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| {{redirect|Boltzmann}}
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| {{Infobox scientist
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| | birth_name = Ludwig Eduard Boltzmann
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| | image = Boltzmann2.jpg|225px
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| | image_size = 225px
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| | caption = Ludwig Boltzmann
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| | birth_date = {{Birth date|1844|2|20|mf=y}}
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| | birth_place= [[Vienna]], [[Austrian Empire]] (present-day Austria)
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| | death_date = {{death date and age|1906|9|5|1844|2|20|mf=y}}
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| | death_place= [[Tybein]] near [[Trieste]], [[Austria-Hungary]] (present-day [[Duino]], Italy)
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| | death_cause = Suicide
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| | residence = Austria, Germany
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| | nationality = Austrian
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| | field = [[Physicist]]
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| | work_institution = [[University of Graz]]<br />[[University of Vienna]]<br />[[University of Munich]]<br />[[University of Leipzig]]
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| | alma_mater = [[University of Vienna]]
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| | doctoral_advisor = [[Josef Stefan]]
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| | doctoral_students = [[Paul Ehrenfest]]<br />[[Philipp Frank]]<br />[[Gustav Herglotz]]<br />[[Franc Hočevar]]<br />[[Ignacij Klemenčič]]
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| | notable_students = [[Lise Meitner]]
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| | known_for = [[Boltzmann constant]]<br />[[Boltzmann equation]]<br />[[Boltzmann distribution]]<br />[[H-theorem]]<br />[[Maxwell-Boltzmann distribution]]<br />[[Stefan–Boltzmann constant]]<br />[[Stefan–Boltzmann law]]
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| | prizes =
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| | religion =
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| | signature = Ludwig Boltzmann signature.svg
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| | footnotes =
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| }}
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| '''Ludwig Eduard Boltzmann''' (February 20, 1844 – September 5, 1906) was an Austrian physicist and philosopher whose greatest achievement was in the development of [[statistical mechanics]], which explains and predicts how the properties of [[atom]]s (such as [[atomic mass|mass]], [[electric charge|charge]], and structure) determine the physical properties of [[matter]] (such as viscosity, [[thermal conductivity]], and diffusion).
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| ==Biography==
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| ===Childhood and education===
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| Boltzmann was born in [[Vienna]], the capital of the [[Austrian Empire]]. His father, Ludwig Georg Boltzmann, was a revenue official. His grandfather, who had moved to Vienna from Berlin, was a clock manufacturer, and Boltzmann’s mother, Katharina Pauernfeind, was originally from [[Salzburg]]. He received his primary education from a private tutor at the home of his parents. Boltzmann attended high school in [[Linz]], [[Upper Austria]]. When Boltzmann was 15 his father died.
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| Boltzmann studied [[physics]] at the [[University of Vienna]], starting in 1863. Among his teachers were [[Johann Josef Loschmidt|Josef Loschmidt]], [[Joseph Stefan]], [[Andreas von Ettingshausen]] and [[Jozef Maximilián Petzval|Jozef Petzval]]. Boltzmann received his PhD degree in 1866 working under the supervision of Stefan; his dissertation was on [[kinetic theory of gases]]. In 1867 he became a [[Privatdozent]] (lecturer). After obtaining his doctorate degree, Boltzmann worked two more years as Stefan’s assistant. It was Stefan who introduced Boltzmann to [[James Clerk Maxwell|Maxwell's]] work.
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| ===Academic career===
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| In 1869 at age 25, thanks to a letter of recommendation written by Stefan,<ref>{{cite journal |url=http://www.kvarkadabra.net/article.php/Ludwig-Boltzmann |title=Ludwig Boltzmann in prva študentka fizike in matematike slovenskega rodu |language=Slovene |trans_title=Ludwig Boltzmann and the First Student of Physics and Mathematics of Slovene Descent |date=December 2001 |last=Južnič |first=Stanislav |work=Kvarkadabra.net |issue=12 |accessdate=17 February 2012}}</ref> he was appointed full Professor of [[Mathematical Physics]] at the [[University of Graz]] in the province of [[Styria]]. In 1869 he spent several months in [[Heidelberg]] working with [[Robert Bunsen]] and [[Leo Königsberger]] and then in 1871 he was with [[Gustav Kirchhoff]] and [[Hermann von Helmholtz]] in Berlin. In 1873 Boltzmann joined the University of Vienna as Professor of Mathematics and there he stayed until 1876.
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| [[File:Boltzmann-grp.jpg|thumb|left|280px|Ludwig Boltzmann and co-workers in Graz, 1887. (standing, from the left) [[Walther Nernst|Nernst]], [[Heinrich Streintz|Streintz]], [[Svante Arrhenius|Arrhenius]], Hiecke, (sitting, from the left) Aulinger, [[Albert von Ettingshausen|Ettingshausen]], Boltzmann, [[Ignacij Klemenčič|Klemenčič]], Hausmanninger]]
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| In 1872, long before women were admitted to Austrian universities, he met Henriette von Aigentler, an aspiring teacher of mathematics and physics in Graz. She was refused permission to audit lectures unofficially. Boltzmann advised her to appeal, which she did, successfully. On July 17, 1876 Ludwig Boltzmann married Henriette; they had three daughters and two sons. Boltzmann went back to [[Graz]] to take up the chair of Experimental Physics. Among his students in Graz were [[Svante Arrhenius]] and [[Walther Nernst]].<ref name="springer">"Paul Ehrenfest (1880–1933) along with Nernst[,] Arrhenius, and Meitner must be considered among Boltzmann’s most outstanding students."—{{Cite journal|last=Jäger|first=Gustav|last2=Nabl|first2=Josef|last3=Meyer|first3=Stephan|date=April 1999|title=Three Assistants on Boltzmann|journal=Synthese |volume=119|issue=1–2|pages=69–84 |doi=10.1023/A:1005239104047}}</ref><ref name="huji">[http://chem.ch.huji.ac.il/history/nernst.htm "Walther Hermann Nernst visited lectures by Ludwig Boltzmann"]</ref> He spent 14 happy years in Graz and it was there that he developed his statistical concept of nature. In 1885 he became a member of the Imperial [[Austrian Academy of Sciences]] and in 1887 he became the President of the [[University of Graz]]. He was elected a member of the [[Royal Swedish Academy of Sciences]] in 1888.
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| Boltzmann was appointed to the Chair of Theoretical Physics at the [[University of Munich]] in [[Bavaria]], Germany in 1890.
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| In 1893, Boltzmann succeeded his teacher [[Joseph Stefan]] as Professor of Theoretical Physics at the University of Vienna.
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| ===Final years===
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| Boltzmann spent a great deal of effort in his final years defending his theories. He did not get along with some of his colleagues in Vienna, particularly [[Ernst Mach]], who became a professor of philosophy and history of sciences in 1895. That same year [[Georg Helm]] and [[Wilhelm Ostwald]] presented their position on ''Energetics'', at a meeting in [[Lübeck]] in 1895. They saw energy, and not matter, as the chief component of the universe. Boltzmann's position carried the day among other physicists who supported his atomic theories in the debate.<ref>{{cite journal|author=Max Planck|title=Gegen die neure Energetik|journal=Annalen der Physik|volume=57|year=1896|pages=72–78}}</ref> In 1900, Boltzmann went to the [[University of Leipzig]], on the invitation of [[Wilhelm Ostwald]].<ref>Ostwald offered to Boltzmann the professorial chair of physics which was vacated upon the death of [[Gustav Heinrich Wiedemann]].</ref> After the retirement of Mach due to bad health, Boltzmann came back to Vienna in 1902.<ref>Upon Boltzmann's resignation, [[Theodor des Coudres]] became his successor in the professorial chair at Leipzig.</ref> In 1903 he founded the [[Austrian Mathematical Society]] together with [[Gustav von Escherich]] and [[Emil Müller (mathematician)|Emil Müller]]. His students included [[Karl Přibram|Karl Przibram]], [[Paul Ehrenfest]] and [[Lise Meitner]].
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| In Vienna, Boltzmann taught physics and also lectured on philosophy. Boltzmann’s lectures on [[natural philosophy]] were very popular, and received a considerable attention at that time. His first lecture was an enormous success. Even though the largest lecture hall had been chosen for it, the people stood all the way down the staircase. Because of the great successes of Boltzmann’s philosophical lectures, the Emperor invited him for a reception at the Palace.
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| Boltzmann was subject to rapid alternation of depressed moods with elevated, expansive or irritable moods, likely the symptoms of undiagnosed [[bipolar disorder]]. He himself jestingly attributed his rapid swings in temperament to the fact that he was born during the night between [[Shrove Tuesday]] and [[Ash Wednesday]].<ref>{{cite news |url=http://www.washingtonpost.com/wp-srv/style/longterm/books/chap1/lisemeitner.htm |title=Lise Meitner, A Life in Physics |accessdate=2009-02-06 |author=Ruth Lewin Sime |publisher=Washington Post | date=May 13, 1997}}</ref> Meitner relates that those who were close to Boltzmann were aware of his bouts of severe depression and his suicide attempts.
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| On September 5, 1906, while on a summer vacation in [[Duino]], near [[Trieste]], Boltzmann hanged himself during an attack of [[Mood disorder|depression]].<ref>{{cite book|last=Boltzmann|first=Ludwig|editor1-first=John T.|editor1-last=Blackmore|others=|title=Ludwig Boltzmann: His Later Life and Philosophy, 1900-1906|url=http://books.google.com/?id=apip-Jm9WuwC&pg=PA207 |volume=2|year=1995|publisher=Springer|isbn=978-0-7923-3464-4|pages=206–207|chapter=Conclusions}}</ref><ref>Upon Boltzmann's death, [[Friedrich Hasenöhrl|Friedrich ("Fritz") Hasenöhrl]] became his successor in the professorial chair of physics at Vienna.</ref> He is buried in the Viennese [[Zentralfriedhof]]; his tombstone bears the inscription of the [[Boltzmann's_entropy_formula|entropy formula]]:
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| :<math>S = k \cdot \log W \,</math>
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| where log stands for the [[natural logarithm]].
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| ==Philosophy==
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| Boltzmann's kinetic theory of gases seemed to presuppose the reality of [[atom]]s and [[molecule]]s, but almost all German philosophers and many scientists like [[Ernst Mach]] and the physical chemist [[Wilhelm Ostwald]] disbelieved their existence. During the 1890s Boltzmann attempted to formulate a compromise position which would allow both atomists and anti-atomists to do physics without arguing over atoms. His solution was to use [[Heinrich Hertz|Hertz]]'s theory that atoms were "Bilder", that is, models or pictures. Atomists could think the pictures were the real atoms while the anti-atomists could think of the pictures as representing a useful but unreal model, but this did not fully satisfy either group. Furthermore, Ostwald and many defenders of "pure thermodynamics" were trying hard to refute the kinetic theory of gases and statistical mechanics because of Boltzmann's assumptions about atoms and molecules and especially statistical interpretation of the second law.
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| Around the turn of the century, Boltzmann's science was being threatened by another philosophical objection. Some physicists, including Mach's student, [[Gustav Jaumann]], interpreted Hertz to mean that all electromagnetic behavior is continuous, as if there were no atoms and molecules, and likewise as if all physical behavior were ultimately electromagnetic. This movement around 1900 deeply depressed Boltzmann since it could mean the end of his kinetic theory and statistical interpretation of the second law of thermodynamics.
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| After Mach's resignation in Vienna in 1901, Boltzmann returned there and decided to become a philosopher himself to refute philosophical objections to his physics, but he soon became discouraged again. In 1904 at a physics conference in St. Louis most physicists seemed to reject atoms and he was not even invited to the physics section. Rather, he was stuck in a section called "applied mathematics," he violently attacked philosophy, especially on allegedly Darwinian grounds but actually in terms of [[Lamarck]]'s theory of the inheritance of acquired characteristics that people inherited bad philosophy from the past and that it was hard for scientists to overcome such inheritance.
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| In 1905 Boltzmann corresponded extensively with the Austro-German philosopher [[Franz Brentano]] with the hope of gaining a better mastery of philosophy, apparently, so that he could better refute its relevancy in science, but he became discouraged about this approach as well. In the following year 1906 his mental condition became so bad that he had to resign his position. He committed suicide in September of that same year by hanging himself while on vacation with his wife and daughter near Trieste, Italy.<ref>"Eureka! Science's greatest thinkers and their key breakthroughs", Hazel Muir, p.152, ISBN 1780873255</ref>
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| ==Physics==
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| Boltzmann's most important scientific contributions were in [[kinetic theory]], including the [[Maxwell–Boltzmann distribution]] for molecular speeds in a gas. In addition, [[Maxwell–Boltzmann statistics]] and the [[Boltzmann distribution]] over energies remain the foundations of [[classical mechanics|classical]] statistical mechanics. They are applicable to the many [[phenomenon|phenomena]] that do not require [[Maxwell–Boltzmann statistics#Limits of applicability|quantum statistics]] and provide a remarkable insight into the meaning of [[thermodynamic temperature|temperature]].
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| [[File:Boltzmanns-molecule.jpg|225px|thumb|right|Boltzmann’s 1898 I<sub>2</sub> molecule diagram showing atomic “sensitive region” (α, β) overlap.]]
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| Much of the [[physics]] establishment did not share his belief in the reality of [[atom]]s and [[molecule]]s — a belief shared, however, by [[James Clerk Maxwell|Maxwell]] in Scotland and [[Josiah Willard Gibbs|Gibbs]] in the United States; and by [[History of chemistry#The dispute about atomism|most]] [[chemistry|chemists]] since the discoveries of [[John Dalton]] in 1808. He had a long-running dispute with the editor of the preeminent German physics journal of his day, who refused to let Boltzmann refer to atoms and molecules as anything other than convenient [[Theory#Science|theoretical]] constructs. Only a couple of years after Boltzmann's death, [[Jean Baptiste Perrin|Perrin's]] studies of [[colloid]]al suspensions (1908–1909), based on [[Albert Einstein|Einstein's]] [[Albert Einstein#Thermodynamic fluctuations and statistical physics|theoretical studies]] of 1905, confirmed the values of [[Avogadro's number]] and [[Boltzmann constant|Boltzmann's constant]], and convinced the world that the tiny particles [[Atomic theory#History|really exist]].
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| To quote [[Max Planck|Planck]], "The [[logarithm]]ic connection between [[entropy]] and [[probability]] was first stated by L. Boltzmann in his [[kinetic theory]] of gases".<ref>Max Planck, p. 119.</ref> This famous formula for entropy ''S'' is<ref>The concept of [[entropy]] was introduced by [[Rudolf Clausius]] in 1865. He was the first to enunciate the [[second law of thermodynamics]] by saying that "entropy always increases".</ref><ref>An alternative is the [[Information entropy#Formal definitions|information entropy]] definition introduced in 1948 by [[Claude Elwood Shannon|Claude Shannon]].[http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html] It was intended for use in communication theory, but is applicable in all areas. It reduces to Boltzmann's expression when all the probabilities are equal, but can, of course, be used when they are not. Its virtue is that it yields immediate results without resorting to [[factorial]]s or [[Stirling's approximation]]. Similar formulas are found, however, as far back as the work of Boltzmann, and explicitly in [[H-theorem#Quantum mechanical H-theorem|Gibbs]] (see reference).</ref>
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| :<math> S = k_B \ln W \, </math>
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| where ''k'' = 1.3806505(24) × 10<sup>−23</sup> [[Joule|J]] [[Kelvin|K<sup>−1</sup>]] is [[Boltzmann constant|Boltzmann's constant]], and ''ln'' is the [[natural logarithm]]. ''W'' is ''Wahrscheinlichkeit'', a German word meaning the [[frequency (statistics)|frequency]] of occurrence of a [[macrostate]]<ref>{{cite book|last=Pauli| first=Wolfgang| title=Statistical Mechanics|publisher=MIT Press|location=Cambridge|year=1973|isbn=0-262-66035-0}}, p. 21</ref> or, more precisely, the number of possible [[microstate (statistical mechanics)|microstates]] corresponding to the macroscopic state of a system — number of (unobservable) "ways" in the (observable) [[thermodynamics|thermodynamic]] state of a system can be realized by assigning different [[coordinate system|positions]] and [[momentum|momenta]] to the various molecules. Boltzmann’s [[Paradigm#Paradigm shifts|paradigm]] was an [[ideal gas]] of ''N'' ''identical'' particles, of which ''N''<sub>''i''</sub> are in the ''i''th microscopic condition (range) of position and momentum. ''W'' can be counted using the formula for [[Maxwell–Boltzmann statistics#A derivation of the Maxwell–Boltzmann distribution|permutations]]
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| :<math> W = \frac{N!}{\prod_i N_i!} </math>
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| where ''i'' ranges over all possible molecular conditions. (<math>!</math> denotes [[factorial]].) The "correction" in the denominator is because identical particles in the same condition are [[Identical particles|indistinguishable]].
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| Boltzmann was also one of the founders of quantum mechanics due to his suggestion in 1877 that the energy levels of a physical system could be discrete.
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| The equation for ''S'' is engraved on Boltzmann's [[headstone|tombstone]] at the Vienna [[Zentralfriedhof]] — his second grave.
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| ==The Boltzmann equation==
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| [[File:Ludwig Boltzmann at U Vienna.JPG|thumb|Boltzmann's bust in the courtyard arcade of the main building, University of Vienna.]]
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| {{main|Boltzmann equation}}
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| The Boltzmann equation was developed to describe the dynamics of an ideal gas.
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| :<math> \frac{\partial f}{\partial t}+ v \frac{\partial f}{\partial x}+ \frac{F}{m} \frac{\partial f}{\partial v} = \frac{\partial f}{\partial t}\left.{\!\!\frac{}{}}\right|_\mathrm{collision} </math>
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| where ''ƒ'' represents the distribution function of single-particle position and momentum at a given time (see the [[Maxwell–Boltzmann distribution]]), ''F'' is a force, ''m'' is the mass of a particle, ''t'' is the time and ''v'' is an average velocity of particles.
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| This equation describes the [[time|temporal]] and [[space|spatial]] variation of the probability distribution for the position and momentum of a density distribution of a cloud of points in single-particle [[phase space]]. (See [[Hamiltonian mechanics]].) The first term on the left-hand side represents the explicit time variation of the distribution function, while the second term gives the spatial variation, and the third term describes the effect of any force acting on the particles. The right-hand side of the equation represents the effect of collisions.
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| [[File:Zentralfriedhof Vienna - Boltzmann.JPG|thumb|left|Boltzmann's grave in the [[Zentralfriedhof]], Vienna, with bust and entropy formula.]]
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| In principle, the above equation completely describes the dynamics of an ensemble of gas particles, given appropriate [[boundary conditions]]. This first-order [[differential equation]] has a deceptively simple appearance, since ''ƒ'' can represent an arbitrary single-particle distribution function. Also, the [[force]] acting on the particles depends directly on the velocity distribution function ''ƒ''. The Boltzmann equation is notoriously difficult to [[Integral|integrate]]. [[David Hilbert]] spent years trying to solve it without any real success.
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| The form of the collision term assumed by Boltzmann was approximate. However for an ideal gas the standard [[Chapman–Enskog theory|Chapman–Enskog]] solution of the Boltzmann equation is highly accurate. It is expected to lead to incorrect results for an ideal gas only under [[shock wave]] conditions.
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| Boltzmann tried for many years to "prove" the [[second law of thermodynamics]] using his gas-dynamical equation — his famous [[H-theorem]]. However the key assumption he made in formulating the collision term was "[[molecular chaos]]", an assumption which breaks [[CPT symmetry|time-reversal symmetry]] as is necessary for ''anything'' which could imply the second law. It was from the probabilistic assumption alone that Boltzmann's apparent success emanated, so his long dispute with [[Johann Josef Loschmidt|Loschmidt]] and others over [[Loschmidt's paradox]] ultimately ended in his failure.
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| Finally, in the 1970s [[E.G.D. Cohen]] and J.R. Dorfman proved that a systematic (power series) extension of the Boltzmann equation to high densities is mathematically impossible. Consequently [[non-equilibrium statistical mechanics|nonequilibrium statistical mechanics]] for dense gases and [[liquid]]s focuses on the [[Green–Kubo relations]], the [[fluctuation theorem]], and other approaches instead.
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| ==The Second Law as a law of disorder==
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| The idea that the [[second law of thermodynamics]] or "entropy law" is a law of disorder (or that dynamically ordered states are "infinitely improbable") is due to Boltzmann's view of the second law. In particular, it was his attempt to reduce it to a [[stochastic]] collision function, or law of probability following from the random collisions of mechanical particles. Following Maxwell,<ref>Maxwell, J. (1871). Theory of heat. London: Longmans, Green & Co.</ref> Boltzmann modeled gas molecules as colliding billiard balls in a box, noting that with each collision nonequilibrium velocity distributions (groups of molecules moving at the same speed and in the same direction) would become increasingly disordered leading to a final state of macroscopic uniformity and maximum microscopic disorder or the state of maximum entropy (where the macroscopic uniformity corresponds to the obliteration of all field potentials or gradients).<ref>Boltzmann, L. (1974). The second law of thermodynamics. Populare Schriften, Essay 3, address to a formal meeting of the Imperial Academy of Science, 29 May 1886, reprinted in Ludwig Boltzmann, Theoretical
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| physics and philosophical problem, S. G. Brush (Trans.). Boston: Reidel. (Original work published 1886)</ref> The second law, he argued, was thus simply the result of the fact that in a world of mechanically colliding particles disordered states are the most probable. Because there are so many more possible disordered states than ordered ones, a system will almost always be found either in the state of maximum disorder – the macrostate with the greatest number of accessible microstates such as a gas in a box at equilibrium – or moving towards it. A dynamically ordered state, one with molecules moving "at the same speed and in the same direction," Boltzmann concluded, is thus "the most improbable case conceivable...an infinitely improbable configuration of energy."<ref>Boltzmann, L. (1974). The second law of thermodynamics. p. 20</ref>
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| Boltzmann accomplished the feat of showing that the second law of thermodynamics is only a statistical fact. The gradual disordering of energy is analogous to the disordering of an initially ordered [[pack of cards]] under repeated shuffling, and just as the cards will finally return to their original order if shuffled a gigantic number of times, so the entire universe must some day regain, by pure chance, the state from which it first set out. (This optimistic coda to the idea of the dying universe becomes somewhat muted when one attempts to estimate the timeline which will probably elapse before it spontaneously occurs.)<ref>"[[Collier's Encyclopedia]]", Volume 19 Phyfe to Reni, Physics, by David Park, p. 15</ref> The tendency for entropy increase seems to cause difficulty to beginners in thermodynamics, but is easy to understand from the standpoint of the theory of probability. Consider two ordinary [[dice]], with both sixes face up. After the dice are shaken, the chance of finding these two sixes face up is small (1 in 36); thus one can say that the random motion (the agitation) of the dice, like the chaotic collisions of molecules because of thermal energy, causes the less probable state to change to one that is more probable. With millions of dice, like the millions of atoms involved in thermodynamic calculations, the probability of their all being sixes becomes so vanishingly small that the system ''must'' move to one of the more probable states.<ref>"Collier's Encyclopedia", Volume 22 Sylt to Uruguay, Thermodynamics, by Leo Peters, p. 275</ref> However, mathematically the odds of all the dice results not being a pair sixes is also as hard as the ones of all of them being sixes, and since statistically the data tend to balance, one in every 36 pairs of dice will tend to be a pair of sixes. And the cards, when shuffled, will sometimes present a certain temporary sequence order even if in its whole they are disordered.
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| ==Energetics of evolution==
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| Boltzmann's views played an essential role in the development of [[energetics]], the scientific study of energy flows under transformation. In 1922, for example, [[Alfred J. Lotka]] referred to Boltzmann as one of the first proponents of the proposition that available energy can be understood as the fundamental object under contention in the biological, or life-struggle and therefore also in the evolution of the organic world.<ref>[[Maximum power principle#Proposals for maximum power principle as 4th thermodynamic law|Maximum power principle]]</ref> Lotka interpreted Boltzmann's view to imply that available energy could be the central concept that unified physics and biology as a quantitative physical principle of evolution. In the foreword to Boltzmann's ''Theoretical Physics and Philosophical Problems'', S.R. de Groot noted that
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| {{cquote|Boltzmann had a tremendous admiration for Darwin and he wished to extend Darwinism from biological to cultural evolution. In fact he considered biological and cultural evolution as one and the same things. ... In short, cultural evolution was a physical process taking place in the brain. Boltzmann included ethics in the ideas which developed in this fashion ... }}
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| [[Howard T. Odum]] later sought to develop these views when looking at the evolution of ecological systems, and suggested that the [[maximum power principle]] was an example of Darwin's law of [[natural selection]].''
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| ==See also==
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| * [[List of things named after Ludwig Boltzmann]]
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| * [[Equipartition theorem|Boltzmann's Energy Equipartition theorem]]
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| * [[Boltzmann brain]]
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| * [[Boltzmann machine]]
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| * [[History of the molecule]]
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| * [[Lattice Boltzmann methods]], used in [[computational fluid dynamics]]
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| * [[Philosophy of thermal and statistical physics]]
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| * [[Ludwig Boltzmann Gesellschaft]]
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| * [[Boltzmann Medal]]
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| * [[Boltzmann (crater)]]
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| ==References==
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| {{reflist|2}}
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| ==Further reading==
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| * Roman Sexl & John Blackmore (eds.), "Ludwig Boltzmann – Ausgewahlte Abhandlungen", (Ludwig Boltzmann Gesamtausgabe, Band 8), Vieweg, Braunschweig, 1982.
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| * John Blackmore (ed.), "Ludwig Boltzmann – His Later Life and Philosophy, 1900–1906, Book One: A Documentary History", Kluwer, 1995. ISBN 978-0-7923-3231-2
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| * John Blackmore, "Ludwig Boltzmann – His Later Life and Philosophy, 1900–1906, Book Two: The Philosopher", Kluwer, Dordrecht, Netherlands, 1995. ISBN 978-0-7923-3464-4
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| * John Blackmore (ed.), "Ludwig Boltzmann – Troubled Genius as Philosopher", in Synthese, Volume 119, Nos. 1 & 2, 1999, pp. 1–232.
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| * Brush, Stephen G. (ed. & tr.), Boltzmann, ''Lectures on Gas Theory'', Berkeley, CA: U. of California Press, 1964
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| * Brush, Stephen G. (ed.), ''Kinetic Theory'', New York: Pergamon Press, 1965
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| * {{cite isbn|9780198501541}}
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| * [[Boltzmann]], ''Ludwig Boltzmann – Leben und Briefe'', ed., Walter Hoeflechner, Akademische Druck- u. Verlagsanstalt. Graz, Oesterreich, 1994
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| * {{cite book | last=Brush | first=Stephen G. | chapter=Boltzmann | editor=Charles Coulston Gillispie (ed.) | title=Dictionary of Scientific Biography | publisher=Scribner | location=New York | year=1970 | isbn=0-684-16962-2 |series=}}
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| * {{cite book| last=Brush | first=Stephen G. | coauthors= | authorlink= | title=The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases | edition= | publisher=North-Holland | location=Amsterdam | year=1986 | isbn=0-7204-0370-7 | series=}}
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| * {{cite journal | last=Everdell | first=William R | year=1988 | title=The Problem of Continuity and the Origins of Modernism: 1870–1913 | journal=History of European Ideas | volume=9 | issue=5 | pages=531–552 | doi=10.1016/0191-6599(88)90001-0 }}
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| * {{cite book | last=Everdell | first=William R | year=1997 | title=The First Moderns | edition= | publisher=University of Chicago Press | location=Chicago }}
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| * [[Paul Ehrenfest|P. Ehrenfest]] & [[Tatyana Afanasyeva|T. Ehrenfest]] (1911) "Begriffliche Grundlagen der statistischen Auffassung in der Mechanik", in [[Klein's encyclopedia|''Encyklopädie der mathematischen Wissenschaften mit Einschluß ihrer Anwendungen'']] Band IV, 2. Teil ( F. Klein and C. Müller (eds.). Leipzig: Teubner, pp. 3–90. Translated as ''The Conceptual Foundations of the Statistical Approach in Mechanics''. New York: Cornell University Press, 1959. ISBN 0-486-49504-3
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| * {{cite book| last=Klein | first=Martin J. | chapter=The Development of Boltzmann’s Statistical Ideas | editor=[[E.G.D. Cohen]] and W. Thirring (eds) | title=The Boltzmann Equation: Theory and Applications | publisher=Springer | location=Wien | year=1973 | isbn=0-387-81137-0 | pages=53–106 | series=Acta physica Austriaca Suppl. 10 }}
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| * {{cite book | last=Tolman | first=Richard C. | title=The Principles of Statistical Mechanics | publisher=Oxford University Press | year=1938}} Reprinted: Dover (1979). ISBN 0-486-63896-0
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| * {{cite book |last=Gibbs |first=Josiah Willard |authorlink=Josiah Willard Gibbs |title=[[Elementary Principles in Statistical Mechanics|Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics]] |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York}}
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| * {{cite book| last=Lindley | first=David | coauthors= | authorlink=David Lindley (Physicist) | title=Boltzmann's Atom: The Great Debate That Launched A Revolution In Physics | edition= | publisher= Free Press| location= New York| year=2001 | isbn=0-684-85186-5 }}
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| * {{cite journal | last=Lotka | first=A. J. | coauthors= | year=1922 | title=Contribution to the Energetics of Evolution | doi= 10.1073/pnas.8.6.147 | journal=Proc. Natl. Acad. Sci. U.S.A. | volume=8 | issue=6 | pages=147–51 | pmid=16576642 | pmc=1085052 |bibcode = 1922PNAS....8..147L }}
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| * {{cite book | last=Bronowski| first=Jacob |coauthors= | authorlink=Jacob_Bronowski | title=The Ascent Of Man | chapter= World Within World | publisher=Little Brown & Co | year=1974 | isbn=978-0-316-10930-7}}
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| * {{cite book | last= Meyer| first=Stefan |coauthors= | authorlink=Stefan Meyer (physicist) | title=Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage 20. Februar 1904 | publisher=J. A. Barth | year=1904 |language = German }}
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| * {{cite book | last=Planck | first=Max | authorlink=Max Planck | title=The Theory of Heat Radiation | publisher=P. Blakiston Son & Co | year=1914}} English translation by Morton Masius of the 2nd ed. of ''Waermestrahlung''. Reprinted by Dover (1959) & (1991). ISBN 0-486-66811-8
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| ==External links==
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| {{Sister project links|wikt=no|commons=Ludwig Boltzmann|b=no|n=no|q=Ludwig Boltzmann|s=Author:Ludwig Eduard Boltzmann|v=no|species=no|voy=no}}
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| * {{cite web | last=Uffink | first=Jos | title=Boltzmann's Work in Statistical Physics | year=2004 | url=http://plato.stanford.edu/entries/statphys-Boltzmann/ | accessdate=2007-06-11 | work=[[Stanford Encyclopedia of Philosophy]]}}
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| * {{MacTutor Biography|id=Boltzmann}}
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| * "[http://www.dieuniversitaet-online.at/beitraege/news/ludwig-boltzmann-leben-und-werk-zu-besichtigen/10.html Ludwig Boltzmann,]" Universität Wien (German).
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| * Ruth Lewin Sime, ''Lise Meitner: A Life in Physics'' [http://www.washingtonpost.com/wp-srv/style/longterm/books/chap1/lisemeitner.htm Chapter One: Girlhood in Vienna] gives [[Lise Meitner]]'s account of Boltzmann's teaching and career.
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| * [[E.G.D. Cohen]], 1996, "[http://xxx.lanl.gov/abs/cond-mat/9608054 Boltzmann and Statistical Mechanics.]"
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| * Eftekhari, Ali, "[http://philsci-archive.pitt.edu/archive/00001717/02/Ludwig_Boltzmann.pdf Ludwig Boltzmann (1844–1906).]" Discusses Boltzmann's philosophical opinions, with numerous quotes.
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| * {{cite arXiv | last = Rajasekar | first = S. | coauthors = Athavan, N. | title = Ludwig Edward Boltzmann | date = 2006-09-07 | eprint = physics/0609047 | class = physics.hist-ph }}
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| * {{MathGenealogy|13105}}
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| * {{ScienceWorldBiography |urlname=Boltzmann |title=Boltzmann, Ludwig (1844–1906)}}
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| * {{Find a Grave|1518}}
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| * [[Jacob Bronowski]] from "[http://www.youtube.com/watch?v=C2p9By0qXms The Ascent Of Man]"
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| {{Scientists whose names are used in physical constants}}
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| {{Authority control|VIAF=68956918}}
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| <!-- Metadata: see [[Wikipedia:Persondata]] -->
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| {{Persondata
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| |NAME= Boltzmann, Ludwig
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| |ALTERNATIVE NAMES=
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| |SHORT DESCRIPTION= [[Austria]]n [[Physicist]]
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| |DATE OF BIRTH= February 20, 1844
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| |PLACE OF BIRTH=[[Vienna]], [[Austrian Empire]]
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| |DATE OF DEATH= September 5, 1906
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| |PLACE OF DEATH= [[Duino]], Italy
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| }}
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| {{DEFAULTSORT:Boltzmann, Ludwig}}
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| [[Category:1844 births]]
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| [[Category:1906 deaths]]
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| [[Category:Scientists from Vienna]]
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| [[Category:Austrian physicists]]
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| [[Category:Thermodynamicists]]
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| [[Category:Scientists who committed suicide]]
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| [[Category:Mathematicians who committed suicide]]
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| [[Category:Burials at the Zentralfriedhof]]
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| [[Category:University of Vienna alumni]]
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| [[Category:Members of the Royal Swedish Academy of Sciences]]
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| [[Category:Corresponding Members of the St Petersburg Academy of Sciences]]
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| [[Category:People with bipolar disorder]]
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| [[Category:Suicides by hanging in Italy]]
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| [[Category:Foreign Members of the Royal Society]]
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