Editing Josiah Willard Gibbs (section)

=== Statistical mechanics ===

Together with [[James Clerk Maxwell]] and [[Ludwig Boltzmann]], Gibbs founded "statistical mechanics", a term that he coined to identify the branch of theoretical physics that accounts for the observed thermodynamic properties of systems in terms of the statistics of [[Statistical ensemble (mathematical physics)|ensembles]] of all possible physical states of a system composed of many particles. He introduced the concept of "[[phase space|phase of a mechanical system]]".<ref name=Nolte2010>{{cite journal |doi=10.1063/1.3397041 |title=The tangled tale of phase space |journal=Physics Today |volume=63 |issue=4 |pages=33–38 |year=2010 |last1=Nolte |first1=David D. |bibcode=2010PhT....63d..33N |s2cid=17205307 }}</ref>{{refn|For a mechanical system composed of ''n'' particles, the phase is represented by a point in a 2''n''-dimensional space, which he called "extension-in-phase" and is equivalent to our modern notion of phase space. However, the phrase "phase space" was not invented by him.<ref name=Nolte2010 />}} He used the concept to define the [[microcanonical ensemble|microcanonical]], [[canonical ensemble|canonical]], and [[grand canonical ensemble]]s; all related to the [[Gibbs measure]], thus obtaining a more general formulation of the statistical properties of many-particle systems than Maxwell and Boltzmann had achieved before him.<ref>Wheeler 1998, pp. 155–159.</ref>

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

According to [[Henri Poincaré]], writing in 1904, even though Maxwell and Boltzmann had previously explained the [[Irreversible process|irreversibility]] of macroscopic physical processes in probabilistic terms, "the one who has seen it most clearly, in a book too little read because it is a little difficult to read, is Gibbs, in his ''Elementary Principles of Statistical Mechanics''".<ref name="Poincare">{{cite book |last=Poincaré |first=Henri |author-link=Henri Poincaré |year=1904 |chapter=[[s:The Principles of Mathematical Physics|The Principles of Mathematical Physics]] |title=The Foundations of Science (The Value of Science) |pages=297–320 |publisher=Science Press |place=New York}}</ref> Gibbs's analysis of irreversibility, and his formulation of Boltzmann's [[H-theorem]] and of the [[ergodic hypothesis]], were major influences on the mathematical physics of the 20th century.<ref name="Wightman">{{cite book | chapter = On the Prescience of J. Willard Gibbs | last = Wightman | first = Arthur S. | author-link = Arthur Wightman | title = Proceedings of the Gibbs Symposium | year = 1990 |pages = 23–38}}</ref><ref name="Wiener">{{cite book | last = Wiener | first = Norbert | title =Cybernetics: or Control and Communication in the Animal and the Machine | chapter = II: Groups and Statistical Mechanics | edition = 2 | publisher = MIT Press | year = 1961 | isbn = 978-0-262-23007-0}}</ref>

Gibbs was well aware that the application of the [[equipartition theorem]] to large systems of classical particles failed to explain the measurements of the [[Heat capacity|specific heats]] of both solids and gases, and he argued that this was evidence of the danger of basing thermodynamics on "hypotheses about the constitution of matter".<ref name=Klein1990 /> Gibbs's own framework for statistical mechanics, based on ensembles of macroscopically indistinguishable [[Microstate (statistical mechanics)|microstates]], could be carried over almost intact after the discovery that the microscopic laws of nature obey quantum rules, rather than the classical laws known to Gibbs and to his contemporaries.<ref name="MacTutor" /><ref>Wheeler 1998, pp. 160–161.</ref> His resolution of the so-called "[[Gibbs paradox]]", about the entropy of the mixing of gases, is now often cited as a prefiguration of the [[Identical particles|indistinguishability of particles]] required by quantum physics.<ref>See, e.g., {{cite book | last = Huang | first = Kerson | author-link = Kerson Huang | title = Statistical Mechanics | publisher = John Wiley & Sons | year = 1987 | edition = 2 | pages = 140–143 | isbn = 978-0-471-81518-1}}</ref>