View source for Ludwig Boltzmann

{{Short description|Austrian theoretical physicist and philosopher (1844–1906)}}
{{Redirect|Boltzmann}}
{{Use dmy dates|date=March 2021}}
{{Infobox scientist
| honorific_suffix  = {{post-nominals|country=GBR|size=100%|ForMemRS}}
| image             = Boltzmann2.jpg
| image_size        = 
| caption           = Boltzmann in 1902
| birth_name        = Ludwig Eduard Boltzmann
| birth_date        = {{birth date|df=y|1844|2|20}}
| birth_place       = [[Vienna]], [[Austrian Empire]], [[German Confederation]]
| death_date        = {{death date and age|1906|9|5|1844|2|20|df=y}}
| death_place       = [[Tybein]], [[Princely County of Gorizia and Gradisca]], [[Austria-Hungary]]
| resting_place     = [[Vienna Central Cemetery]]
| alma_mater        = [[University of Vienna]] ([[doctorate]], 1866; [[Dr. habil.]], 1869)
| known_for         = {{plainlist|
* Formulating the [[Boltzmann distribution]] (1868)
* Formulating [[Boltzmann's entropy formula]] (1872–1875)
* Formulating the [[Boltzmann equation]] (1872)
* Introducing the [[H-theorem|''H''-theorem]] (1872)
* Expanding the [[equipartition theorem]] (1876)
* Describing [[canonical ensemble]] (1884)
* Formulating the [[Stefan–Boltzmann law]] (1884)
}}
| spouse            = {{marriage|Henriette von Aigentler|1876}}
| children          = 4
| awards            = [[Fellow of the Royal Society#Foreign member|ForMemRS]] (1899)
| fields            = [[Theoretical physics]]
| work_institutions = {{plainlist|
* [[University of Graz]] (1869–1873, 1876–1890)
* University of Vienna (1873–1876, 1894–1900, 1902–1906)
* [[University of Munich]] (1890–1894)
* [[University of Leipzig]] (1900–1902)
}}
| thesis_title      = {{lang|de|Über die mechanische Bedeutung des zweiten Hauptsatzes der mechanischen Wärmetheorie}}
| thesis_url        = https://www.digitale-sammlungen.de/en/view/bsb10133426?page=3
| thesis_year       = 1866
| doctoral_advisor  = [[Josef Stefan]]
| academic_advisors = 
| doctoral_students = {{ubl|[[Paul Ehrenfest]]|[[Philipp Frank]]|[[Gustav Herglotz]]|[[Franc Hočevar]]|[[Ignacij Klemenčič]]}}
| notable_students  = [[Lise Meitner]]<br>[[Stefan Meyer (physicist)|Stefan Meyer]]
| signature         = Ludwig Boltzmann signature.svg
}}
{{Thermodynamics}}

'''Ludwig Eduard Boltzmann''' ({{IPAc-en|ˈ|b|ɒ|l|t|s|m|ə|n}},<ref name="OED">{{OED|Boltzmann|6830903157}}</ref> {{IPAc-en|US|ˈ|b|oʊ|l|-|,_|ˈ|b|ɔː|l|-}};<ref name="OED"/><ref>{{cite Merriam-Webster|Boltzmann constant}}</ref> {{IPA|de|ˈluːtvɪk ˈbɔltsman|lang}}; 20 February 1844 – 5 September 1906) was an Austrian [[theoretical physicist]] and [[philosopher]]. His greatest achievements were the development of [[statistical mechanics]] and the statistical explanation of the [[second law of thermodynamics]]. In 1877 he provided the current definition of [[entropy]], <math>S = k_{\rm B} \ln \Omega</math>, where Ω is the number of microstates whose energy equals the system's energy, interpreted as a measure of the statistical disorder of a system.<ref name="EncycloBritan">
{{cite book 
 |last1= Klein |first1= Martin
 |year= 1970
 |orig-year= 1768
 |chapter= Boltzmann, Ludwig
 |editor1-last= Preece |editor1-first= Warren E.
 |title= Encyclopædia Britannica
 |type=hard cover
 |language= en
 |volume= 3
 |edition= Commemorative Edition for Expo 70
 |location= Chicago
 |publisher= William Benton
 |publication-date= 1970
 |page= 893a
 |isbn= 0-85229-135-3
}}
</ref> [[Max Planck]] named the constant {{math|''k''<sub>B</sub>}} the [[Boltzmann constant]].<ref>
{{citation
 | last = Partington | first = J.R. | author-link = J.R. Partington
 | title = An Advanced Treatise on Physical Chemistry
 | volume = 1, ''Fundamental Principles'', ''The Properties of Gases''
 | place = London
 | publisher = [[Longman|Longmans, Green and Co.]]
 | year = 1949
 | page = 300
}}</ref>

Statistical mechanics is one of the pillars of modern [[physics]]. It describes how [[macroscopic]] observations (such as [[temperature]] and [[pressure]]) are related to [[microscopic]] parameters that fluctuate around an average. It connects thermodynamic quantities (such as [[heat capacity]]) to microscopic behavior, whereas, in [[classical thermodynamics]], the only available option would be to measure and tabulate such quantities for various materials.<ref name="gibbs">{{cite book |last=Gibbs |first=Josiah Willard |author-link=Josiah Willard Gibbs |title=Elementary Principles in Statistical Mechanics |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York |title-link=Elementary Principles in Statistical Mechanics }}</ref>

== Biography ==
=== Childhood and education ===
Boltzmann was born in Erdberg, a suburb of [[Vienna]] into a [[Catholic]] family. 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]]. Boltzmann was home-schooled until the age of ten,<ref>
{{cite book
 |title=The Scientific 100
 |first1=John |last1=Simmons
 |first2=Lynda |last2=Simmons
 |isbn=978-0-8065-3678-1
 |page=123
 |publisher=Kensington
 |year=2000
}}</ref> and then attended high school in [[Linz]], [[Upper Austria]]. When Boltzmann was 15, his father died.<ref name=james2004>
{{cite book
 |title=Remarkable Physicists: From Galileo to Yukawa
 |url=https://archive.org/details/remarkablephysic00jame
 |url-access=limited
 |first1=Ioan  |last1=James
 |year=2004
 |isbn=978-0-521-01706-0
 |page=[https://archive.org/details/remarkablephysic00jame/page/n185 169]
 |publisher=Cambridge University Press
}}</ref>

Starting in 1863, Boltzmann studied [[mathematics]] and [[physics]] at the [[University of Vienna]]. He received his doctorate in 1866 and his [[venia legendi]] in 1869. Boltzmann worked closely with [[Josef Stefan]], director of the institute of physics. It was Stefan who introduced Boltzmann to [[James Clerk Maxwell|Maxwell's]] work.<ref name=james2004 />

=== Academic career ===
In 1869 at age 25, thanks to a [[letter of recommendation]] written by [[Josef Stefan]],<ref>{{cite journal |url=http://www.kvarkadabra.net/2001/12/ludwig-boltzmann/ |title=Ludwig Boltzmann in prva študentka fizike in matematike slovenskega rodu |language=sl |trans-title=Ludwig Boltzmann and the First Student of Physics and Mathematics of Slovene Descent |date=December 2001 |last=Južnič |first=Stanislav |journal=Kvarkadabra |issue=12 |access-date=17 February 2012}}</ref> Boltzmann 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 in 1871 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.

[[File:Boltzmann-grp.jpg|thumb|left|upright=1.2|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]]
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 supported her decision to appeal, which was successful. On 17 July 1876 Ludwig Boltzmann married Henriette; they had three daughters: Henriette (1880), Ida (1884) and Else (1891); and a son, Arthur Ludwig (1881).<ref>{{cite web |last1=Fasol |first1=Gerhard |title=Ludwig Boltzmann biography (20 Feb 1844 - 5 Sept 1906) |url=https://www.boltzmann.com/ludwig-boltzmann/biography/ |website=Ludwig Boltzmann. |access-date=20 May 2024}}</ref> 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">{{Cite journal |quote=Paul Ehrenfest (1880–1933) along with Nernst, Arrhenius, and Meitner must be considered among Boltzmann's most outstanding students. |last1=Jäger |first1=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|s2cid=30499879 }}</ref><ref name="huji">{{cite web |url=http://chem.ch.huji.ac.il/history/nernst.htm |title=Walther Hermann Nernst |quote=Walther Hermann Nernst visited lectures by Ludwig Boltzmann |archive-url=https://web.archive.org/web/20080612133921/http://chem.ch.huji.ac.il/history/nernst.htm |archive-date=12 June 2008 }}</ref> He spent 14 happy years in Graz and it was there that he developed his statistical concept of nature.

Boltzmann was appointed to the Chair of Theoretical Physics at the [[University of Munich]] in [[Bavaria]], Germany in 1890.

In 1894, Boltzmann succeeded his teacher [[Joseph Stefan]] as Professor of Theoretical Physics at the University of Vienna.<ref>{{Cite web |title=Ludwig Boltzmann: Viennese Inventor of a new Theoretical Physics |url=https://www.iqoqi-vienna.at/blogs/blog/ludwig-boltzmann-viennese-inventor-of-a-new-theoretical-physics |access-date=2024-10-15 |website=www.iqoqi-vienna.at |language=en-US}}</ref>

=== Final years and death ===

Boltzmann spent a great deal of effort in his final years defending his theories.<ref name ="Carlo">Cercignani, Carlo (1998). ''Ludwig Boltzmann: The Man Who Trusted Atoms''. Oxford University Press. {{ISBN|978-0-19-850154-1}}</ref> 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 [[energeticism|energetics]] at a meeting in [[Lübeck]]. 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|issue=1|year=1896|pages=72–78|doi=10.1002/andp.18962930107 |bibcode = 1896AnP...293...72P |url=https://zenodo.org/record/1423910}}</ref> In 1900, Boltzmann went to the [[University of Leipzig]], on the invitation of [[Wilhelm Ostwald]]. Ostwald offered Boltzmann the professorial chair in physics, which became vacant when [[Gustav Heinrich Wiedemann]] died. After Mach retired due to bad health, Boltzmann returned to Vienna in 1902.<ref name ="Carlo"/> In 1903, Boltzmann, together with [[Gustav von Escherich]] and [[Emil Müller (mathematician)|Emil Müller]], founded the [[Austrian Mathematical Society]]. His students included [[Karl Přibram]], [[Paul Ehrenfest]] and [[Lise Meitner]].<ref name ="Carlo"/>

In Vienna, Boltzmann taught physics and also lectured on philosophy. Boltzmann's lectures on [[natural philosophy]] were very popular and received considerable attention. His first lecture was an enormous success: people stood all the way down the staircase outside the largest available lecture hall, and the Emperor invited him to a reception{{when|date=May 2022}}.<ref>''The Boltzmann Equation: Theory and Applications'', E. G. D. Cohen, W. Thirring, ed., Springer Science & Business Media, 2012</ref>

In 1905, he gave an invited course of lectures in the summer session at the [[University of California in Berkeley]], which he described in a popular essay ''A German professor's trip to El Dorado''.<ref>{{Cite journal |last=Boltzmann |first=Ludwig |date=1992-01-01 |title=A German Professor's trip to El Dorado |url=https://ui.adsabs.harvard.edu/abs/1992PhT....45a..44B |journal=Physics Today |volume=45 |issue=1 |pages=44–51 |doi=10.1063/1.881339 |bibcode=1992PhT....45a..44B |issn=0031-9228}}</ref>

In May 1906, Boltzmann's deteriorating mental condition (described in a letter by the Dean as "a serious form of [[neurasthenia]]") forced him to resign his position. His symptoms indicate he experienced what might today be diagnosed as [[bipolar disorder]].<ref name="Carlo" /><ref name="Paperpile">{{cite web | last = Nina Bausek and Stefan Washietl | title = Tragic deaths in science: Ludwig Boltzmann – a mind in disorder | publisher = [[Paperpile]] | date = 13 February 2018 | url = https://paperpile.com/blog/ludwig-boltzmann| access-date = 26 April 2020 }}</ref> Four months later he died by suicide on 5 September 1906, by [[hanging]] himself while on vacation with his wife and daughter in [[Duino]], near [[Trieste]] (then Austria).<ref>Muir, Hazel, ''Eureka! Science's greatest thinkers and their key breakthroughs'', p.152, {{ISBN|1-78087-325-5}}</ref><ref>{{cite book|last=Boltzmann|first=Ludwig|editor1-first=John T.|editor1-last=Blackmore|title=Ludwig Boltzmann: His Later Life and Philosophy, 1900–1906|chapter-url=https://books.google.com/books?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><ref name="Paperpile" />
He is buried in the Viennese [[Zentralfriedhof]]. His tombstone bears the inscription of [[Boltzmann's entropy formula]]: <math>S = k \cdot \log W </math>.<ref name="Carlo" />

== Philosophy ==
Boltzmann's [[kinetic theory of gases]] seemed to presuppose the reality of [[atom]]s and [[molecule]]s, but almost all [[German philosophy|German philosophers]] and many scientists like [[Ernst Mach]] and the [[physical chemistry|physical chemist]] [[Wilhelm Ostwald]] disbelieved their existence.<ref>{{cite book | last=Bronowski | first=Jacob | author-link=Jacob Bronowski | title=The Ascent Of Man | chapter=World Within World | publisher=Little Brown & Co | year=1974 | isbn=978-0-316-10930-7 | page=265 | chapter-url=https://archive.org/details/ascentofmanbron00bron }}</ref> Boltzmann had been exposed to molecular theory by [[James Clerk Maxwell]]’s paper, "Illustrations of the Dynamical Theory of Gases," which described temperature as dependent on the speed of the molecules. This inspired Boltzmann to embrace atomism, introducing statistics into physics and extending the theory.<ref>Nancy Forbes, Basil Mahon (2019). ''Faraday, Maxwell, and the Electromagnetic Field''. Chapter 11. {{ISBN|978-1633886070}}.{{full citation needed|reason=publisher?|date=October 2023}}</ref>

Boltzmann wrote treatises on philosophy such as "On the question of the objective existence of processes in inanimate nature" (1897). He was a [[Philosophical realism|realist.]]<ref>Cercignani, Carlo. ''Ludwig Boltzmann: The Man Who Trusted Atoms''. {{ISBN|978-0198570646}}.{{full citation needed|reason=publisher? year?|date=October 2023}}</ref> In his work "On Thesis of Schopenhauer's", Boltzmann refers to his philosophy as [[materialism]] and says further: "Idealism asserts that only the ego exists, the various ideas, and seeks to explain matter from them. Materialism starts from the existence of matter and seeks to explain sensations from it."<ref>{{Cite book |last=Cercignani |first=Carlo |title=Ludwig Boltzmann: the man who trusted atoms |date=2008 |publisher=Oxford Univ. Press |isbn=978-0-19-850154-1 |edition=Repr |location=Oxford |page=176}}</ref>

== Physics ==
Boltzmann's most important scientific contributions were in the [[kinetic theory of gases]] based upon the [[Second law of thermodynamics]]. This was important because Newtonian mechanics did not differentiate between past and future [[Newton's laws of motion|motion]], but [[Rudolf Clausius]]’ invention of entropy to describe the second law was based on [[disgregation]] or dispersion at the molecular level so that the future was one-directional. Boltzmann was twenty-five years of age when he came upon [[James Clerk Maxwell]]'s work on the kinetic theory of gases which hypothesized that [[temperature]] was caused by collision of molecules. Maxwell used statistics to create a curve of molecular kinetic energy distribution from which Boltzmann clarified and developed the ideas of kinetic theory and entropy based upon statistical atomic theory creating the [[Maxwell–Boltzmann distribution]] as a description of molecular speeds in a gas.<ref>Ludwig Boltzmann, ''Lectures on the Theory of Gases'', translated by Stephen G. Brush, "Translator's Introduction", 1968.</ref> It was Boltzmann who derived the first equation to model the dynamic evolution of the probability distribution Maxwell and he had created.<ref>Penrose, Roger. "Foreword". In Cercignani, Carlo, ''Ludwig Boltzmann: The Man Who Trusted Atoms'', {{ISBN|978-0198570646}}.</ref> Boltzmann's key insight was that dispersion occurred due to the statistical probability of increased molecular "states". Boltzmann went beyond Maxwell by applying his distribution equation to not solely gases, but also liquids and solids. Boltzmann also extended his theory in his 1877 paper beyond Carnot, [[Rudolf Clausius]], [[James Clerk Maxwell]] and [[Lord Kelvin]] by demonstrating that entropy is contributed to by heat, spatial separation, and radiation.<ref>Boltzmann, Ludwig (1877). Translated by Sharp, K.; Matschinsky, F. "On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium". ''Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe''. Part II, LXXVI. 76:373–435. Vienna. Reprinted in ''Wissenschaftliche Abhandlungen'', Vol. II, reprint 42, pp. 164–223, Barth, Leipzig, 1909. ''Entropy'' 2015, 17, 1971–2009. {{doi|10.3390/e17041971}}</ref> [[Maxwell–Boltzmann statistics]] and the [[Boltzmann distribution]] remain central in the foundations of [[classical mechanics|classical]] statistical mechanics. They are also applicable to other [[phenomenon|phenomena]] that do not require [[Maxwell–Boltzmann statistics#Applicability|quantum statistics]] and provide insight into the meaning of [[thermodynamic temperature|temperature]].

He made multiple attempts to explain the second law of thermodynamics, with the attempts ranging over many areas. He tried [[Hermann von Helmholtz|Helmholtz]]'s monocycle model,<ref>{{Citation |last=Príncipe |first=João |title=Henri Poincaré: The Status of Mechanical Explanations and the Foundations of Statistical Mechanics |date=2014 |work=Poincaré, Philosopher of Science: Problems and Perspectives |series=The Western Ontario Series in Philosophy of Science |volume=79 |pages=127–151 |editor-last=de Paz |editor-first=María |url=https://doi.org/10.1007/978-94-017-8780-2_8 |access-date=2024-05-28 |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-94-017-8780-2_8 |isbn=978-94-017-8780-2 |editor2-last=DiSalle |editor2-first=Robert|hdl=10174/13352 |hdl-access=free }}</ref><ref>{{Citation |last=Klein |first=Martin J. |title=Boltzmann, Monocycles and Mechanical Explanation |date=1974 |work=Philosophical Foundations of Science |series=Boston Studies in the Philosophy of Science |volume=11 |pages=155–175 |editor-last=Seeger |editor-first=Raymond J. |url=http://link.springer.com/10.1007/978-94-010-2126-5_8 |access-date=2024-05-28 |place=Dordrecht |publisher=Springer Netherlands |doi=10.1007/978-94-010-2126-5_8 |isbn=978-90-277-0376-7 |editor2-last=Cohen |editor2-first=Robert S.}}</ref> a pure ensemble approach like Gibbs, a pure mechanical approach like ergodic theory, the combinatorial argument, the ''[[Molecular chaos|Stoßzahlansatz]]'', etc.<ref>{{Citation |last=Uffink |first=Jos |title=Boltzmann's Work in Statistical Physics |date=2022 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/sum2022/entries/statphys-Boltzmann/ |access-date=2024-05-28 |edition=Summer 2022 |publisher=Metaphysics Research Lab, Stanford University}}</ref>

[[File:Boltzmanns-molecule.jpg|thumb|right|Boltzmann's 1898 I<sub>2</sub> molecule diagram showing atomic "sensitive region" (α, β) overlap]]

[[History of chemistry#19th century|Most chemist]]s, since the discoveries of [[John Dalton]] in 1808, and [[James Clerk Maxwell]] in Scotland and [[Josiah Willard Gibbs]] in the United States, shared Boltzmann's belief in [[atom]]s and [[molecule]]s, but much of the [[physics]] establishment did not share this belief until decades later. Boltzmann 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#Thermodynamic fluctuations and statistical physics|Einstein's theoretical studies]] of 1905, confirmed the values of the [[Avogadro constant]] and the [[Boltzmann constant]], convincing the world that the tiny particles [[Atomic theory#History|really exist]].

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> 
<math display="block"> S = k_{\mathrm{B}} \ln W </math>
where {{mvar|k}}{{sub|B}} is the [[Boltzmann constant]], and ln is the [[natural logarithm]]. {{mvar|W}} (for {{lang|la|Wahrscheinlichkeit}}, a German word meaning "[[probability theory|probability]]") is the probability of occurrence of a [[macrostate]]<ref>{{cite book|last=Pauli| first=Wolfgang| title=Statistical Mechanics|publisher=MIT Press|location=Cambridge|year=1973|isbn=978-0-262-66035-8}}, p. 21</ref> or, more precisely, the number of possible [[microstate (statistical mechanics)|microstates]] corresponding to the macroscopic state of a system – the number of (unobservable) "ways" in the (observable) [[thermodynamics|thermodynamic]] state of a system that 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 {{mvar|N}} ''identical'' particles, of which {{math|{{var|N}}{{sub|{{var|i}} }} }} are in the {{mvar|i}}th microscopic condition (range) of position and momentum. {{mvar|W}}&nbsp;can be counted using the formula for [[Maxwell–Boltzmann statistics#Derivations|permutations]]
<math display="block"> W = N! \prod_i \frac{1}{N_i!} </math>
where {{mvar|i}} ranges over all possible molecular conditions, and where <math>!</math> denotes [[factorial]]. The "correction" in the denominator account for [[Identical particles|indistinguishable]] particles in the same condition.

Boltzmann could also be considered one of the forerunners of quantum mechanics due to his suggestion in 1877 that the energy levels of a physical system could be discrete, although Boltzmann used this as a mathematical device with no physical meaning.<ref>Boltzmann, Ludwig (1877). Translated by Sharp, K.; Matschinsky, F. [https://www.mdpi.com/1099-4300/17/4/1971 "On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium"]. ''Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe''. Part II, LXXVI, 76:373-435. Vienna. Reprinted in {{lang|de|Wissenschaftliche Abhandlungen}}, Vol. II, reprint 42, p. 164–223, Barth, Leipzig, 1909. ''Entropy'' 2015, 17, 1971–2009. {{doi|10.3390/e17041971}}.</ref>

An alternative to Boltzmann's formula for entropy, above, is the [[Information entropy#Formal definitions|information entropy]] definition introduced in 1948 by [[Claude Elwood Shannon|Claude Shannon]].<ref>{{cite web |title=A Mathematical Theory of Communication by Claude E. Shannon |url=http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html |website=cm.bell-labs.com |archive-url=https://archive.today/20070503225307/http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html |archive-date=3 May 2007}}</ref> Shannon's definition 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).

== Boltzmann equation ==
{{main|Boltzmann equation}}
[[File:Ludwig Boltzmann at U Vienna.JPG|thumb|Boltzmann's bust in the courtyard arcade of the main building, University of Vienna]]

The Boltzmann equation was developed to describe the dynamics of an ideal gas.
<math display="block"> \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>
where {{mvar|ƒ}} represents the distribution function of single-particle position and momentum at a given time (see the [[Maxwell–Boltzmann distribution]]), {{mvar|F}} is a force, {{mvar|m}} is the mass of a particle, {{mvar|t}} is the time and {{mvar|v}} is an average velocity of particles.

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.

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 {{mvar|f}} can represent an arbitrary single-particle [[Distribution function (physics)|distribution function]]. Also, the [[force]] acting on the particles depends directly on the velocity distribution function&nbsp;{{mvar|f}}. The Boltzmann equation is notoriously difficult to [[Integral|integrate]]. [[David Hilbert]] spent years trying to solve it without any real success.

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.

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.

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 liquids focuses on the [[Green–Kubo relations]], the [[fluctuation theorem]], and other approaches instead.

== Second thermodynamics law as a law of disorder ==
[[File:Zentralfriedhof Vienna - Boltzmann.JPG|thumb|right|Boltzmann's grave in the [[Zentralfriedhof]], Vienna, with bust and entropy formula]]
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 of thermodynamics.

In particular, it was Boltzmann's 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 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>

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>

== Legacy and impact on modern science ==
Ludwig Boltzmann's contributions to physics and philosophy have left a lasting impact on modern science. His pioneering work in statistical mechanics and thermodynamics laid the foundation for some of the most fundamental concepts in physics. For instance, [[Max Planck]] in quantizing resonators in his [[Black Body]] theory of radiation used the [[Boltzmann constant]] to describe the entropy of the system to arrive at his formula in 1900.<ref>A. Douglas Stone, “Einstein and the Quantum “, Chapter 1 “An Act of Desperation.” 2015.</ref> However, Boltzmann's work was not always readily accepted during his lifetime, and he faced opposition from some of his contemporaries, particularly in regard to the existence of atoms and molecules. Nevertheless, the validity and importance of his ideas were eventually recognized, and they have since become cornerstones of modern physics. Here, we delve into some aspects of Boltzmann's legacy and his influence on various areas of science.

=== Atomic theory and the existence of atoms and molecules ===
Boltzmann's kinetic theory of gases was one of the first attempts to explain macroscopic properties, such as pressure and temperature, in terms of the behaviour of individual atoms and molecules. Although many chemists were already accepting the existence of atoms and molecules, the broader physics community took some time to embrace this view. Boltzmann's long-running dispute with the editor of a prominent German physics journal over the acceptance of atoms and molecules underscores the initial resistance to this idea.

It was only after experiments, such as Jean Perrin's studies of colloidal suspensions, confirmed the values of the Avogadro constant and the Boltzmann constant that the existence of atoms and molecules gained wider acceptance. Boltzmann's kinetic theory played a crucial role in demonstrating the reality of atoms and molecules and explaining various phenomena in gases, liquids, and solids.

=== Statistical mechanics and the Boltzmann constant ===
Statistical mechanics, which Boltzmann pioneered, connects macroscopic observations with microscopic behaviors. His statistical explanation of the second law of thermodynamics was a significant achievement, and he provided the current definition of entropy (<math>S = k_{\rm B} \ln \Omega</math>), where {{math|''k''<sub>B</sub> }} is the Boltzmann constant and Ω is the number of microstates corresponding to a given macrostate.

Max Planck later named the constant {{math|''k''<sub>B</sub>}} as the Boltzmann constant in honor of Boltzmann's contributions to statistical mechanics. The Boltzmann constant is now a fundamental constant in physics and across many scientific disciplines.

=== Boltzmann equation and modern uses ===
Because the [[Boltzmann equation]] is practical in solving problems in rarefied or dilute gases, it has been used in many diverse areas of technology. It is used to calculate [[Space Shuttle]] re-entry in the upper atmosphere.<ref>Neunzert, H., Gropengießer, F., Struckmeier, J. (1991). Computational Methods for the Boltzmann Equation. In: Spigler, R. (eds) Applied and Industrial Mathematics. Mathematics and Its Applications, vol 56. Springer, Dordrecht. {{doi|10.1007/978-94-009-1908-2_10}}</ref> It is the basis for [[Neutron transport]] theory, and ion transport in [[Semiconductors]].<ref>Advanced Theory of Semiconductors and Semiconductor Devices Numerical Methods and Simulation / Umberto Ravaioli http://transport.ece.illinois.edu/ECE539S12-Lectures/Chapter2-DriftDiffusionModels.pdf</ref><ref>AN OVERVIEW OF THE BOLTZMANN TRANSPORT EQUATION SOLUTION FOR NEUTRONS, PHOTONS AND ELECTRONS IN CARTESIAN GEOMETRY, 
Ba ́rbara D. do Amaral Rodriguez, Marco Tu ́llio Vilhena, 2009 International Nuclear Atlantic Conference - INAC 2009 Rio de Janeiro, RJ, Brazil, September 27 to October 2, 2009 ASSOCIAC ̧A ̃OBRASILEIRADEENERGIANUCLEAR-ABEN {{ISBN|978-85-99141-03-8}}</ref>

== Influence on quantum mechanics ==
{{unreferenced section|date=July 2023}}
Boltzmann's work in statistical mechanics laid the groundwork for understanding the statistical behavior of particles in systems with a large number of degrees of freedom. In his 1877 paper, he used discrete energy levels of physical systems as a mathematical device and went on to show that the same approach could be applied to continuous systems. This might be seen as a forerunner to the development of quantum mechanics.<ref>Sharp, K.; Matschinsky, F. Translation of Ludwig Boltzmann’s Paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium” Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI 1877, pp 373-435 (Wien. Ber. 1877, 76:373-435). Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, p. 164-223, Barth, Leipzig, 1909. Entropy 2015, 17, 1971-2009. https://doi.org/10.3390/e17041971 https://www.mdpi.com/1099-4300/17/4/1971</ref> One biographer of Boltzmann says that Boltzmann’s approach “pav[ed] the way for Planck.”<ref>Carlo Cercignani, “Ludwig Boltzmann: The Man Who Trusted Atoms,” Chap. 12.3 Black-Body Radiation, 2006, {{ISBN|978-0198570646}}.</ref>

Quantization of energy levels became a fundamental postulate in quantum mechanics, leading to groundbreaking theories like [[quantum electrodynamics]] and [[quantum field theory]]. Thus, Boltzmann's early insights into the quantization of energy levels had a profound influence on the development of quantum physics.

== Works ==
* {{cite book|title=Verhältniss zur Fernwirkungstheorie, Specielle Fälle der Elektrostatik, stationären Strömung und Induction|volume=2|publisher=Johann Ambrosius Barth|location=Leipzig|year=1893|language=de|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=6713304}}
* {{cite book|title=Theorie van der Waals, Gase mit zusammengesetzten Molekülen, Gasdissociation, Schlussbemerkungen|volume=2|publisher=Johann Ambrosius Barth|location=Leipzig|year=1896|language=de|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=10946008}}
* {{cite book|title=Theorie der Gase mit einatomigen Molekülen, deren Dimensionen gegen die mittlere Weglänge verschwinden|volume=1|publisher=Johann Ambrosius Barth|location=Leipzig|year=1896|language=de|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=10990650}}
* {{cite book|title=Abteilung der Grundgleichungen für ruhende, homogene, isotrope Körper |volume=1|publisher=Johann Ambrosius Barth|location=Leipzig|year=1908|language=de|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=6713871}}
* {{cite book|title=Vorlesungen über Gastheorie|volume=|publisher=Gauthier-Villars|location=Paris|year=1922|language=fr|url=https://gutenberg.beic.it/webclient/DeliveryManager?pid=6714366}}
<gallery>
File:Boltzmann-1.jpg|Volumes I and II of ''Vorlesungen über Gastheorie'' (1896-1898)
File:Boltzmann-2.jpg|Title page to volumes I and II of ''Vorlesungen über Gastheorie'' (1896-1898)
File:Boltzmann-5.jpg|Table of contents to volumes I and II of ''Vorlesungen über Gastheorie'' (1896-1898)
File:Boltzmann-7.jpg|Introduction to volumes I and II of ''Vorlesungen über Gastheorie'' (1896-1898)
</gallery>

== Awards and honours ==
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 and a [[List of Fellows of the Royal Society elected in 1899|Foreign Member of the Royal Society (ForMemRS) in 1899]].<ref name=frs>{{cite web|archive-url=https://web.archive.org/web/20150316060617/https://royalsociety.org/about-us/fellowship/fellows/|archive-date=16 March 2015|url=https://royalsociety.org/about-us/fellowship/fellows/|publisher=[[Royal Society]]|location=London|title=Fellows of the Royal Society}}</ref> [[List of things named after Ludwig Boltzmann|Numerous things]] are named in his honour.

== References ==
{{reflist|30em}}

== Further reading ==
* Roman Sexl & John Blackmore (eds.), "Ludwig Boltzmann – Ausgewahlte Abhandlungen", (Ludwig Boltzmann Gesamtausgabe, Band 8), Vieweg, Braunschweig, 1982.
* 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}}
* 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}}
* John Blackmore (ed.), "Ludwig Boltzmann – Troubled Genius as Philosopher", in Synthese, Volume 119, Nos. 1 & 2, 1999, pp.&nbsp;1–232.
* {{cite book|last1=Blundell|first1=Stephen|last2=Blundell|first2=Katherine M.|title=Concepts in Thermal Physics|url=https://books.google.com/books?id=vuBHXwAACAAJ|year=2006|publisher=Oxford University Press|isbn=978-0-19-856769-1|page=29}}
* Boltzmann, ''Ludwig Boltzmann – Leben und Briefe'', ed., Walter Hoeflechner, Akademische Druck- u. Verlagsanstalt. Graz, Oesterreich, 1994
* Brush, Stephen G. (ed. & tr.), Boltzmann, ''Lectures on Gas Theory'', Berkeley, California: U. of California Press, 1964
* Brush, Stephen G. (ed.), ''Kinetic Theory'', New York: Pergamon Press, 1965
* {{cite book | last=Brush | first=Stephen G. | chapter=Boltzmann | editor=Charles Coulston Gillispie | title=Dictionary of Scientific Biography | chapter-url=https://archive.org/details/dictionaryofsci001gill | chapter-url-access=registration | publisher=Scribner | location=New York | year=1970 | isbn=978-0-684-16962-0 }}
* {{cite book| last=Brush | first=Stephen G. | title=The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases | publisher=North-Holland | location=Amsterdam | year=1986 | isbn=978-0-7204-0370-1 }}
* {{cite book | last = Cercignani | first = Carlo | author-link = Carlo Cercignani | title = Ludwig Boltzmann: The Man Who Trusted Atoms | publisher = Oxford University Press | year = 1998 | isbn = 978-0-19-850154-1 | url-access = registration | url = https://archive.org/details/ludwigboltzmannm0000cerc }}
* {{Cite book |last=Darrigol |first = Olivier |title = Atoms, Mechanics, and Probability: Ludwig Boltzmann's Statistico-Mechanical |publisher = [[Oxford University Press]] |year = 2018 |url = https://books.google.com/books?id=APBIDwAAQBAJ&pg=PA376 |isbn = 978-0-19-881617-1}}
* [[Paul Ehrenfest|Ehrenfest, P.]] & [[Tatyana Afanasyeva|Ehrenfest, T.]] (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.&nbsp;3–90. Translated as ''The Conceptual Foundations of the Statistical Approach in Mechanics''. New York: Cornell University Press, 1959. {{ISBN|0-486-49504-3}}
* {{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 }}
* {{cite book | last=Everdell | first=William R | year=1997 | title=The First Moderns | url=https://archive.org/details/firstmodernsprof00ever | url-access=registration | publisher=University of Chicago Press | location=Chicago | isbn=9780226224800 }}
* {{cite book |last=Gibbs |first=Josiah Willard |author-link=Josiah Willard Gibbs |title=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|title-link=Elementary Principles in Statistical Mechanics }}
* {{cite book | last = Johnson | first = Eric | title = ''Anxiety and the Equation: Understanding Boltzmann's Entropy'' | publisher = The MIT Press | year = 2018 | isbn = 978-0-262-03861-4}}
* {{cite book| last=Klein | first=Martin J. | chapter=The Development of Boltzmann's Statistical Ideas | editor=E.G.D. Cohen | editor-link=E.G.D. Cohen |editor2=W. Thirring | title=The Boltzmann Equation: Theory and Applications | url=https://archive.org/details/boltzmannequatio00schm | url-access=limited | publisher=Springer | location=Wien | year=1973 | isbn=978-0-387-81137-6 | pages=[https://archive.org/details/boltzmannequatio00schm/page/n65 53]–106 | series=Acta physica Austriaca Suppl. 10 }}
* {{cite book | last=Lindley | first=David | author-link=David Lindley (Physicist) | title=Boltzmann's Atom: The Great Debate That Launched A Revolution In Physics | publisher=Free Press | location=New York | year=2001 | isbn=978-0-684-85186-0 | url=https://archive.org/details/isbn_9780684851860 }}
* {{cite journal | last=Lotka | first=A. J. | 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 | doi-access=free }}
* {{cite book | last= Meyer| first=Stefan  | author-link=Stefan Meyer (physicist) | title=Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage 20. Februar 1904  | url= https://archive.org/details/festschriftludw00meyegoog| publisher=J. A. Barth | year=1904  |language = de }}
* {{cite book | last=Planck | first=Max | author-link=Max Planck | title=The Theory of Heat Radiation | url=https://archive.org/details/theoryofheatradi00planrich | 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}}
* Sharp, Kim (2019). ''Entropy and the Tao of Counting: A Brief Introduction to Statistical Mechanics and the Second Law of Thermodynamics'' (SpringerBriefs in Physics). Springer Nature. {{ISBN|978-3030354596}}
* {{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}}

== External links ==
{{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}}
* {{cite encyclopedia | last=Uffink | first=Jos | title=Boltzmann's Work in Statistical Physics | year=2004 | url=http://plato.stanford.edu/entries/statphys-Boltzmann/ | access-date=11 June 2007 | encyclopedia=[[Stanford Encyclopedia of Philosophy]]}}
*[https://www.youtube.com/watch?v=hENRIAx-6D4 Ludwig Boltzmann - The genius of disorder (Youtube)]
* {{MacTutor Biography|id=Boltzmann}}
* [[Ruth Lewin Sime]], ''Lise Meitner: A Life in Physics'' [https://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.
* 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.
* {{cite arXiv | last = Rajasekar | first = S. |author2=Athavan, N. | title = Ludwig Edward Boltzmann | eprint = physics/0609047 | date = 7 September 2006 }}
* {{MathGenealogy|13105}}
* {{ScienceWorldBiography |urlname=Boltzmann |title=Boltzmann, Ludwig (1844–1906)}}

{{Scientists whose names are used in physical constants}}

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