamics by the Methods of Willard Gibbs'' (1933), by Edward A. Guggenheim
Edward Armand Guggenheim FRS (11 August 1901 in Manchester – 9 August 1970) was an English physical chemist, noted for his contributions to thermodynamics.
Life
Guggenheim was born in Manchester 11 August 1901, the son of Armand Guggenheim and ...

.
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. According to mathematical physicist Arthur Wightman
Arthur Strong Wightman (March 30, 1922 – January 13, 2013) was an American mathematical physicist. He was one of the founders of the axiomatic approach to quantum field theory, and originated the set of Wightman axioms. With his rigorous treatm ...

,
Initially unaware of Gibbs's contributions in that field, Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

wrote three papers on statistical mechanics, published between 1902 and 1904. After reading Gibbs's textbook (which was translated into German by Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...

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.
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
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Convex sets
A subset C \subseteq X of s ...

", including ideas that, according to Barry Simon
Barry Martin Simon (born 16 April 1946) is an American mathematical physicist and was the IBM professor of Mathematics and Theoretical Physics at Caltech, known for his prolific contributions in spectral theory, functional analysis, and no ...

, "lay dormant for about seventy-five years". Important mathematical concepts based on Gibbs's work on thermodynamics and statistical mechanics include the Gibbs lemma 200px, Josiah Willard Gibbs
In game theory and in particular the study of Blotto games and operational research, the Gibbs lemma is a result that is useful in maximization problems. It is named for Josiah Willard Gibbs.
Consider \phi=\sum_^n f_i( ...

in game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

, the Gibbs inequality in information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...

, as well as Gibbs sampling
In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is dif ...

in computational statistics
Computational statistics, or statistical computing, is the bond between statistics and computer science. It means statistical methods that are enabled by using computational methods. It is the area of computational science (or scientific computi ...

.
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
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...

'', 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.
At Yale, Gibbs was also mentor to Lee De Forest, who went on to invent the triode
A triode is an electronic amplifying vacuum tube (or ''valve'' in British English) consisting of three electrodes inside an evacuated glass envelope: a heated filament or cathode, a grid, and a plate (anode). Developed from Lee De Forest's 19 ...

amplifier and has been called the "father of radio".Seeger 1974, p. 18 De Forest credited Gibbs's influence for the realization "that the leaders in electrical development would be those who pursued the higher theory of waves and oscillations and the transmission by these means of intelligence and power." Another student of Gibbs who played a significant role in the development of radio technology was Lynde Wheeler.
Gibbs also had an indirect influence on mathematical economics. He supervised the thesis of Irving Fisher
Irving Fisher (February 27, 1867 – April 29, 1947) was an American economist, statistician, inventor, eugenicist and progressive social campaigner. He was one of the earliest American neoclassical economists, though his later work on debt def ...

, 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
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...

of markets, and he used Gibbs's vectorial notation. Gibbs's protégé Edwin Bidwell Wilson became, in turn, a mentor to leading American economist and Nobel Laureate Paul Samuelson
Paul Anthony Samuelson (May 15, 1915 – December 13, 2009) was an American economist who was the first American to win the Nobel Memorial Prize in Economic Sciences. When awarding the prize in 1970, the Swedish Royal Academies stated that he "h ...

. In 1947, Samuelson published ''Foundations of Economic Analysis
''Foundations of Economic Analysis'' is a book by Paul A. Samuelson published in 1947 (Enlarged ed., 1983) by Harvard University Press. It is based on Samuelson's 1941 doctoral dissertation at Harvard University. The book sought to demonstrate a ...

'', based on his doctoral dissertation, in which he used as epigraph a remark attributed to Gibbs: "Mathematics is a language." Samuelson later explained that in his understanding of prices his "debts were not primarily to Pareto or Slutsky Slutsky is a Belarusian language, Belarusian, Russian language, Russian, and Ashkenazi Jewish surnames, Ashkenazi Jewish surname that derived from Slutsk in Belarus. The Jewish ethnicity, Jewish family name 'Slutsky' is an Ashkenazized form of ( he, ...

, but to the great thermodynamicist, Willard Gibbs of Yale."
Mathematician Norbert Wiener
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...

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
Cybernetics is a wide-ranging field concerned with circular causality, such as feedback, in regulatory and purposive systems. Cybernetics is named after an example of circular causal feedback, that of steering a ship, where the helmsperson m ...

. 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."
Commemoration

When the German physical chemist Walther Nernst visited Yale in 1906 to give theSilliman lecture
The Silliman Memorial lectures series has been published by Yale University since 1901. The lectures were established by the university on the foundation of a bequest of $80,000, left in 1883 by Augustus Ely Silliman, in memory of his mother, Mrs. ...

, 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
Lee Oscar Lawrie (October 16, 1877 – January 23, 1963) was an American architectural sculptor and a key figure in the American art scene preceding World War II. Over his long career of more than 300 commissions Lawrie's style evolved through ...

, installed in the Sloane Physics Laboratory.Seeger 1974, p. 21 In 1910, the American Chemical Society
The American Chemical Society (ACS) is a scientific society based in the United States that supports scientific inquiry in the field of chemistry. Founded in 1876 at New York University, the ACS currently has more than 155,000 members at all d ...

established the Willard Gibbs Award
The Willard Gibbs Award, presented by thChicago Sectionof the American Chemical Society, was established in 1910 by William A. Converse (1862–1940), a former Chairman and Secretary of the Chicago Section of the society and named for Professor Jo ...

for eminent work in pure or applied chemistry. In 1923, the American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

endowed the Josiah Willard Gibbs Lectureship The Josiah Willard Gibbs Lectureship (also called the Gibbs Lecture) of the American Mathematical Society is an annually awarded mathematical prize, named in honor of Josiah Willard Gibbs. The prize is intended not only for mathematicians, but also ...

, "to show the public some idea of the aspects of mathematics and its applications".
In 1945, Yale University created the J. Willard Gibbs Professorship in Theoretical Chemistry, held until 1973 by Lars Onsager
Lars Onsager (November 27, 1903 – October 5, 1976) was a Norwegian-born American physical chemist and theoretical physicist. He held the Gibbs Professorship of Theoretical Chemistry at Yale University. He was awarded the Nobel Prize in C ...

. Onsager, who much like Gibbs, focused on applying new mathematical ideas to problems in physical chemistry, won the 1968 Nobel Prize in chemistry. 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. Rutgers University
Rutgers University (; RU), officially Rutgers, The State University of New Jersey, is a Public university, public land-grant research university consisting of four campuses in New Jersey. Chartered in 1766, Rutgers was originally called Queen's ...

endowed a J. Willard Gibbs Professorship of Thermomechanics, held as of 2014 by Bernard Coleman.
Gibbs was elected in 1950 to the Hall of Fame for Great Americans
The Hall of Fame for Great Americans is an outdoor sculpture gallery located on the grounds of Bronx Community College (BCC) in the Bronx, New York City. It is the first such hall of fame in the United States. Built in 1901 as part of the U ...

. The oceanographic research ship
Oceanography (), also known as oceanology and ocean science, is the scientific study of the oceans. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynami ...

USNS ''Josiah Willard Gibbs'' (T-AGOR-1) was in service with the United States Navy
The United States Navy (USN) is the maritime service branch of the United States Armed Forces and one of the eight uniformed services of the United States. It is the largest and most powerful navy in the world, with the estimated tonnage ...

from 1958 to 1971. Gibbs crater, near the eastern limb
Limb may refer to:
Science and technology
*Limb (anatomy), an appendage of a human or animal
*Limb, a large or main branch of a tree
*Limb, in astronomy, the curved edge of the apparent disk of a celestial body, e.g. lunar limb
*Limb, in botany, ...

of the Moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...

, was named in the scientist's honor in 1964.
Edward Guggenheim introduced the symbol ''G'' for the Gibbs free energy in 1933, and this was used also by Dirk ter Haar
Dirk ter Haar FRSE FIP DSc (; Oosterwolde, 19 April 1919 – Drachten, 3 September 2002) was an Anglo-Dutch physicist.
Life
Dirk ter Haar was born at Oosterwolde in the province Friesland in the north of the Netherlands on 19 April 1919.
He s ...

in 1966.Seeger 1974, p. 96 This notation is now universal and is recommended by the IUPAC
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...

. In 1960, William Giauque and others suggested the name "gibbs" (abbreviated gbs.) for the unit of entropy calorie
The calorie is a unit of energy. For historical reasons, two main definitions of "calorie" are in wide use. The large calorie, food calorie, or kilogram calorie was originally defined as the amount of heat needed to raise the temperature of on ...

per kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...

, but this usage did not become common, and the corresponding SI unit joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applied ...

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: "Lorentz
Lorentz is a name derived from the Roman surname, Laurentius, which means "from Laurentum". It is the German form of Laurence. Notable people with the name include:
Given name
* Lorentz Aspen (born 1978), Norwegian heavy metal pianist and keyboar ...

", adding "I never met Willard Gibbs; perhaps, had I done so, I might have placed him beside Lorentz." Author Bill Bryson
William McGuire Bryson (; born 8 December 1951) is an American–British journalist and author. Bryson has written a number of nonfiction books on topics including travel, the English language, and science. Born in the United States, he has b ...

in his bestselling popular science
''Popular Science'' (also known as ''PopSci'') is an American digital magazine carrying popular science content, which refers to articles for the general reader on science and technology subjects. ''Popular Science'' has won over 58 awards, incl ...

book '' A Short History of Nearly Everything'' ranks Gibbs as "perhaps the most brilliant person that most people have never heard of".
In 1958, USS ''San Carlos'' was renamed USNS ''Josiah Willard Gibbs'' and re-designated as an oceanographic research ship.
In literature

In 1909, the American historian and novelistHenry Adams
Henry Brooks Adams (February 16, 1838 – March 27, 1918) was an American historian and a member of the Adams political family, descended from two U.S. Presidents.
As a young Harvard graduate, he served as secretary to his father, Charles Fr ...

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
William James (January 11, 1842 – August 26, 1910) was an American philosopher, historian, and psychologist, and the first educator to offer a psychology course in the United States.
James is considered to be a leading thinker of the lat ...

, 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. 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
Brooks may refer to:
Places
;Antarctica
*Cape Brooks
;Canada
*Brooks, Alberta
;United States
* Brooks, Alabama
* Brooks, Arkansas
*Brooks, California
*Brooks, Georgia
* Brooks, Iowa
* Brooks, Kentucky
* Brooks, Maine
* Brooks Township, Michigan ...

.
In the 1930s, feminist poet Muriel Rukeyser
Muriel Rukeyser (December 15, 1913 – February 12, 1980) was an American poet and political activist, best known for her poems about equality, feminism, social justice, and Judaism. Kenneth Rexroth said that she was the greatest poet of her "e ...

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). According to Rukeyser:
In 1946, ''Fortune
Fortune may refer to:
General
* Fortuna or Fortune, the Roman goddess of luck
* Luck
* Wealth
* Fortune, a prediction made in fortune-telling
* Fortune, in a fortune cookie
Arts and entertainment Film and television
* ''The Fortune'' (1931 film) ...

'' 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"Rukeyser 1988, p. 203 and the magazine saw in it "the abstract creation of a great American scientist that lends itself to the symbolism of contemporary art forms." The artwork by Arthur Lidov also included Gibbs's mathematical expression of the phase rule for heterogeneous mixtures, as well as a radar
Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...

screen, an oscilloscope
An oscilloscope (informally a scope) is a type of electronic test instrument that graphically displays varying electrical voltages as a two-dimensional plot of one or more signals as a function of time. The main purposes are to display repetiti ...

waveform, Newton's apple, and a small rendition of a three-dimensional phase diagram.
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. Rukeyser's approach to Gibbs was also sharply criticized by Gibbs's former student and protégé Edwin Wilson. With Van Name's and Wilson's encouragement, physicist Lynde Wheeler published a new biography of Gibbs in 1951.Wheeler 1998, pp. ix–xiii
Both Gibbs and Rukeyser's biography of him figure prominently in the poetry collection ''True North'' (1997) by Stephanie Strickland
Stephanie Strickland (born February 22, 1942) is a poet living in New York City. She has published ten volumes of print poetry and co-authored twelve digital poems. Her files and papers are being collected by the David M. Rubenstein Rare Book And ...

. In fiction, Gibbs appears as the mentor to character Kit Traverse in Thomas Pynchon
Thomas Ruggles Pynchon Jr. ( , ; born May 8, 1937) is an American novelist noted for his dense and complex novels. His fiction and non-fiction writings encompass a vast array of subject matter, genres and themes, including history, music, scie ...

's novel ''Against the Day
''Against the Day'' is an epic historical novel by Thomas Pynchon, published in 2006. The narrative takes place between the 1893 Chicago World's Fair and the time immediately following World War I and features more than a hundred characters spr ...

'' (2006). That novel also prominently discusses the birefringence of Iceland spar
Iceland spar, formerly called Iceland crystal ( is, silfurberg , ) and also called optical calcite, is a transparent variety of calcite, or crystallized calcium carbonate, originally brought from Iceland, and used in demonstrating the polarizati ...

, an optical phenomenon that Gibbs investigated.
Gibbs stamp (2005)

In 2005, theUnited States Postal Service
The United States Postal Service (USPS), also known as the Post Office, U.S. Mail, or Postal Service, is an independent agency of the executive branch of the United States federal government responsible for providing postal service in the U ...

issued the ''American Scientists'' commemorative postage stamp
A postage stamp is a small piece of paper issued by a post office, postal administration, or other authorized vendors to customers who pay postage (the cost involved in moving, insuring, or registering mail), who then affix the stamp to the fa ...

series designed by artist Victor Stabin, depicting Gibbs, John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

, Barbara McClintock
Barbara McClintock (June 16, 1902 – September 2, 1992) was an American scientist and cytogeneticist who was awarded the 1983 Nobel Prize in Physiology or Medicine. McClintock received her PhD in botany from Cornell University in 1927. There s ...

, and Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...

. The first day of issue
A first day of issue cover or first day cover (FDC) is a postage stamp on a cover, postal card or stamped envelope franked on the first day the issue is authorized for useBennett, Russell and Watson, James; ''Philatelic Terms Illustrated'', Stanl ...

ceremony for the series was held on May 4 at Yale University's Luce Hall and was attended by John Marburger
John Harmen "Jack" Marburger III (February 8, 1941 – July 28, 2011) was an American physicist who directed the Office of Science and Technology Policy in the administration of President George W. Bush, serving as the Science Advisor to the Pre ...

, 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.
Kenneth R. Jolls, a professor of chemical engineering at Iowa State University
Iowa State University of Science and Technology (Iowa State University, Iowa State, or ISU) is a public land-grant research university in Ames, Iowa. Founded in 1858 as the Iowa Agricultural College and Model Farm, Iowa State became one of the n ...

and an expert on graphical methods in thermodynamics, consulted on the design of the stamp honoring Gibbs. 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. Microprinting
Microprinting is the production of recognizable patterns or characters in a printed medium at a scale that requires magnification to read with the naked eye. To the unaided eye, the text may appear as a solid line. Attempts to reproduce by meth ...

on the collar
Collar may refer to:
Human neckwear
*Clerical collar (informally ''dog collar''), a distinctive collar used by the clergy of some Christian religious denominations
*Collar (clothing), the part of a garment that fastens around or frames the neck
...

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.
Outline of principal work

*Physical chemistry
Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical mecha ...

: free energy, phase diagram
A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseous ...

, phase rule
In thermodynamics, the phase rule is a general principle governing "pVT" systems, whose thermodynamic states are completely described by the variables pressure (), volume () and temperature (), in thermodynamic equilibrium. If is the number of d ...

, transport phenomena
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechan ...

* Statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...

: statistical ensemble
In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...

, phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...

, chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...

, Gibbs entropy
The concept entropy was first developed by German physicist Rudolf Clausius in the mid-nineteenth century as a thermodynamic property that predicts that certain spontaneous processes are irreversible or impossible. In statistical mechanics, entropy ...

, Gibbs paradox
In statistical mechanics, a semi-classical derivation of entropy that does not take into account the indistinguishability of particles yields an expression for entropy which is not extensive (is not proportional to the amount of substance in qu ...

* Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

: ''convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Convex sets
A subset C \subseteq X of s ...

, Gibbs phenomenon
In mathematics, the Gibbs phenomenon, discovered by Available on-line at:National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. and rediscovered by , is the oscillatory behavior of the Fourier series of a piecewise continuousl ...

* Electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

: Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...

, birefringence
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefring ...

See also

* Concentration of measure in physics * Thermodynamics of crystal growth *Governor (device)
A governor, or speed limiter or controller, is a device used to measure and regulate the speed of a machine, such as an engine.
A classic example is the centrifugal governor, also known as the Watt or fly-ball governor on a reciprocating stea ...

* List of notable textbooks in statistical mechanics
A list of notable textbooks in thermodynamics and statistical mechanics, arranged by category and date.
Only or mainly thermodynamics
*
*
*
*
*
*
Both thermodynamics and statistical mechanics
*
*
* 2e Kittel, Charles; and Kroemer, Herb ...

* List of theoretical physicists
The following is a partial list of notable theoretical physicists. Arranged by century of birth, then century of death, then year of birth, then year of death, then alphabetically by surname. For explanation of symbols, see Notes at end of this art ...

* List of things named after Josiah W. Gibbs
* Timeline of United States discoveries
* Timeline of thermodynamics
A timeline of events in the history of thermodynamics.
Before 1800
* 1650 – Otto von Guericke builds the first vacuum pump
* 1660 – Robert Boyle experimentally discovers Boyle's Law, relating the pressure and volume of a gas (pu ...

References

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). . This contains previously unpublished work by Gibbs, from the period between 1863 and 1871. * J. W. Gibbs, "On the Equilibrium of Heterogeneous Substances
In the history of thermodynamics, ''On the Equilibrium of Heterogeneous Substances'' is a 300-page paper written by American chemical physicist Willard Gibbs. It is one of the founding papers in thermodynamics, along with German physicist Hermann ...

", ''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 901. * J. W. Gibbs, '' Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics'', (New York: Dover Publications, 1960

902
__NOTOC__
Year 902 (Roman numerals, CMII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* Spring – Adalbert II, Margrave of Tuscany, Adalbert II, margr ...

.
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 906
__NOTOC__
Year 906 ( CMVI) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* February 27 – Battle of Fritzlar: The Conradines defeat the Babenberg co ...

. . For scans of the 1906 printing, sevol. I

an

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

928
Year 928 (Roman numerals, CMXXVIII) was a leap year starting on Tuesday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* King Rudolph of France, Rudolph I loses the support of Herbert II, Count of ...

. For scans of the 1928 printing, sevol. I

an

vol. II

Secondary

* . Reprinted with some additions in both ''The Scientific Papers'', vol. Ipp. xiii–xxviiii

(1906) and ''The Collected Works of J. Willard Gibbs'', vol. I, pp. xiii–xxviiii (1928). Also available her

* 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. * M. J. Crowe, ''A History of Vector Analysis: The Evolution of the Idea of a Vectorial System'', (New York: Dover, 1994 967. * J. G. Crowther, ''Famous American Men of Science'', (Freeport, NY: Books for Libraries Press, 1969

937
Year 937 ( CMXXXVII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* A Hungarian army invades Burgundy, and burns the city of Tournus. Then they go southward ...

.
* 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 936
Year 936 ( CMXXXVI) was a leap year starting on Friday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* June 19 – At Laon, Louis IV, the 14-year old son of the late King Charles the Simp ...

. . Onlvol 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

942
Year 942 ( CMXLII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* Summer – The Hungarians invade Al-Andalus (modern Spain) and besiege the fortress ...

.
* R. J. Seeger, ''J. Willard Gibbs, American mathematical physicist'' par excellence, (Oxford and New York: Pergamon Press, 1974).
* L. P. Wheeler, ''Josiah Willard Gibbs, The History of a Great Mind'', (Woodbridge, CT: Ox Bow Press, 1998 951
Year 951 ( CMLI) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar.
Events
By place
Europe
* King Berengar II of Italy seizes Liguria, with help from the feudal lord Oberto I. He re ...

.
* 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.
* E. B. Wilson,"Reminiscences of Gibbs by a student and colleague"

''Bulletin of the American Mathematical Society'', 37, 401–416 (1931).

External links

* *, in ''Selected Papers of Great American Scientists'', American Institute of Physics, (2003

976
Year 976 ( CMLXXVI) was a leap year starting on Saturday (link will display the full calendar) of the Julian calendar.
Events
By place Byzantine Empire
* January 10 – Emperor John I Tzimiskes dies at Constantinople, after re ...

*
"Gibbs"

by

Muriel Rukeyser
Muriel Rukeyser (December 15, 1913 – February 12, 1980) was an American poet and political activist, best known for her poems about equality, feminism, social justice, and Judaism. Kenneth Rexroth said that she was the greatest poet of her "e ...

Reflections on Gibbs: From Statistical Physics to the Amistad

by

Leo Kadanoff
Leo Philip Kadanoff (January 14, 1937 – October 26, 2015) was an American physicist. He was a professor of physics (emeritus from 2004) at the University of Chicago and a former President of the American Physical Society (APS). He contributed t ...

, Prof.
National Academy of Sciences, Biography, Josiah Willard Gibbs

* Josiah Willard Gibbs Papers. General Collection, Beinecke Rare Book and Manuscript Library, Yale University. {{DEFAULTSORT:Gibbs, Josiah Willard 1839 births 1903 deaths Thermodynamicists Mathematical analysts American physical chemists 19th-century American mathematicians 20th-century American mathematicians Foreign Members of the Royal Society Yale School of Engineering & Applied Science alumni Heidelberg University alumni Yale University faculty Hopkins School alumni Scientists from New Haven, Connecticut Recipients of the Copley Medal Burials at Grove Street Cemetery Theoretical physicists Fluid dynamicists Hall of Fame for Great Americans inductees Connecticut Republicans Philosophers of science Yale College alumni Statistical physicists