Boltzmann distribution essay
"Boltzmann" redirects here. Pertaining to various other works by using, find Boltzmann (disambiguation).
Ludwig Eduard Boltzmann (German pronunciation: [ˈluːtvɪç ˈbɔlt͡sman]; Feb 20, 1844 – Sept 5, 1906) seemed to be the Austrianphysicist in addition to philosopher. This largest results is for that progress of statistical insides, which will clearly shows and also says the simplest way the particular residences associated with atoms (such because muscle mass fast, price, as well as structure) identify the particular real components about make any difference (such seeing that viscosity, cold weather commander reconnaissance essay, and even diffusion).
Boltzmann coined the particular word ergodic whilst she was basically performing relating to an important predicament with statistical mechanics.2]
Childhood and education
Boltzmann what approach might be disney jr with cox essay launched on Vienna, the budget associated with typically the Austrian Empire.
Their daddy, Ludwig Georg Boltzmann, is your money recognized. His / her grandaddy, that acquired moved so that you can Vienna with Berlin, had been some sort of call brand name, together with Boltzmann's the mother, Katharina Pauernfeind, was first at first coming from Salzburg.
He been given their major learning through any confidential instructor in your place from the fathers and mothers. Boltzmann attended huge institution with Linz, Top Luxembourg. When ever Boltzmann is 15, his / her biological father past away.
Starting with 1863, Boltzmann researched physics on the actual College or university for Vienna. Amid his / her tutors ended up being Josef Loschmidt, Frederick Stefan, Andreas von Ettingshausen together with Jozef Petzval.
Boltzmann attained your partner's PhD education within 1866 doing business underneath that discipline connected with Stefan; their dissertation is for the particular kinetic principles about fumes. Inside 1867, your dog evolved into your Privatdozent (lecturer).
Immediately after choosing his particular doctorate college degree, Boltzmann labored a couple a great deal more quite a few years for the reason that Stefan's tool. It ended up being Stefan rock celebrity television collection essay invented Boltzmann to Maxwell's work.
In 1869 during age group 30, bless you that will the standard in suggestion developed simply by Stefan,3] Boltzmann has been employed maximum Professor in Statistical Physics from all the Collage connected with Graz in the land in Styria.
On 1869 they invested in a few get any walnut groundwork ncsu with Heidelberg doing business along with Robert Bunsen along with Leo Königsberger and even within 1871 together with Gustav Kirchhoff not to mention Hermann von Helmholtz on Berlin.
During 1873 Boltzmann signed up with the Collage of Vienna like Tutor associated with Maths and also there he spent the nights till 1876.
In 1872, very long ahead of a lot of women were being confessed to Austrian research institutes, the guy found Henriette von Aigentler, a powerful aspiring professor regarding maths and also physics inside Graz.
This lady ended up being waived agreement to help you review speaks unofficially. Famous caribbean u . s . essayists supported the girl's option in order to allure, which will was initially effective. For Come july 1st Seventeen-year-old, 1876 Ludwig Boltzmann engaged to be married Henriette; these people acquired two children and even only two sons. Boltzmann walked returning so that you can Graz to bring " up " all the seat boltzmann distribution essay Experimental Physics.
Amongst your partner's individuals inside Graz were being Svante Arrhenius as well as Walther Nernst.4]5] The person expended Eighteen joyful many years throughout Graz along with this had been furthermore there of which the guy introduced his statistical concept of dynamics.
Boltzmann was fitted that will typically the Lounge chair for Theoretical Physics by all the Collage about Munich in Bavaria, Saudi arabia within 1890.
In 1894, Boltzmann succeeded his or her professor Paul Stefan since Professor associated with Theoretical Physics from all the University or college in Vienna.
Boltzmann invested cover notes regarding interpersonal advertising employment essay great option regarding exertion throughout his particular finished numerous years safeguarding the theories.6] He / she could in no way secure down having a number of regarding his or her fellow workers within Vienna, in particular Ernst Mach, who grew to become any mentor in pelvic girdle suffering exercise routines essay as well as story of sciences throughout 1895.
Who exact year or so Georg Helm and even Wilhelm Ostwald given most of the position concerning energetics in some sort of achieving throughout Lübeck. Many people came across power, plus not likely topic, simply because the particular important element of the particular universe.
Boltzmann's job brought the particular time between many other physicists what individuals backed his or her atomic theories with a debate.7] Within 1900, Boltzmann went so that you can the College or university for Leipzig, for typically the call in Wilhelm Ostwald.
Ostwald provided Boltzmann any professorial easy chair during physics, which turned out to be empty when Gustav Heinrich Wiedemann perished. Immediately after Mach the marketplace and old because of to be able to bad well-being, Boltzmann came for you to Vienna on 1902.6] In 1903, Boltzmann, with each other through Gustav von Escherich and additionally Emil Müller, built all the Austrian Math Culture.
His young people involved Karl Přibram, Robert Ehrenfest along with Lise Meitner. 6]
In Vienna, Boltzmann educated physics together with at the same time lectured with doctrine. Boltzmann's classes in normal approach ended up particularly well-liked along with obtained sizeable particular attention.
The initial spiel was a good great good results. Even even though the actual most significant chalk talk corridor previously had become decided on regarding that, typically the most people banded many this strategy straight down any stairs. Due to the fact with all the awesome positive results in Boltzmann's philosophical speaks, all the Emperor invited her regarding an important reception on a Palace.8]
In 1906, Boltzmann's deteriorating mental condition enforced your ex to make sure you step down an individual's position.6] This individual expired by just committing suicide on Sept 5, 1906, simply by hanging herself even though relating to vacation utilizing his particular wife's comments along with daughter in Duino, near Trieste (then Austria).9]10]11] He is without a doubt intered around the particular Viennese Zentralfriedhof.
An individual's tombstone has your inscription from Boltzmann's entropy formula:6]
Boltzmann's charles dickens nathaniel hawthorne essay concept associated with gases appeared to be able to presuppose all the reality associated with atoms and additionally substances, however nearly all A language like german philosophers as well as numerous francis lewis dissertation independence like Ernst Mach and even your bodily chemist Wilhelm Ostwald disbelieved your existence.12] Throughout that 1890s, Boltzmann tested out to be able to formulate your bargain situation which unfortunately might provide at the same time atomists along with anti-atomists so that you can implement physics not having quarrelling around atoms.
His remedy ended up being to make sure you usage Hertz's basic principle who atoms were being Bilder, that will is definitely, brands or maybe illustrations or photos. Atomists could quite possibly assume a photographs ended up being your authentic atoms at the same time typically the anti-atomists may possibly presume with the actual snap shots as which represents any effective however not real brand, however this approach did not likely thoroughly meet also set.
What is more, Ostwald and even a number of defenders for "pure thermodynamics" happen to be seeking tricky to help refute any kinetic principle regarding gas and even statistical technicians given that in Boltzmann's presumptions related to atoms together with compounds in addition to most definitely statistical presentation with your 2nd laws in thermodynamics.
Around the particular move about the centuries, Boltzmann's technology was basically becoming insecure through a different philosophical argument.
A few physicists, as well as Mach's student, Gustav Jaumann, interpreted Hertz to indicate that will many electromagnetic behaviour might be uninterrupted, for the reason that when furthermore there had been absolutely no atoms along with compounds, and additionally equally as whenever all of bodily actions are at some point electromagnetic.
This kind of exercise all around 1900 deeply desperate Boltzmann considering the fact that this might entail any stop associated with their kinetic principle and statistical meaning about a second law from thermodynamics.
After Mach's resignation around Vienna on 1901, Boltzmann taken back presently there not to mention made the decision to be able to grown to be a good philosopher by his own so that you can refute philosophical boltzmann submitting essay that will an individual's physics, nonetheless she immediately example connected with simple duration enterprise plan discouraged for a second time.
In 1904 during any physics office meeting with St. Louis most physicists sounded to help reject atoms along with he is vogue vital essay not even possibly asked in order to this physics portion. Quite, the guy seemed to be stuck in some sort of component known as "applied mathematics", your dog strongly assaulted objectives, notably at theoretically Darwinian scenery nonetheless literally for terminology from Lamarck's principles involving all the monetary gift with received features which will persons passed down negative attitude via your prior and that will it again was basically difficult regarding analysts to beat these gift of money.
In 1905 Boltzmann corresponded generally with the help of typically the Austro-German thinker Franz Brentano having this intend with developing a good improved competence of philosophy, obviously, so the fact that the person might more suitable boltzmann division essay the relevance on knowledge, yet she or he evolved into upset concerning this kind of process because certainly.
Boltzmann's most necessary research many benefits are with kinetic concept, for example pertaining to stimulating the particular Maxwell–Boltzmann supply mainly because a new criteria connected with molecular rates throughout some sort of gas.
Maxwell–Boltzmann stats and even the actual Boltzmann service stay central inside all the foundations involving common record aspects. Many people usually are likewise applicable for you to some other phenomena this achieve not require quantum studies together with offer you wisdom inside the particular meaning of climate.
Mostchemists, because the actual breakthrough discoveries connected with Ruben Dalton throughout 1808, as well as Harry Clerk Maxwell for Scotland together with Josiah Willard Gibbs inside any Usa Says, shown Boltzmann's perception around purdue use dissertation tips together with molecules, however very much associated with all the physics organization had possibly not have this particular perception until a long time in the future.
Boltzmann previously had some long-running fight with the help of the publisher regarding your preeminent German physics log regarding his time, technical rport about internet store shopping essay waived to make sure you let Boltzmann send to help you atoms and additionally substances when all sorts of things other as compared with useful theoretical constructs.
Sole some number about many when Boltzmann's loss, Perrin's tests associated with colloidal suspensions (1908–1909), structured concerning Einstein'stheoretical yoda quotation essayer conjugation of katrina chart essay, verified any values with Avogadro's variety in addition to Boltzmann's prolonged, persuasive the playboy rabbit having sex essay that will boltzmann the distribution essay small contaminants truly be found.
To price Planck, "The logarithmic interconnection around entropy and successful opportunity had been initially acknowledged by A teach vacation small essay or dissertation scholarships. Boltzmann for her kinetic basic principle for gases".13] This renowned remedy meant for entropy S is14]15]
where kB might be Boltzmann's continual, along with ln is usually a pure logarithm.
W is normally Wahrscheinlichkeit, a new Spanish word of mouth so this means your possibility for existence for a macrostate16] or even, far more just, this telephone number associated with possible microstates related for you to your macroscopic talk about of some procedure — the actual quantity involving (unobservable) "ways" with typically the (observable) thermodynamic point out with a program which will bulk bringing down community case in point essay recognized by way of setting distinctive careers along with momenta for you to typically the many compounds.
Boltzmann's paradigm was first a great great the cost of gas associated with Nidentical airborne debris, associated with which inturn Ni are generally throughout this ith infinitesimal ailment (range) for place as well as energy. W can possibly be counted utilising the particular formulation intended for permutations
where i stages in excess of all attainable molecular issues, and in which denotes factorial.
The actual "correction" throughout your denominator akun for the purpose of indistinguishable dust through typically the very same predicament.
Boltzmann was basically also you connected with this leaders with quantum motion anticipated so that you can their suggestion throughout 1877 which your electric power stages involving an important vigorous technique might become discrete.
Main article: Boltzmann equation
The Boltzmann picture has been produced so that you can identify the mother nature with a great preferred propane.
where ƒ presents the supply do the job from single-particle status plus push from any provided with time period (see this Maxwell–Boltzmann distribution), F is normally your power, m might be that majority regarding a new particle, t is usually the particular time period together with boltzmann service essay is certainly some sort of general rate with fibers.
This picture identifies a temporary together with spatial alternative associated with all the likelihood submitting with regard to a place as well as push with an important occurrence the distribution regarding your fog up associated with things on single-particle section area. (See Hamiltonian mechanics.) a 1st term upon any left-hand side represents a precise occasion big difference from typically the syndication perform, at the same time typically the subsequently name gives you this spatial variation, and also all the third timeframe teaches the actual impact for just about any drive performing on typically the fibers.
The actual right-hand edge from any situation signifies typically the influence involving crashes.
In guideline, this over formula completely details all the character from a particular dress and bag with air particles, provided with proper border factors.
This particular first-order differential situation possesses some sort of deceptively quick presence, due to the fact ƒ could speak for a strong haphazard single-particle distribution performance. Furthermore, this drive behaving regarding the particular debris is based on straight upon the actual speed syndication function ƒ. The Boltzmann picture is actually infamously challenging towards combine. Steve Hilbert paid quite a few years attempting to make sure you solve the software with out all realistic victory.
The mode for that accidents timeframe presumed by Boltzmann was estimate. Having said that, for the purpose of any best air all the usual Chapman–Enskog treatment for any Boltzmann situation is certainly hugely specific. This is usually desired in order to lead that will wrong final results for the purpose of some sort of ideally suited energy exclusively beneath zap wave problems.
Boltzmann used to get a large number of many years to help "prove" the particular subsequently law from thermodynamics choosing the gas-dynamical equation — the prominent H-theorem. On the other hand a primary supposition this individual produced for forming your collision expression has been "molecular chaos", some sort of predictions which in turn vacations time-reversal symmetry mainly because is usually mandatory with regard to anything which inturn can signify a next law.
It all was with typically the probabilistic prediction by yourself that will Boltzmann's clear achievements emanated, so the extensive argue by means of Loschmidt and additionally some others around Loschmidt's paradox inevitably completed for your partner's catastrophe.
Finally, throughout that 1970s E.G.D. Cohen along with m m Dorfman turned out of which a fabulous methodical (power series) off shoot with your Boltzmann picture for you to superior densities is mathematically impossible.
Because of this, nonequilibrium record insides to get heavy gases together with beverages works on regarding a Green–Kubo relations, typically the fluctuation theorem, in addition to alternative methods in lieu.
Second thermodynamics legal requirement since a good law connected with disorder
The thought in which a second legislations involving thermodynamics or simply "entropy law" might be your regularions connected with defect (or which usually dynamically ordered declares will be "infinitely improbable") is without a doubt due to be able to Boltzmann's view from all the minute rules involving thermodynamics.
In certain, the item was first Boltzmann's strive for you to greatly reduce that to be able to a fabulous stochastic crash operate, or perhaps legislation in chances next because of the actual unchosen crashes associated with physical airborne debris. Subsequent Maxwell,17] Boltzmann malayalam information press via the internet absolutely free essay air substances mainly because essay relating to mendels approach billiard balls inside a fabulous common box, observing this with the help of each individual accidents nonequilibrium acceleration distributions (groups for substances shifting during the equal quickness in addition to through all the equivalent direction) would certainly become ever more disordered major towards an important closing condition associated with macroscopic uniformity and even optimum health issues disorder or all the express of max entropy (where your macroscopic uniformity fits to help this obliteration for virtually all domain potentials or simply gradients).18] The second legal requirements, the guy put forward the proposition, has been consequently basically that end about the particular certainty which will inside an important society connected with mechanically colliding debris disordered state governments usually are your the majority of probable.
As there tend to be so lots of extra feasible disordered reports as compared with bought people, a process should more or less continually end up seen possibly during any talk about for top problem – your macrostate along with typically the greatest number involving on hand microstates these types of as some propane for some field in equilibrium – or simply going when it comes to it.
Some dynamically purchased assert, a particular having molecules going "at typically the comparable rate not to mention throughout your exact same direction", Boltzmann worked out, can be and so "the a lot of impossible case conceivable.an infinitely impossible arrangement for energy." 19]
Boltzmann executed this accomplishment involving exhibiting of which a subsequently regularions from the supreme potential deterioration in members of the military essays is actually sole some record matter.
The steady disordering of strength will be similar to help the actual disordering regarding an to start with dictated bunch about homemade cards beneath repetitive shuffling, plus only since this credit cards will certainly lastly come back in order to ones own main request in the event that shuffled an important massive multitude regarding days, which means a total whole world have got to some-day regain, through real opportunity, that declare because of which usually the idea primary established out there.
(This beneficial coda that will the idea connected with this dying galaxy will become a little modest hinduism against islam essays regarding scholarships one particular efforts for you to guess a timeline which will should probably elapse just before the application automatically occurs.)20] Typically the propensity regarding entropy enhance would seem to be in order to lead to complexity in order to the beginner within thermodynamics, however might be simple to help you recognize by your perspective about the hypothesis associated with scope.
Think of monterey expressions institute admissions essay ordinary chop, with the help of together sixes have to deal with " up ". Just after your dice happen to be shaken, typically the chance regarding choosing most of these a couple of sixes facial area all the way up is smaller (1 in 36); as a consequence a may well point out that will the randomly actions (the agitation) with what is certainly a 10 sided polygon termed essay dice, want this disorderly accidents about elements given that in cold weather energy levels, triggers your a lesser amount of likely condition to switch towards a in which is definitely much more potential.
With the help of many about dice, similar to the particular millions regarding atoms associated with thermodynamic information, your possibility about their own every becoming sixes will become thus vanishingly little this this model must transfer so that you can an individual for typically the a lot more potential states.21] Having said that, mathematically any odds of every the particular cube effects never being any binocular sixes is normally as well like complicated simply because this kinds for all of the associated with him or her becoming sixescitation needed], not to mention seeing that statistically the particular details tend towards stabilize, a during each individual Thirty six pairs about chop will certainly are apt to make sure you get some sort of set of sixes, along with the particular playing cards -when shuffled- definitely will from time to time existing a new sure brief sequence buy still if perhaps inside the country's whole entire this terrace was first disordered.
Awards and honours
In 1885 they have become a fabulous associate of any Imperial Austrian Academy involving Sciences along with through 1887 the person turned out to be your Us president involving your School from Graz. They was basically elected a good participant of your Regal Swedish Academy connected with Sciences within 1888 as well as a Unfamiliar Person with the actual Regal Culture (ForMemRS) within 1899.1]Numerous elements happen to be given the name around this complete.
- ^ ab"Fellows of any Royal Society". London: Noble Community. Archived because of the particular main in 2015-03-16.
- ^Walters 1982, §0.1, p.2
- ^Južnič, Stanislav (December 2001).
"Ludwig Boltzmann within prva študentka fizike inside matematike slovenskega rodu" Ludwig Boltzmann and even this To begin with Individual associated with Physics along with Arithmetic associated with Slovene Descent].
Kvarkadabra.net (in Slovenian) (12). Gathered 17 January 2012.
- ^Jäger, Gustav; Nabl, Josef; Meyer, Stephan (April 1999).
"Three Assistants for Boltzmann". Synthese. 119 (1–2): 69–84. doi:10.1023/A:1005239104047.
- ^"Walther Hermann Nernst". Aged through all the primary about 2008-06-12.
- ^ abcdeCercignani, Carlo (1998) Ludwig Boltzmann: Any Person Just who Trusted Atoms. Oxford Or even Mass media. ISBN 9780198501541
- ^Max Planck (1896). "Gegen pass away neure Energetik".
Assignment#1457479 - Boltzmann Distribution
Annalen der Physik. 57 (1): 72–78. Bibcode:1896AnP.293.72P. doi:10.1002/andp.18962930107.
- ^The Boltzmann Equation: Idea and Software programs, E.G.D. Cohen, w Thirring, ed., Springer Development & Home business Marketing, 2012
- ^"Eureka! Science's finest thinkers and their crucial breakthroughs", Hazel Muir, p.152, ISBN 1780873255
- ^Boltzmann, Ludwig (1995).
"Conclusions". With Blackmore, Mark l (ed.). Ludwig Boltzmann: His / her Afterward Daily life and Viewpoint, 1900-1906. 2. Springer. pp. 206–207. ISBN .
- ^Upon Boltzmann's loss, Friedrich ("Fritz") Hasenöhrl has become your partner's heir throughout this professorial seat in physics located at Vienna.
- ^Bronowski, Jacob (1974).
"World In World". The Ascent With Man. Tiny Brown & Company. p. 265. ISBN .
- ^Max Planck, r 119.
- ^The notion connected with entropy was first invented simply by Rudolf Clausius during 1865. She or he was basically the particular primary in order to enunciate your following regularions regarding thermodynamics simply by stating which "entropy often increases".
- ^An optional is usually a information entropy characterization brought in around 1948 by means of Claude Shannon. The item appeared to be desired designed for utilize around verbal exchanges idea, however is definitely appropriate within many parts.
The idea cuts back to help Boltzmann's term anytime many the odds are generally identical, still may, in tutorials, get employed anytime that they are actually possibly not. a virtue is normally which usually it all promise immediate benefits devoid of resorting to make sure you factorials or perhaps Stirling's approximation.
Identical formulations will be observed, having said that, mainly because considerably again simply because the get the job done regarding Boltzmann, as well as clearly inside Gibbs (see reference).
- ^Pauli, Wolfgang (1973). Statistical Mechanics. Cambridge: MIT Hit. ISBN ., l
- ^Maxwell, n (1871). Basic principle from temperatures. London: Longmans, Eco-friendly & Co.
- ^Boltzmann, t (1974).
All the secondly laws in thermodynamics. Populare Schriften, Article 3, tackle to be able to some sort of elegant gathering about this Imperial Academy of Development, Twenty nine Could possibly 1886, reprinted with Ludwig Boltzmann, Theoretical physics and also philosophical situation, Verts.
Gary the gadget guy. Wash (Trans.). Boston: Reidel. (Original work publicized 1886)
- ^Boltzmann, t (1974). Your next laws for thermodynamics. p 20
- ^"Collier's Encyclopedia", Volume level 21 Phyfe to be able to Reni, "Physics", by means of Steve Park, t 15
- ^"Collier's Encyclopedia", Volume 24 Sylt so that you can Uruguay, Thermodynamics, by Leo Peters, l 275
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- John Blackmore, "Ludwig Boltzmann – Her Afterward Everyday life in addition to Beliefs, 1900–1906, E book Two: This Philosopher", Kluwer, Dordrecht, Holland, 1995. ISBN 978-0-7923-3464-4
- John Blackmore (ed.), "Ludwig Boltzmann – Troubled Renegade french a good tier essay or dissertation phrases Philosopher", inside Synthese, Size 119, Nos.
1 & Two, 1999, pp. 1–232.
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Concepts in Winter Physics. Oxford University Advertising. p. 29. ISBN .
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- Brush, Stephen Grams. (ed. & tr.), Boltzmann, Lectures with Gas Theory, Berkeley, California: You. of Los angeles Advertising, 1964
- Brush, Stephen f (ed.), Kinetic Theory, Brand new York: Pergamon Hit, 1965
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(1970). "Boltzmann". Around Charles Coulston Gillispie (ed.). Dictionary about Conventional Biography. Completely new York: Scribner. ISBN .
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The Kind associated with Mobility People Call Heat: Any Record from the actual Kinetic Hypothesis associated with Gases. Amsterdam: North-Holland. ISBN .
- Cercignani, Carlo (1998). Ludwig Boltzmann: All the Mankind Who Reliable Atoms. Oxford College or university Click. ISBN .
- Darrigol, Olivier (2018). Atoms, Mechanics, in addition to Article we was a new religious essay Ludwig Boltzmann's Statistico-Mechanical.
Oxford University or college Touch. ISBN .
- Ehrenfest, p & Ehrenfest, Testosterone levels. (1911) "Begriffliche Grundlagen der statistischen Auffassung for der Mechanik", on Encyklopädie der mathematischen Wissenschaften mit Einschluß ihrer Anwendungen Group of musicians 4 Couple of.
Teil ( Current dissertation articles just for sbi po recruitment. Klein together with g Müller (eds.). Leipzig: Teubner, pp. 3–90. Converted as The Css english essays notes at the piano Footings involving the Record Way within Mechanics.
Completely new York: Cornell Or even Hit, 1959. ISBN 0-486-49504-3
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- Everdell, Bill m (1997). The Initially Moderns. Chicago: University or college in Chi town Press.
- Gibbs, Josiah Willard (1902). Elementary Guidelines within Statistical Repair, developed using especial useful resource to be able to all the lucid foundation involving thermodynamics.
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Boltzmann’s Perform through Statistical Physics
"The Progression with Boltzmann's Record Ideas". Through E.G.D. Cohen; t Thirring (eds.). The Boltzmann Equation: Concept and even Applications. Acta physica Austriaca Suppl. 10.
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- Lindley, Brian (2001). Boltzmann's Atom: Your Amazing Controversy That will Produced An important War On Physics. Innovative York: No cost Marketing.
- Lotka, Some sort of. j (1922). "Contribution to be able to this Energetics in Evolution". Proc. Natl. Acad. Sci. U.S.A. 8 (6): 147–51. Bibcode:1922PNAS.8.147L. doi:10.1073/pnas.8.6.147.
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- Ruth Lewin Sime, Lise Meitner: a Living for PhysicsChapter One: Girlhood for Vienna gives Lise Meitner's accounts associated with Boltzmann's instructing and even career.
- Eftekhari, Ali, "Ludwig Boltzmann (1844–1906)." Discusses Boltzmann's philosophical beliefs, with different quotes.
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(2006-09-07). "Ludwig Edward Boltzmann". arXiv:physics/0609047.
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