98-505: The Lars Onsager Prize is a prize in theoretical statistical physics awarded annually by the American Physical Society . Prize recipients receive a medal, certificate, and $ 10,000. It was established in 1993 by Drs. Russell and Marian Donnelly in memory of Lars Onsager . This science awards article is a stub . You can help Misplaced Pages by expanding it . Statistical physics In physics , statistical mechanics
196-458: A vector calculus well-suited to the needs of physicists. With this object in mind, Gibbs distinguished between the dot and cross products of two vectors and introduced the concept of dyadics . Similar work was carried out independently, and at around the same time, by the British mathematical physicist and engineer Oliver Heaviside . Gibbs sought to convince other physicists of the convenience of
294-456: 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 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 to large systems of classical particles failed to explain
392-458: A central role in Claude Shannon 's information theory and is therefore often seen as the basis of the modern information-theoretical interpretation of thermodynamics. According to Henri Poincaré , writing in 1904, even though Maxwell and Boltzmann had previously explained the irreversibility of macroscopic physical processes in probabilistic terms, "the one who has seen it most clearly, in
490-688: A chapter on Gibbs's work in the next edition of his Theory of Heat , published in 1875. He explained the usefulness of Gibbs's graphical methods in a lecture to the Chemical Society of London and even referred to it in the article on "Diagrams" that he wrote for the Encyclopædia Britannica . 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
588-402: A conservative Democrat , 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 , one of
686-601: A few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level. Many physical phenomena involve quasi-thermodynamic processes out of equilibrium, for example: All of these processes occur over time with characteristic rates. These rates are important in engineering. The field of non-equilibrium statistical mechanics
784-455: A finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics. For systems containing many particles (the thermodynamic limit ), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles
882-478: A mass of isolated facts and observations. 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 , who translated Gibbs's monograph into German, referred to Gibbs as the "founder of chemical energetics". According to modern commentators, It is universally recognised that its publication
980-467: A punishing regimen of study, Gibbs caught a serious cold and a doctor, fearing tuberculosis, advised him to rest on the Riviera , where he and his sisters spent several months and where he made a full recovery. Moving to Berlin , Gibbs attended the lectures taught by mathematicians Karl Weierstrass and Leopold Kronecker , as well as by chemist Heinrich Gustav Magnus . In August 1867, Gibbs's sister Julia
1078-403: A quotation from Rudolf Clausius that expresses what would later be called the first and second laws of thermodynamics : "The energy of the world is constant. The entropy of the world tends towards a maximum." Gibbs's monograph rigorously and ingeniously applied his thermodynamic techniques to the interpretation of physico-chemical phenomena, explaining and relating what had previously been
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#17328876046301176-424: A role in materials science, nuclear physics, astrophysics, chemistry, biology and medicine (e.g. study of the spread of infectious diseases). Analytical and computational techniques derived from statistical physics of disordered systems, can be extended to large-scale problems, including machine learning, e.g., to analyze the weight space of deep neural networks . Statistical physics is thus finding applications in
1274-407: A state with a balance of forces that has ceased to evolve.) The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems. The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics)
1372-477: A surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion. The founding of the field of statistical mechanics is generally credited to three physicists: In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius , Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave
1470-624: A textbook, Vector Analysis , published in 1901. That book helped to popularize the " del " notation that is widely used today in electrodynamics and fluid mechanics . In other mathematical work, he re-discovered the " Gibbs phenomenon " in the theory of Fourier series (which, unbeknownst to him and to later scholars, had been described fifty years before by an obscure English mathematician, Henry Wilbraham ). From 1882 to 1889, Gibbs wrote five papers on physical optics , in which he investigated birefringence and other optical phenomena and defended Maxwell's electromagnetic theory of light against
1568-561: A three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was a professor of mathematical physics from 1871 until his death in 1903. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised by Albert Einstein as "the greatest mind in American history". In 1901, Gibbs received what
1666-404: Is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics , its applications include many problems in the fields of physics, biology , chemistry , neuroscience , computer science , information theory and sociology . Its main purpose is to clarify
1764-550: Is chiefly remembered today as the abolitionist who found an interpreter for the African passengers of the ship Amistad , allowing them to testify during the trial that followed their rebellion against being sold as slaves. Willard Gibbs was educated at the Hopkins School and entered Yale College in 1854 at the age of 15. At Yale, Gibbs received prizes for excellence in mathematics and Latin , and he graduated in 1858, near
1862-499: Is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.) In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent,
1960-504: Is firmly entrenched. Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics , a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous. Gibbs' methods were initially derived in the framework classical mechanics , however they were of such generality that they were found to adapt easily to
2058-417: Is however a disconnect between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and
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#17328876046302156-449: Is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics. Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections. One approach to non-equilibrium statistical mechanics
2254-469: Is primarily concerned with thermodynamic equilibrium , statistical mechanics has been applied in non-equilibrium statistical mechanics to the issues of microscopically modeling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions and flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study
2352-432: Is simply related to the free energy change when the reactants are in their standard states : Chemical potential is usually defined as partial molar Gibbs free energy: Gibbs also obtained what later came to be known as the " Gibbs–Duhem equation ". In an electrochemical reaction characterized by an electromotive force ℰ and an amount of transferred charge Q , Gibbs's starting equation becomes The publication of
2450-400: Is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium . Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. (By contrast, mechanical equilibrium is
2548-431: Is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium , and the microscopic behaviours and motions occurring inside the material. Whereas statistical mechanics proper involves dynamics, here
2646-411: Is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes , a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within
2744-417: Is usual for probabilities, the ensemble can be interpreted in different ways: These two meanings are equivalent for many purposes, and will be used interchangeably in this article. However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in
2842-403: Is very useful in diverse areas, such as metallurgy, mineralogy, and petrology. It can also be applied to various research problems in physical chemistry. Together with James Clerk Maxwell and Ludwig Boltzmann , Gibbs founded "statistical mechanics", a term that he coined to identify the branch of theoretical physics that accounts for the observed thermodynamic properties of systems in terms of
2940-647: The Civil War of 1861–65. He was not conscripted and he remained at Yale for the duration of the war. In 1863, Gibbs received the first Doctorate of Philosophy (PhD) in engineering granted in the US, for a thesis entitled "On the Form of the Teeth of Wheels in Spur Gearing", in which he used geometrical techniques to investigate the optimum design for gears . In 1861, Yale had become
3038-497: The Legendre transform of this expression, he defined the concepts of enthalpy H and Gibbs free energy G : This compares to the expression for Helmholtz free energy A : 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
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3136-432: The von Neumann equation . These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. These ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy
3234-560: The "interesting" information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors ), where the electrons are indeed analogous to a rarefied gas. Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations,
3332-753: 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. After his term as tutor ended, Gibbs traveled to Europe with his sisters. They spent the winter of 1866–67 in Paris, where Gibbs attended lectures at the Sorbonne and the Collège de France , given by such distinguished mathematical scientists as Joseph Liouville and Michel Chasles . Having undertaken
3430-425: 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." Gibbs did supervise the doctoral thesis on mathematical economics written by Irving Fisher in 1891. After Gibbs's death, Fisher financed the publication of his Collected Works . Another distinguished student
3528-412: The application of Maxwell's equations to problems in physical optics . As a mathematician, he created modern vector calculus (independently of the British scientist Oliver Heaviside , who carried out similar work during the same period) and described the Gibbs phenomenon in the theory of Fourier analysis. In 1863, Yale University awarded Gibbs the first American doctorate in engineering . After
3626-448: The area of medical diagnostics . Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems . In quantum mechanics, a statistical ensemble (probability distribution over possible quantum states ) is described by a density operator S , which is a non-negative, self-adjoint , trace-class operator of trace 1 on the Hilbert space H describing
3724-625: The attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving ( mechanical equilibrium ), rather, only that the ensemble is not evolving. A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.). There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why
3822-650: The characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities. There are some cases which allow exact solutions. Although some problems in statistical physics can be solved analytically using approximations and expansions, most current research utilizes
3920-408: The classical laws known to Gibbs and to his contemporaries. His resolution of the so-called " Gibbs paradox ", about the entropy of the mixing of gases, is now often cited as a prefiguration of the indistinguishability of particles required by quantum physics. British scientists, including Maxwell, had relied on Hamilton's quaternions in order to express the dynamics of physical quantities, like
4018-412: The concept of the chemical potential μ {\displaystyle \mu } of a given chemical species, defined to be the rate of the increase in U associated with the increase in the number N of molecules of that species (at constant entropy and volume). Thus, it was Gibbs who first combined the first and second laws of thermodynamics by expressing the infinitesimal change in
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4116-626: The conductance of an electronic system is the use of the Green–Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method. The ensemble formalism can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in: Statistical physics explains and quantitatively describes superconductivity , superfluidity , turbulence , collective phenomena in solids and plasma , and
4214-408: 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 (1876), and statistical ensemble (1902). Gibbs's derivation of the laws of thermodynamics from the statistical properties of systems consisting of many particles
4312-425: The electric and magnetic fields, having both a magnitude and a direction in three-dimensional space. Following W. K. Clifford in his Elements of Dynamic (1888), Gibbs noted that the product of quaternions could be separated into two parts: a one-dimensional (scalar) quantity and a three-dimensional vector , so that the use of quaternions involved mathematical complications and redundancies that could be avoided in
4410-531: The ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state. One special class of ensemble
4508-524: The ensemble for a given system should have one form or another. A common approach found in many textbooks is to take the equal a priori probability postulate . This postulate states that The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate: Other fundamental postulates for statistical mechanics have also been proposed. For example, recent studies shows that
4606-476: The entropy of an arbitrary ensemble as where k B {\displaystyle k_{\text{B}}} is the Boltzmann constant , while the sum is over all possible microstates i {\displaystyle i} , with p i {\displaystyle p_{i}} the corresponding probability of the microstate (see Gibbs entropy formula ). This same formula would later play
4704-405: The first US university to offer a PhD degree and Gibbs's was only the fifth PhD granted in the US in any subject. 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 a railway brake and read a paper before
4802-410: The fluctuation–dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics. A few of the theoretical tools used to make this connection include: An advanced approach uses a combination of stochastic methods and linear response theory . As an example, one approach to compute quantum coherence effects ( weak localization , conductance fluctuations ) in
4900-425: 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, Gibbs was not an advertiser for personal renown nor a propagandist for science; he was a scholar, scion of an old scholarly family, living before the days when research had become ré search ... Gibbs was not a freak, he had no striking ways, he
4998-459: The ideal of the unselfish, Christian gentleman. In the minds of those who knew him, the greatness of his intellectual achievements will never overshadow the beauty and dignity of his life. Gibbs's papers from the 1870s introduced the idea of expressing the internal energy U of a system in terms of the entropy S , in addition to the usual state variables of volume V , pressure p , and temperature T . He also introduced
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#17328876046305096-454: The internal energy, d U , of a closed system in the form where T is the absolute temperature , p is the pressure, d S is an infinitesimal change in entropy and d V is an infinitesimal change of volume. The last term is the sum, over all the chemical species in a chemical reaction, of the chemical potential, μ i , of the i -th species, multiplied by the infinitesimal change in the number of moles, d N i of that species. By taking
5194-479: The journal had few readers capable of understanding Gibbs's work, he shared reprints with correspondents in Europe and received an enthusiastic response from James Clerk Maxwell at Cambridge . Maxwell even made, with his own hands, a clay model illustrating Gibbs's construct . He then produced two plaster casts of his model and mailed one to Gibbs. That cast is on display at the Yale physics department. Maxwell included
5292-518: The large processing power of modern computers to simulate or approximate solutions. A common approach to statistical problems is to use a Monte Carlo simulation to yield insight into the properties of a complex system . Monte Carlo methods are important in computational physics , physical chemistry , and related fields, and have diverse applications including medical physics , where they are used to model radiation transport for radiation dosimetry calculations. The Monte Carlo method examines just
5390-409: The later quantum mechanics , and still form the foundation of statistical mechanics to this day. In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics . For both types of mechanics, the standard mathematical approach is to consider two concepts: Using these two concepts, the state at any other time, past or future, can in principle be calculated. There
5488-437: The measurements of the specific heats of both solids and gases, and he argued that this was evidence of the danger of basing thermodynamics on "hypotheses about the constitution of matter". Gibbs's own framework for statistical mechanics, based on ensembles of macroscopically indistinguishable microstates , could be carried over almost intact after the discovery that the microscopic laws of nature obey quantum rules, rather than
5586-436: The mechanical theories of Lord Kelvin 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
5684-512: The new Johns Hopkins University in Baltimore, Maryland offered him a position paying $ 3,000 per year. In response, Yale offered him an annual salary of $ 2,000, which he was content to accept. In 1879, Gibbs derived the Gibbs–Appell equation of motion , rediscovered in 1900 by Paul Émile Appell . From 1880 to 1884, Gibbs worked on developing the exterior algebra of Hermann Grassmann into
5782-479: The only time that Gibbs spent outside New Haven. He joined Yale's College Church (a Congregational church ) at the end of his freshman year and remained a regular attendant for the rest of his life. Gibbs generally voted for the Republican candidate in presidential elections but, like other " Mugwumps ", his concern over the growing corruption associated with machine politics led him to support Grover Cleveland ,
5880-412: The opinion of biographers, Gibbs's principal mentor and champion, both at Yale and in the Connecticut Academy, was probably the astronomer and mathematician Hubert Anson Newton , a leading authority on meteors , who remained Gibbs's lifelong friend and confidant. After the death of his father in 1861, Gibbs inherited enough money to make him financially independent. Recurrent pulmonary trouble ailed
5978-617: The paper " On the Equilibrium of Heterogeneous Substances " (1874–1878) is now regarded as a landmark in the development of chemistry . In it, Gibbs developed a rigorous mathematical theory for various transport phenomena , including adsorption , electrochemistry , and the Marangoni effect in fluid mixtures. He also formulated the phase rule for the number F of variables that may be independently controlled in an equilibrium mixture of C components existing in P phases . The phase rule
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#17328876046306076-427: The practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in. Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble , which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of
6174-417: 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 , 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". Gibbs
6272-507: The proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H -theorem . The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in 1884. According to Gibbs, the term "statistical", in the context of mechanics, i.e. statistical mechanics,
6370-471: The properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics , a field for which it was successful in explaining macroscopic physical properties—such as temperature , pressure , and heat capacity —in terms of microscopic parameters that fluctuate about average values and are characterized by probability distributions . While classical thermodynamics
6468-473: The proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. Five years later, in 1864, Ludwig Boltzmann , a young student in Vienna, came across Maxwell's paper and spent much of his life developing
6566-433: The quantum system. This can be shown under various mathematical formalisms for quantum mechanics . One such formalism is provided by quantum logic . Josiah Willard Gibbs Josiah Willard Gibbs ( / ɡ ɪ b z / ; 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
6664-432: The response can be analysed in linear response theory . A remarkable result, as formalized by the fluctuation–dissipation theorem , is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in
6762-408: The same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium. This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations,
6860-413: The simplest non-equilibrium situation of a steady state current flow in a system of many particles. In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases . In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on
6958-445: The size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system. Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from
7056-627: 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 , and grand canonical ensembles ; all related to the Gibbs measure , thus obtaining a more general formulation of the statistical properties of many-particle systems than Maxwell and Boltzmann had achieved before him. Gibbs generalized Boltzmann's statistical interpretation of entropy S {\displaystyle S} by defining
7154-419: The structural features of liquid . It underlies the modern astrophysics . In solid state physics, statistical physics aids the study of liquid crystals , phase transitions , and critical phenomena . Many experimental studies of matter are entirely based on the statistical description of a system. These include the scattering of cold neutrons , X-ray , visible light , and more. Statistical physics also plays
7252-455: The subject further. Statistical mechanics was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory . Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem , transport theory , thermal equilibrium , the equation of state of gases, and similar subjects, occupy about 2,000 pages in
7350-462: The system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier. The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where
7448-413: The system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinate axes. In quantum statistical mechanics, the ensemble is a probability distribution over pure states and can be compactly summarized as a density matrix . As
7546-403: The theory of statistical mechanics can be built without the equal a priori probability postulate. One such formalism is based on the fundamental thermodynamic relation together with the following set of postulates: where the third postulate can be replaced by the following: There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside
7644-573: The top of his class. He remained at Yale as a graduate student at the Sheffield Scientific School . At age 19, soon after his graduation from college, Gibbs was inducted into the Connecticut Academy of Arts and Sciences , a scholarly institution composed primarily of members of the Yale faculty. Relatively few documents from the period survive and it is difficult to reconstruct the details of Gibbs's early career with precision. In
7742-474: The vectorial approach over the quaternionic calculus of William Rowan Hamilton , which was then widely used by British scientists. This led him, in the early 1890s, to a controversy with Peter Guthrie Tait and others in the pages of Nature . Gibbs's lecture notes on vector calculus were privately printed in 1881 and 1884 for the use of his students, and were later adapted by Edwin Bidwell Wilson into
7840-458: The wife of American founding father Roger Sherman ; and he was the second cousin of Roger Sherman Baldwin , see the Amistad case below. The elder Gibbs was generally known to his family and colleagues as "Josiah", while the son was called "Willard". Josiah Gibbs was a linguist and theologian who served as professor of sacred literature at Yale Divinity School from 1824 until his death in 1861. He
7938-474: The young Gibbs and his physicians were concerned that he might be susceptible to tuberculosis , which had killed his mother. He also suffered from astigmatism , whose treatment was then still largely unfamiliar to oculists , so that Gibbs had to diagnose himself and grind his own lenses. Though in later years he used glasses only for reading or other close work, Gibbs's delicate health and imperfect eyesight probably explain why he did not volunteer to fight in
8036-482: Was Lee De Forest , 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 . In May, Yale organized a memorial meeting at the Sloane Laboratory. The eminent British physicist J. J. Thomson
8134-553: 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 ", 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, begins with
8232-728: Was a careful investor and financial manager, and at his death in 1903 his estate was valued at $ 100,000 (roughly $ 3.39 million today ). For many years, he served as trustee, secretary, and treasurer of his alma mater, the Hopkins School. US President Chester A. Arthur appointed him as one of the commissioners to the National Conference of Electricians, which convened in Philadelphia in September 1884, and Gibbs presided over one of its sessions. A keen and skilled horseman, Gibbs
8330-445: Was a kindly dignified gentleman. According to Lynde Wheeler , who had been Gibbs's student at Yale, in his later years Gibbs was always neatly dressed, usually wore a felt hat on the street, and never exhibited any of the physical mannerisms or eccentricities sometimes thought to be inseparable from genius ... His manner was cordial without being effusive and conveyed clearly the innate simplicity and sincerity of his nature. He
8428-421: Was an event of the first importance in the history of chemistry ... Nevertheless it was a number of years before its value was generally known, this delay was due largely to the fact that its mathematical form and rigorous deductive processes make it difficult reading for anyone, and especially so for students of experimental chemistry whom it most concerns. Gibbs continued to work without pay until 1880, when
8526-462: Was born in New Haven, Connecticut. He belonged to an old Yankee family that had produced distinguished American clergymen and academics since the 17th century. He was the fourth of five children and the only son of Josiah Willard Gibbs Sr. , and his wife Mary Anna, née Van Cleve. On his father's side, he was descended from Samuel Willard , who served as acting President of Harvard College from 1701 to 1707. On his mother's side, one of his ancestors
8624-446: Was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology. Important cases where the thermodynamic ensembles do not give identical results include: In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in
8722-491: Was first used by the Scottish physicist James Clerk Maxwell in 1871: "In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus." "Probabilistic mechanics" might today seem a more appropriate term, but "statistical mechanics"
8820-590: Was hired without salary. 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
8918-567: Was in attendance and delivered a brief address. Gibbs never married, living all his life in his childhood home with his sister Julia and her husband Addison Van Name, who was the Yale librarian. Except for his customary summer vacations in the Adirondacks (at Keene Valley, New York ) and later at the White Mountains (in Intervale, New Hampshire ), his sojourn in Europe in 1866–1869 was almost
9016-402: Was instrumental in transforming physical chemistry into a rigorous deductive science. Together with James Clerk Maxwell and Ludwig Boltzmann , he created statistical mechanics (a term that he coined), explaining the laws of thermodynamics as consequences of the statistical properties of ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on
9114-684: 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 , Gibbs was exposed to the work of physicists Gustav Kirchhoff and Hermann von Helmholtz , and chemist Robert Bunsen . At the time, German academics were the leading authorities in the natural sciences, especially chemistry and thermodynamics . Gibbs returned to Yale in June 1869 and briefly taught French to engineering students. It
9212-407: 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
9310-402: Was probably also around this time that he worked on a new design for a steam-engine governor , his last significant investigation in mechanical engineering. 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
9408-513: Was seen habitually in New Haven driving his sister's carriage . In an obituary published in the American Journal of Science , Gibbs's former student Henry A. Bumstead referred to Gibbs's personal character: Unassuming in manner, genial and kindly in his intercourse with his fellow-men, never showing impatience or irritation, devoid of personal ambition of the baser sort or of the slightest desire to exalt himself, he went far toward realizing
9506-591: Was the Rev. Jonathan Dickinson , the first president of the College of New Jersey (later Princeton University ). Gibbs's given name, which he shared with his father and several other members of his extended family, derived from his ancestor Josiah Willard, who had been Secretary of the Province of Massachusetts Bay in the 18th century. His paternal grandmother, Mercy (Prescott) Gibbs, was the sister of Rebecca Minot Prescott Sherman,
9604-534: Was then considered the highest honor awarded by the international scientific community, the Copley Medal of the Royal Society 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 and the great international impact of his ideas. Though his work was almost entirely theoretical,
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