Wys bron van Ludwig Boltzmann

{{Inligtingskas Persoon
| beeld           = Boltzmann2.jpg
| onderskrif      = Ludwig Boltzmann
| geboortenaam    = Ludwig Eduard Boltzmann
| geboortedatum   = {{geboortedatum|1844|2|20|}}
| geboorteplek    = [[Wene]], [[Oostenryk]]
| sterfdatum      = {{sterfdatum en ouderdom|1906|9|5|1844|2|20|}}
| sterfplek       = [[Tybein]], [[Triest]], [[Oostenryk-Hongarye]]
| nasionaliteit   = Oostenryk
| alma_mater      = [[Universiteit van Wene]]
| bekend_vir = {{Plainlist|
* Boltzmann konstante
* Boltzmann-vergelyking
* Boltzmann distribusie
* Gedetailleerde balans
* H-stelling
* Maxwell–Boltzmann distribusie
* Stefan–Boltzmann konstante
* Stefan–Boltzmann wet
* Maxwell-Boltzmann statistieke
* Boltzmann faktor
* Epistemologiese idealisme
}}
| handtekening  = Ludwig Boltzmann signature.svg
}}
'''Ludwig Eduard Boltzmann''' ([[Duits]]e fonetiese uitspraak: |ˈluːtvɪç ˈbɔlt͡sman; [[20 Februarie]] [[1844]] – [[5 September]] [[1906]]) was 'n [[Oostenryk]]se [[fisikus]] en [[filosoof]]. Sy grootste prestasie was die ontwikkeling van statistiese meganika, wat 'n verduideliking bied van, en voorspel hoe, die eienskappe van [[Atoom|atome]] (soos bv. [[atoommassa]], [[elektriese lading]] en struktuur) die fisiese eienskappe van [[materie]] bepaal (soos bv. die viskositeit, termiese geleidingsvermoë en diffusie of verspreiding daarvan). Boltzmann het die woord [[ergodisiteit]] uitgedink terwyl hy besig was om aan 'n probleem in statistiese meganika te werk.

==Biografie==
{{brei uit}}
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===Childhood and education===
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.

Starting in 1863, Boltzmann studied [[physics]] at the [[University of Vienna]]. 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 the [[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.

===Academic career===
In 1869 at age 25, thanks to a letter of recommendation written by 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=Slovenian |trans-title=Ludwig Boltzmann and the First Student of Physics and Mathematics of Slovene Descent |date=December 2001 |last=Južnič |first=Stanislav |journal=Kvarkadabra.net |issue=12 |accessdate=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|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]]
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 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">{{Cite journal |quote=Paul Ehrenfest (1880–1933) along with Nernst, Arrhenius, and Meitner must be considered among Boltzmann's most outstanding students. |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">{{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 |language=en |archive-url=https://web.archive.org/web/20091226181239/http://chem.ch.huji.ac.il:80/history/nernst.htm |archive-date=26 Desember 2009 |url-status=dead|df=dmy-all}}</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.

===Final years===

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|9780198501541}}</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 [[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. 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.<ref>The Boltzmann Equation: Theory and Applications, E.G.D. Cohen, W. Thirring, ed., Springer Science & Business Media, 2012</ref>

In 1906, Boltzmann's deteriorating mental condition forced him to resign his position.<ref name ="Carlo"/> He died by suicide on September 5, 1906, by hanging himself while on vacation with his wife and daughter in [[Duino]], near [[Trieste]] (then Austria).<ref>"Eureka! Science's greatest thinkers and their key breakthroughs", Hazel Muir, p.152, {{ISBN|1780873255}}</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/?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 [[Boltzmann's entropy formula]]:<ref name ="Carlo"/>

:<math>S = k \cdot \log W </math>

==Philosophy==
{{Unreferenced section|date=December 2018}}
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 chemist [[Wilhelm Ostwald]] disbelieved their existence.<ref>{{cite book | last=Bronowski | first=Jacob | authorlink=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> 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 of thermodynamics]].

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.

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.

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.

==Physics==
Boltzmann's most important scientific contributions were in [[kinetic theory of gases|kinetic theory]], including for motivating the [[Maxwell–Boltzmann distribution]] as a description of molecular speeds in a gas. [[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#Limits of applicability|quantum statistics]] and provide insight into the meaning of [[thermodynamic temperature|temperature]].

[[File:Boltzmanns-molecule.jpg|225px|thumb|right|Boltzmann's 1898 I<sub>2</sub> molecule diagram showing atomic "sensitive region" (α, β) overlap.]]

[[History of chemistry#The dispute about atomism|Most]] [[chemistry|chemists]], 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|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]], 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|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]].[https://archive.is/20070503225307/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>

:<math> S = k_B \ln W </math>

where ''k<sub>B</sub>'' is [[Boltzmann constant|Boltzmann's constant]], and ''ln'' is the [[natural logarithm]]. ''W'' is ''Wahrscheinlichkeit'', a German word  meaning the [[probability theory|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 ''N'' ''identical'' particles, of which ''N''<sub>''i''</sub> are in the ''i''th microscopic condition (range) of position and momentum.  ''W''&nbsp;can be counted using the formula for [[Maxwell–Boltzmann statistics#A derivation of the Maxwell–Boltzmann distribution|permutations]]

:<math> W = N! \prod_i \frac{1}{N_i!} </math>

where ''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 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.

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

The Boltzmann equation was developed to describe the dynamics of an ideal gas.

:<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>

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.

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 ''ƒ'' can represent an arbitrary single-particle distribution function. Also, the [[force]] acting on the particles depends directly on the velocity distribution function&nbsp;''ƒ''. 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> 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{{Citation needed|date=January 2019}}, 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 the deck was disordered.

==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 |url=https://royalsociety.org/about-us/fellowship/fellows/ |publisher=[[Royal Society]] |location=London |title=Fellows of the Royal Society |language=en |archive-url=https://web.archive.org/web/20150626113817/https://royalsociety.org/about-us/fellowship/fellows/ |archive-date=26 Junie 2015 |url-status=live|df=dmy-all}}</ref> [[List of things named after Ludwig Boltzmann|Numerous things]] are named in his honour.
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==Verwysings==
{{Verwysings}}

==Verdere leeswerk==
* Roman Sexl & John Blackmore (eds.), "Ludwig Boltzmann – Ausgewahlte Abhandlungen", (Ludwig Boltzmann Gesamtausgabe, Band 8), Vieweg, [[Brunswyk|Braunschweig]], 1982.
* John Blackmore (red.), "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]], [[Nederland]] 1995. {{ISBN|978-0-7923-3464-4}}
* John Blackmore (red.), "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'', red., Walter Hoeflechner, Akademische Druck- u. Verlagsanstalt. Graz, Oesterreich, 1994
* Brush, Stephen G. (red. & vertaler), Boltzmann, ''Lectures on Gas Theory'', Berkeley, [[Kalifornië]]: U. of California Press, 1964
* Brush, Stephen G. (red.), ''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 |series=}}
* {{cite book| last=Brush | first=Stephen G. | 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=978-0-7204-0370-1 | series=}}
* {{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 = 9780198501541 | 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/?id=APBIDwAAQBAJ&pg=PA376&lpg=PA376 |isbn = 978-0-19-881617-1}}
* Paul Ehrenfest & Ehrenfest, T. (1911) "Begriffliche Grundlagen der statistischen Auffassung in der Mechanik", in ''Encyklopädie der mathematischen Wissenschaften mit Einschluß ihrer Anwendungen'' Band IV, 2. Teil ( F. Klein and C. Müller (reds.). Leipzig: Teubner, pp.&nbsp;3–90. Vertaal 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 | edition= | publisher=University of Chicago Press | location=Chicago }}
* {{cite book |last=Gibbs |first=Josiah Willard |authorlink= |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= |author-link=Josiah Willard Gibbs}}
* {{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 |editor2=W. Thirring | title=The Boltzmann Equation: Theory and Applications | publisher=Springer | location=Wien | year=1973 | isbn=978-0-387-81137-6 | pages=53–106 | series=Acta physica Austriaca Suppl. 10 }}
* {{cite book | last=Lindley | first=David | authorlink= | title=Boltzmann's Atom: The Great Debate That Launched A Revolution In Physics | edition= | 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 }}
* {{cite book | last= Meyer| first=Stefan  | authorlink= | title=Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage 20. Februar 1904  | url= https://archive.org/details/festschriftludw00meyegoog| publisher=J. A. Barth | year=1904  |language = Duits }}
* {{cite book | last=Planck | first=Max | authorlink=Max Planck | title=The Theory of Heat Radiation | url=https://archive.org/details/theoryofheatradi00planrich | publisher=P. Blakiston Son & Co | year=1914}} Engelse vertaling deur Morton Masius van die 2de uitgawe van ''Waermestrahlung''. Heruitgereik deur Dover (1959) & (1991). {{ISBN|0-486-66811-8}}
* {{cite book | last=Tolman | first=Richard C. | title=The Principles of Statistical Mechanics | publisher=Oxford University Press | year=1938}} Heruitgereik: Dover (1979). {{ISBN|0-486-63896-0}}

==Eksterne skakels==
* {{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 | encyclopedia=Stanford Encyclopedia of Philosophy}}
* {{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 Hoofstuk Een: ''Girlhood in Vienna''] gee Lise Meitner se relaas oor Boltzmann se onderrig en loopbaan.
* Eftekhari, Ali, "[http://philsci-archive.pitt.edu/archive/00001717/02/Ludwig_Boltzmann.pdf Ludwig Boltzmann (1844–1906).]" Bespreek Boltzmann se filosofiese opinies, met verskeie aanhalings.
* {{cite arXiv | last = Rajasekar | first = S. |author2=Athavan, N. | title = Ludwig Edward Boltzmann | eprint = physics/0609047 | date = 2006-09-07 }}

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[[Kategorie:Geboortes in 1844]]
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