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第117行： − 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>+ − − 统计力学是现代物理学的支柱之一。它描述了宏观观察量(如温度和压力)如何与围绕平均值波动的微观参数相联系。它将热力学量(比如热容)与微观行为联系起来，而在经典热力学中，唯一可行的选择就是测量各种材料的热力学量，然后列成表格。<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> − Boltzmann was born in Erdberg, a suburb of [[Vienna]]. 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 at the home of his parents.<ref>{{cite book − |title=The Scientific 100 − |first1=John − |last1=Simmons − |first2=Lynda − |last2=Simmons − |isbn=9780806536781 − |page=123 − |publisher=Kensington Publishing Corp. − |year=2000 − − }}</ref> Boltzmann 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 − |isbn=9780521017060 − |page=[https://archive.org/details/remarkablephysic00jame/page/n185 169] − |publisher=Cambridge University Press − |year=2004 − − }}</ref> − 第158行：
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第142行： − − 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" />+ − − 从1863年开始，玻尔兹曼在维也纳大学学习数学和物理。他于1866年获得博士学位，1869年获得任教资格(Venia legendi)。波尔兹曼与物理研究所所长约瑟夫·斯特凡 [[Josef Stefan]]密切合作。正是斯特凡把玻尔兹曼引入了麦克斯韦的研究。 − 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 |website=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. − − 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: Henriette (1880), Ida (1884) and Else (1891); and a son, Arthur Ludwig (1881).<ref>https://www.boltzmann.com/ludwig-boltzmann/biography/</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=2008-06-12 }}</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. − − 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, and his symptoms indicate he experienced what would 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 = February 13, 2018 | url = https://paperpile.com/blog/ludwig-boltzmann/ | accessdate = 2020-04-26 }}</ref> Four months later 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/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" />+ − − 1906年，玻尔兹曼日渐恶化的精神状况迫使他辞去教职。后人对他当时的症状进行分析得出了躁郁症的结论。<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 = February 13, 2018 | url = https://paperpile.com/blog/ludwig-boltzmann/ | accessdate = 2020-04-26 }}</ref>四个月后，在和妻女的度假过程中，他选择上吊自杀结束自己的痛苦和生命，终年63岁。 − − 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" /> 第217行：
第171行： − 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. − − 在世纪之交，玻尔兹曼的理论受到另一种哲学的威胁。一些物理学家，包括马赫的学生古斯塔夫·乔曼，把赫兹的理论解释为——所有的电磁行为都是连续的，就好像没有原子和分子的存在，同样所有的物理行为最终都是电磁性质的。1900年左右的这一运动使玻尔兹曼深感沮丧，因为这可能意味着他的动力学理论和热力学第二定律的统计解释的终结。 − 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.+ − − − 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]]. − 第241行：
第185行： − − [[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]]".<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'' 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 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. 第265行：
第195行： − − 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 ''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. 第299行：
第213行： − − 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>+ − 具体来讲，玻尔兹曼试图将其简化为随机碰撞函数，或机械粒子随机碰撞后的概率定律。继麦克斯韦之后，玻尔兹曼把气体分子模拟成一个盒子里碰撞的台球，并指出每一次碰撞非平衡态速度分布(多组分子运动的速度和方向相同)会越来越无序，并导致最终的宏观均匀和微观最无序或最大熵状态(宏观均匀的状态对应所有场势或梯度的消失)。因此，他认为，热力学第二定律描述了这样一个事实结果：在一个机械碰撞粒子的世界里，无序状态是最有可能的。因为可能的无序状态比有序状态要多得多，一个系统几乎总是会处于最无序状态——具有最多可接近的微观状态的宏观状态，如处于平衡状态的盒子里的气体——或向最无序状态移动。因此，玻尔兹曼总结道，一个动态有序的状态，即分子以“相同的速度和相同的方向”运动，是“最不可思议的可能情况…… 一种无限不可能的能量配置”。+ − − 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. − − 玻尔兹曼完成了证明热力学第二定律只是一个统计事实的壮举。能量的逐渐无序化类似于一叠最初有序排列的卡牌重复洗牌过程中变得无序，也正如经历了无数次洗牌而重归初始有序状态的卡牌一样，我们的宇宙最终将在某一时刻回到最初的状态。（当人们试图估计宇宙正在消亡的时间线时，这种乐观的结论就显得有些平淡了，因为时间线很可能在它自然发生之前就已经过去了。）熵增加的趋势似乎对初学热力学的人们造成困难，但从概率论的观点出发就很容易理解。考虑两个普通的骰子，都是正面朝上的。摇骰子后，发现这两个6面都朝上的概率很小(1/36)；因此，我们可以说，骰子的随机运动，就像分子由于热能的混乱碰撞，会导致系统从概率较小的状态变为概率较大的状态。数以百万计的骰子，就像热力学计算中数以百万计的原子一样，它们全部为6的概率变得如此之小，以至于系统必须移动到一个更可能的状态。然而，数学上出现所有骰子结果都是6或者都不是6的几率都很小，由于统计数据会趋于平衡，每36对骰子会有一对6出现，而洗过的牌有时会呈现出某种临时的顺序，即使整个牌是无序的。 − 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|archiveurl=https://web.archive.org/web/20150316060617/https://royalsociety.org/about-us/fellowship/fellows/|archivedate=2015-03-16|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.+ − − 1885年，他成为奥地利帝国科学院的成员，1887年，他成为格拉茨大学的校长。1888年，他被选为瑞典皇家科学院成员，1899年被选为皇家学会外国成员。另外许多事物都以他的名字命名。 − − *[[Energetics]] − *[[Boltzmann brain]]
无编辑摘要
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统计力学是现代[[物理学]]的支柱之一。它描述了宏观观察量(如温度和压力)如何与围绕平均值波动的微观参数相联系。它将热力学量(比如[[热容]])与微观行为联系起来，而在[[经典热力学]]中，唯一可行的选择就是测量各种材料的热力学量，然后列成表格。<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>
==个人简介==
==个人简介==
===童年时光与教育经历===
===童年时光与教育经历===
玻尔兹曼出生在维也纳郊区的埃尔德伯格。祖父是一名钟表制造商，从柏林搬到维也纳。父亲路德维希·格奥尔格·玻尔兹曼是一名税务官员，母亲卡塔琳娜·波恩芬德来自萨尔茨堡。他在家里接受了启蒙教育。<ref>{{cite book
玻尔兹曼出生在维也纳郊区的埃尔德伯格。祖父是一名钟表制造商，从柏林搬到维也纳。父亲路德维希·格奥尔格·玻尔兹曼是一名税务官员，母亲卡塔琳娜·波恩芬德来自萨尔茨堡。他在家里接受了启蒙教育。<ref>{{cite book
|title=The Scientific 100
|title=The Scientific 100
|publisher=Kensington Publishing Corp.
|publisher=Kensington Publishing Corp.
|year=2000
|year=2000
}}</ref>高中时光在上奥地利的林茨度过。玻尔兹曼15岁时，他的父亲去世了。<ref name="james2004">{{cite book
}}</ref> 高中时光在上奥地利的林茨度过。玻尔兹曼15岁时，他的父亲去世了。<ref name="james2004">{{cite book
|title=Remarkable Physicists: From Galileo to Yukawa
|title=Remarkable Physicists: From Galileo to Yukawa
|url=https://archive.org/details/remarkablephysic00jame
|url=https://archive.org/details/remarkablephysic00jame
|publisher=Cambridge University Press
|publisher=Cambridge University Press
|year=2004
|year=2004
}}</ref>
}}</ref>
从1863年开始，玻尔兹曼在维也纳大学学习数学和物理。他于1866年获得博士学位，1869年获得任教资格(Venia legendi)。波尔兹曼与物理研究所所长约瑟夫·斯特凡 [[Josef Stefan]]密切合作。正是斯特凡把玻尔兹曼引入了麦克斯韦的研究。<ref name="james2004" />
===学术生涯===
===学术生涯===
1869年，25岁的玻尔兹曼经斯特凡的推荐，被任命为斯蒂里亚省格拉茨大学的数学物理正教授。<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 |website=Kvarkadabra.net |issue=12 |accessdate=17 February 2012}}</ref>1869年，他在海德堡与罗伯特·本森 [[Robert Bunsen]]和里奥·柯尼斯堡 [[Leo Königsberger]]一起工作数月，1871年在柏林与古斯塔夫·基尔霍夫 [[Gustav Kirchhoff]]和赫尔曼·冯·赫姆霍尔兹 [[Hermann von Helmholtz]]进行合作。1873年玻尔兹曼加入维也纳大学担任数学教授，直至1876年。
1869年，25岁的玻尔兹曼经斯特凡的推荐，被任命为斯蒂里亚省格拉茨大学的数学物理正教授。<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 |website=Kvarkadabra.net |issue=12 |accessdate=17 February 2012}}</ref>1869年，他在海德堡与罗伯特·本森 [[Robert Bunsen]]和里奥·柯尼斯堡 [[Leo Königsberger]]一起工作数月，1871年在柏林与古斯塔夫·基尔霍夫 [[Gustav Kirchhoff]]和赫尔曼·冯·赫姆霍尔兹 [[Hermann von Helmholtz]]进行合作。1873年玻尔兹曼加入维也纳大学担任数学教授，直至1876年。
[[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
[[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
1887年，玻尔兹曼与他的合作者们在格拉茨（后排左起：能斯特，施特莱恩，阿伦尼乌斯，海耶克；前排左起：奥林格尔，埃廷斯豪森，玻尔兹曼，克莱门契奇，豪斯曼宁格）|链接=Special:FilePath/Boltzmann-grp.jpg]]
1887年，玻尔兹曼与他的合作者们在格拉茨（后排左起：能斯特，施特莱恩，阿伦尼乌斯，海耶克；前排左起：奥林格尔，埃廷斯豪森，玻尔兹曼，克莱门契奇，豪斯曼宁格）|链接=Special:FilePath/Boltzmann-grp.jpg]]
1872年，远在女性可以被奥地利大学录取之前，他遇到了亨里埃特·冯·艾根特勒 Henriette von Aigentler，这是一位有抱负的格拉茨数学和物理女教师。然而她被拒绝旁听大学讲座。玻尔兹曼支持她上诉的决定，之后上诉成功。1876年7月17日，路德维希·玻尔兹曼与亨里埃特结婚；他们共育有三女一子：亨里埃特 Henriette (1880年)、艾达 Ida (1884年)、爱尔莎 Else (1891年)、亚瑟·路德维希 Arthur Ludwig (1881年)。<ref>https://www.boltzmann.com/ludwig-boltzmann/biography/</ref>玻尔兹曼回到格拉茨担任实验物理学的主席。他在格拉茨的学生中有斯凡特·阿伦尼乌斯 [[Svante Arrhenius]]和瓦尔特·能斯特 [[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=2008-06-12 }}</ref>
1872年，远在女性可以被奥地利大学录取之前，他遇到了亨里埃特·冯·艾根特勒 Henriette von Aigentler，这是一位有抱负的格拉茨数学和物理女教师。然而她被拒绝旁听大学讲座。玻尔兹曼支持她上诉的决定，之后上诉成功。1876年7月17日，路德维希·玻尔兹曼与亨里埃特结婚；他们共育有三女一子：亨里埃特 Henriette (1880年)、艾达 Ida (1884年)、爱尔莎 Else (1891年)、亚瑟·路德维希 Arthur Ludwig (1881年)。<ref>https://www.boltzmann.com/ludwig-boltzmann/biography/</ref>玻尔兹曼回到格拉茨担任实验物理学的主席。他在格拉茨的学生中有斯凡特·阿伦尼乌斯 [[Svante Arrhenius]]和瓦尔特·能斯特 [[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=2008-06-12 }}</ref>
他十分享受在格拉茨度过的14个春秋，正是在那里，他发展了他的自然界的统计概念。
他十分享受在格拉茨度过的14个春秋，正是在那里，他发展了他的自然界的统计概念。
玻尔兹曼于1890年被任命为德国巴伐利亚慕尼黑大学理论物理学主席。
玻尔兹曼于1890年被任命为德国巴伐利亚慕尼黑大学理论物理学主席。
1894年，波尔兹曼接替他的老师约瑟夫·斯特凡 [[Joseph Stefan]] 成为维也纳大学理论物理学教授。
1894年，波尔兹曼接替他的老师约瑟夫·斯特凡 [[Joseph Stefan]] 成为维也纳大学理论物理学教授。
===晚年生活与逝世缘由===
===晚年生活与逝世缘由===
晚年的玻尔兹曼致力于维护自己所创立的理论。<ref name="Carlo">Cercignani, Carlo (1998) Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford University Press. {{ISBN|9780198501541}}</ref> 在维也纳，他和一些同事相处不很融洽，尤其是恩斯特·马赫Ernst Mach (1895年成为哲学和科学史教授)。1895年，格奥尔格·赫尔姆 Georg Helm 和威廉·奥斯特瓦尔德 Wilhelm Ostwald 在吕贝克的一次会议上提出了他们关于能量学的观点。他们认为宇宙的主要组成部分是能量，而不是物质。会议辩论中，玻尔兹曼的观点赢得了其他支持原子理论的物理学家的赞同。<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> 在威廉·奥斯特瓦尔德的邀请下，玻尔兹曼于1900年前往莱比锡大学，填补辞世的古斯塔夫·海因里希·维德曼 [[Gustav Heinrich Wiedemann]] 留下的空缺，出任物理学首席教授。在马赫因健康原因退休以后，玻尔兹曼于1902年返回维也纳。<ref name="Carlo" /> 1903年，玻尔兹曼与古斯塔夫·冯·埃舍里希[[Gustav von Escherich]]、埃米尔·穆勒 [[Emil Müller (mathematician)|Emil Müller]]一同创立了奥地利数学学会。他的门生包括：卡尔·普里贝拉姆 [[Karl Přibram]]（经济学家），保罗·埃伦费斯特 [[Paul Ehrenfest]]（理论物理学家），丽斯·迈特纳 [[Lise Meitner]]（物理学家，被爱因斯坦称为“德国居里夫人”）。<ref name="Carlo" />
晚年的玻尔兹曼致力于维护自己所创立的理论。<ref name="Carlo">Cercignani, Carlo (1998) Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford University Press. {{ISBN|9780198501541}}</ref> 在维也纳，他和一些同事相处不很融洽，尤其是恩斯特·马赫Ernst Mach (1895年成为哲学和科学史教授)。1895年，格奥尔格·赫尔姆 Georg Helm 和威廉·奥斯特瓦尔德 Wilhelm Ostwald 在吕贝克的一次会议上提出了他们关于能量学的观点。他们认为宇宙的主要组成部分是能量，而不是物质。会议辩论中，玻尔兹曼的观点赢得了其他支持原子理论的物理学家的赞同。<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> 在威廉·奥斯特瓦尔德的邀请下，玻尔兹曼于1900年前往莱比锡大学，填补辞世的古斯塔夫·海因里希·维德曼 [[Gustav Heinrich Wiedemann]] 留下的空缺，出任物理学首席教授。在马赫因健康原因退休以后，玻尔兹曼于1902年返回维也纳。<ref name="Carlo" /> 1903年，玻尔兹曼与古斯塔夫·冯·埃舍里希[[Gustav von Escherich]]、埃米尔·穆勒 [[Emil Müller (mathematician)|Emil Müller]]一同创立了奥地利数学学会。他的门生包括：卡尔·普里贝拉姆 [[Karl Přibram]]（经济学家），保罗·埃伦费斯特 [[Paul Ehrenfest]]（理论物理学家），丽斯·迈特纳 [[Lise Meitner]]（物理学家，被爱因斯坦称为“德国居里夫人”）。<ref name="Carlo" />
在维也纳大学，玻尔兹曼教授物理和哲学。他有关自然哲学的讲座颇受欢迎。开课当天，即使已经事先安排了最大的讲堂，现场依然是座无虚席，观者如山。讲座取得巨大成功，轰动全国，奥匈帝国皇帝也邀请玻尔兹曼入宫进行招待。<ref>The Boltzmann Equation: Theory and Applications, E.G.D. Cohen, W. Thirring, ed., Springer Science & Business Media, 2012</ref>
在维也纳大学，玻尔兹曼教授物理和哲学。他有关自然哲学的讲座颇受欢迎。开课当天，即使已经事先安排了最大的讲堂，现场依然是座无虚席，观者如山。讲座取得巨大成功，轰动全国，奥匈帝国皇帝也邀请玻尔兹曼入宫进行招待。<ref>The Boltzmann Equation: Theory and Applications, E.G.D. Cohen, W. Thirring, ed., Springer Science & Business Media, 2012</ref>
1906年，玻尔兹曼日渐恶化的精神状况迫使他辞去教职。后人对他当时的症状进行分析得出了躁郁症的结论。<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 = February 13, 2018 | url = https://paperpile.com/blog/ludwig-boltzmann/ | accessdate = 2020-04-26 }}</ref>四个月后，在和妻女的度假过程中，他选择上吊自杀结束自己的痛苦和生命，终年63岁。<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/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" />
他葬在维也纳中央弗里德霍夫墓园。墓碑上刻着玻尔兹曼熵公式: <math>S = k \cdot \log W </math><ref name="Carlo" />
他葬在维也纳中央弗里德霍夫墓园。墓碑上刻着玻尔兹曼熵公式: <math>S = k \cdot \log W </math><ref name="Carlo" />
==哲学观点==
==哲学观点==
{{Unreferenced section|date=December 2018}}
{{Unreferenced section|date=December 2018}}
玻尔兹曼的气体动力学理论似乎假定了原子和分子的真实性，但几乎所有德国哲学家和许多科学家，如恩斯特·马赫和物理化学家威廉·奥斯特瓦尔德，都不相信它们的存在。<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> 在19世纪90年代，玻尔兹曼试图形成一个折中立场，使原子主义者和反原子主义者都可以在不争论原子存在与否的情况下研究物理学。他的解决方案是使用赫兹的理论，即原子是‘Bilder’，即模型或图片。原子论者可以认为这些图像是真实的原子，而反原子论者则认为这些图像代表了一种有用但不真实的模型，然而这两类人都不完全满意。此外，由于玻尔兹曼关于原子和分子真实存在的假设，特别是对于[[热力学第二定律]]的统计解释，奥斯特瓦尔德和许多“纯热力学”的捍卫者试图努力驳斥气体动力学理论和统计力学。
玻尔兹曼的气体动力学理论似乎假定了原子和分子的真实性，但几乎所有德国哲学家和许多科学家，如恩斯特·马赫和物理化学家威廉·奥斯特瓦尔德，都不相信它们的存在。<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> 在19世纪90年代，玻尔兹曼试图形成一个折中立场，使原子主义者和反原子主义者都可以在不争论原子存在与否的情况下研究物理学。他的解决方案是使用赫兹的理论，即原子是‘Bilder’，即模型或图片。原子论者可以认为这些图像是真实的原子，而反原子论者则认为这些图像代表了一种有用但不真实的模型，然而这两类人都不完全满意。此外，由于玻尔兹曼关于原子和分子真实存在的假设，特别是对于热力学第二定律的统计解释，奥斯特瓦尔德和许多“纯热力学”的捍卫者试图努力驳斥气体动力学理论和统计力学。
在世纪之交，玻尔兹曼的理论受到另一种哲学的威胁。一些物理学家，包括马赫的学生[[古斯塔夫·乔曼]]，把赫兹的理论解释为——所有的电磁行为都是连续的，就好像没有原子和分子的存在，同样所有的物理行为最终都是电磁性质的。1900年左右的这一运动使玻尔兹曼深感沮丧，因为这可能意味着他的动力学理论和热力学第二定律的统计解释的终结。
1901年马赫在维也纳辞职后，玻尔兹曼回到学校，并决定自己成为一名哲学家，以驳斥哲学上对物理学的异议，但他很快又受到了打击。1904年，在圣路易斯举行的一次物理会议上，大多数物理学家似乎都拒绝接受原子，他甚至没有被邀请参加物理部分。相反，他被困在“应用数学”的讨论会，他猛烈地攻击哲学，特别是在所谓的达尔文主义的基础上，但实际上是根据拉马克的后天特征遗传理论，人们从过去继承了糟糕的哲学，科学家很难克服这种遗传。'''<font color="#32CD32">相反，他被困在“应用数学”的讨论会，他猛烈地攻击哲学，特别是在所谓的达尔文主义的基础上，但实际上是根据拉马克的后天特征遗传理论，人们从过去继承了糟糕的哲学，科学家很难克服这种遗传＋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</font>'''（感觉这句话翻译的很奇怪，希望有大佬帮助看看）。
1901年马赫在维也纳辞职后，玻尔兹曼回到学校，并决定自己成为一名哲学家，以驳斥哲学上对物理学的异议，但他很快又受到了打击。1904年，在圣路易斯举行的一次物理会议上，大多数物理学家似乎都拒绝接受原子，他甚至没有被邀请参加物理部分。相反，他被困在“应用数学”的讨论会，他猛烈地攻击哲学，特别是在所谓的达尔文主义的基础上，但实际上是根据拉马克的后天特征遗传理论，人们从过去继承了糟糕的哲学，科学家很难克服这种遗传。'''
1905年，玻尔兹曼与德-奥哲学家弗朗茨·布伦塔诺 Franz Brentano进行了广泛的通信，希望能更好地掌握哲学，显然，这样他就能更好地驳斥哲学在科学上的相关性，但他对这种方法也逐渐感到沮丧。
1905年，玻尔兹曼与德-奥哲学家弗朗茨·布伦塔诺 [[Franz Brentano]] 进行了广泛的通信，希望能更好地掌握哲学，显然，这样他就能更好地驳斥哲学在科学上的相关性，但他对这种方法也逐渐感到沮丧。
==物理学贡献==
==物理学贡献==
玻尔兹曼最重要的科学贡献是在动力学理论上，包括推动应用'''麦克斯韦-玻尔兹曼分布 [[Maxwell–Boltzmann distribution]]''' 描述气体中分子的速度。直至今日，'''麦克斯韦-玻尔兹曼统计 [[Maxwell–Boltzmann statistics]]''' 和'''玻尔兹曼分布 [[Boltzmann distribution]]''' 仍然是经典统计力学基石之一。他们也适用于解释其他不需要量子统计的现象，另外提供了关于温度内涵的洞见。
玻尔兹曼最重要的科学贡献是在动力学理论上，包括推动应用'''麦克斯韦-玻尔兹曼分布 [[Maxwell–Boltzmann distribution]]''' 描述气体中分子的速度。直至今日，'''麦克斯韦-玻尔兹曼统计 [[Maxwell–Boltzmann statistics]]''' 和'''玻尔兹曼分布 [[Boltzmann distribution]]''' 仍然是经典统计力学基石之一。他们也适用于解释其他不需要量子统计的现象，另外提供了关于温度内涵的洞见。
1898年玻尔兹曼所作碘分子示意图，其中原子“敏感区域”(α，β)发生重叠。|链接=Special:FilePath/Boltzmanns-molecule.jpg]]
1898年玻尔兹曼所作碘分子示意图，其中原子“敏感区域”(α，β)发生重叠。|链接=Special:FilePath/Boltzmanns-molecule.jpg]]
自1808年，约翰·道尔顿 John Dalton 提出原子理论以来、包括詹姆斯·克拉克·麦克斯韦James Clerk Maxwell（苏格兰）和乔赛亚·威拉德·吉布斯Josiah Willard Gibbs（美）在内，大多数化学家都认同玻尔兹曼对原子和分子的看法，但很多物理学家直到几十年后才认同这一观点。玻尔兹曼与当时著名的德国物理学杂志的编辑有长期的争执，后者拒绝让玻尔兹曼将原子和分子作为方便的理论结构之外的任何东西。玻尔兹曼去世几年后，佩兰 [[Jean Baptiste Perrin|Perrin]] 基于爱因斯坦 [[Albert Einstein|Einstein]] 1905年的理论研究，对胶体悬浮液进行了研究(1908-1909)，证实了阿伏伽德罗数 [[Avogadro's number]] 和玻尔兹曼常数的值，使世界相信微小粒子确实存在。
自1808年，约翰·道尔顿 John Dalton 提出原子理论以来、包括詹姆斯·克拉克·麦克斯韦James Clerk Maxwell（苏格兰）和乔赛亚·威拉德·吉布斯Josiah Willard Gibbs（美）在内，大多数化学家都认同玻尔兹曼对原子和分子的看法，但很多物理学家直到几十年后才认同这一观点。玻尔兹曼与当时著名的德国物理学杂志的编辑有长期的争执，后者拒绝让玻尔兹曼将原子和分子作为方便的理论结构之外的任何东西。玻尔兹曼去世几年后，佩兰 [[Jean Baptiste Perrin|Perrin]] 基于爱因斯坦 [[Albert Einstein|Einstein]] 1905年的理论研究，对胶体悬浮液进行了研究(1908-1909)，证实了阿伏伽德罗数 [[Avogadro's number]] 和玻尔兹曼常数的值，使世界相信微小粒子确实存在。
普朗克曾说：“熵和概率之间的对数关系是由玻尔兹曼在他的气体动力学理论中首次提出的”。<ref>Max Planck, p. 119.</ref> 也就是著名的熵公式：<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>，其中''k<sub>B</sub>'' 是玻尔兹曼常数，''W'' 代表德文中宏观状态出现的概率，<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>更准确一些来说，是对应于系统宏观状态的可能微观状态的数量——在一个系统的(可观察的)热力学状态下的(不可观测的)“方式”的数量，可以通过分配不同的位置和动量给不同的分子来实现。玻尔兹曼的范式是N个相同粒子的理想气体，其中Ni处于第i个微观位置和动量条件(范围)。''W'' 可以用排列公式计算：<math> W = N! \prod_i \frac{1}{N_i!} </math>，其中''i'' 的范围包含所有可能的分子状态，<math>!</math>代表阶乘。分母中的“修正”解释了相同条件下难以区分的粒子。
普朗克曾说：“熵和概率之间的对数关系是由玻尔兹曼在他的气体动力学理论中首次提出的”。<ref>Max Planck, p. 119.</ref> 也就是著名的熵公式：<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>，其中''k<sub>B</sub>'' 是玻尔兹曼常数，''W'' 代表德文中宏观状态出现的概率，<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>更准确一些来说，是对应于系统宏观状态的可能微观状态的数量——在一个系统的(可观察的)热力学状态下的(不可观测的)“方式”的数量，可以通过分配不同的位置和动量给不同的分子来实现。玻尔兹曼的范式是N个相同粒子的理想气体，其中Ni处于第i个微观位置和动量条件(范围)。''W'' 可以用排列公式计算：<math> W = N! \prod_i \frac{1}{N_i!} </math>，其中''i'' 的范围包含所有可能的分子状态，<math>!</math>代表阶乘。分母中的“修正”解释了相同条件下难以区分的粒子。
玻尔兹曼也被认为是量子力学的先驱之一，因为他在1877年提出了物理系统的能级可以是离散的。
玻尔兹曼也被认为是量子力学的先驱之一，因为他在1877年提出了物理系统的能级可以是离散的。
[[File:Ludwig Boltzmann at U Vienna.JPG|thumb|Boltzmann's bust in the courtyard arcade of the main building, University of Vienna.玻尔兹曼的半身像陈列在维也纳大学主楼的庭院拱廊上。|链接=Special:FilePath/Ludwig_Boltzmann_at_U_Vienna.JPG]]
[[File:Ludwig Boltzmann at U Vienna.JPG|thumb|Boltzmann's bust in the courtyard arcade of the main building, University of Vienna.玻尔兹曼的半身像陈列在维也纳大学主楼的庭院拱廊上。|链接=Special:FilePath/Ludwig_Boltzmann_at_U_Vienna.JPG]]
{{main|Boltzmann equation}}
{{main|Boltzmann equation}}
建立玻尔兹曼方程是用来描述理想气体的动力学规律的。
建立玻尔兹曼方程是用来描述理想气体的动力学规律的。
其中 ''ƒ'' 代表某一时刻单个粒子位置和动量的函数分布（见[[麦克斯韦-玻尔兹曼方程|麦克斯韦-玻尔兹曼分布]]），''F'' 代表力，''m'' 代表粒子的质量，''t'' 是时间，''v'' 是粒子的平均速度。
其中 ''ƒ'' 代表某一时刻单个粒子位置和动量的函数分布（见[[麦克斯韦-玻尔兹曼方程|麦克斯韦-玻尔兹曼分布]]），''F'' 代表力，''m'' 代表粒子的质量，''t'' 是时间，''v'' 是粒子的平均速度。
该方程描述了单粒子相空间中一团点密度分布的位置和动量概率分布的时空变化(见'''[[哈密顿力学]] [[Hamiltonian mechanics]]''')。左边的第一项表示分布函数的显式时间变化，而第二项给出空间变化，第三项描述作用于粒子的任何力响。方程右边表示碰撞的影响。
该方程描述了单粒子相空间中一团点密度分布的位置和动量概率分布的时空变化(见'''[[哈密顿力学]] [[Hamiltonian mechanics]]''')。左边的第一项表示分布函数的显式时间变化，而第二项给出空间变化，第三项描述作用于粒子的任何力响。方程右边表示碰撞的影响。
原则上，在给定适当的边界条件下，上述方程完全描述了气体粒子系综的动力学。这个一阶微分方程看似简单的，因为ƒ可以表示一个任意的单粒子分布函数。另外，作用在粒子上的力直接取决于速度分布函数ƒ。然而玻尔兹曼方程是出了名的难以进行积分。大卫·希尔伯特 [[David Hilbert]] 花了数年时间试图解决这个问题，但没有实现任何真正的突破。
原则上，在给定适当的边界条件下，上述方程完全描述了气体粒子系综的动力学。这个一阶微分方程看似简单的，因为ƒ可以表示一个任意的单粒子分布函数。另外，作用在粒子上的力直接取决于速度分布函数ƒ。然而玻尔兹曼方程是出了名的难以进行积分。大卫·希尔伯特 [[David Hilbert]] 花了数年时间试图解决这个问题，但没有实现任何真正的突破。
玻尔兹曼假设的碰撞项的形式是近似的。然而，对于理想气体，玻尔兹曼方程的标准[[Chapman–Enskog theory|查普曼-恩斯库格]]解是非常精确的；只有在激波条件下才会得到错误的结果。
玻尔兹曼假设的碰撞项的形式是近似的。然而，对于理想气体，玻尔兹曼方程的标准[[Chapman–Enskog theory|查普曼-恩斯库格]]解是非常精确的；只有在激波条件下才会得到错误的结果。
玻尔兹曼多年来一直试图用他的气体动力学方程——著名的“H”定理来“证明”热力学第二定律。然而，他在公式化碰撞项时所做的关键假设是“分子混沌”，该假设打破了时间反转对称性，这对于任何可能指向第二定律的内容都是必要的。玻尔兹曼表面上的成功仅仅来自概率假设，所以他与洛施密特 [[Johann Josef Loschmidt|Loschmidt]] 和其他人就'''洛施密特悖论 [[Loschmidt's paradox]]''' 的长期争论最终以他的失败告终。
玻尔兹曼多年来一直试图用他的气体动力学方程——著名的“H”定理来“证明”热力学第二定律。然而，他在公式化碰撞项时所做的关键假设是“分子混沌”，该假设打破了时间反转对称性，这对于任何可能指向第二定律的内容都是必要的。玻尔兹曼表面上的成功仅仅来自概率假设，所以他与洛施密特 [[Johann Josef Loschmidt|Loschmidt]] 和其他人就'''洛施密特悖论 [[Loschmidt's paradox]]''' 的长期争论最终以他的失败告终。
最后，在20世纪70年代科恩 E.G.D. Cohen和多夫曼 J. R. Dorfman证明了玻尔兹曼方程在高密度上的系统(幂级数)推广在数学上是不可能的。因此，稠密气体和液体的'''非平衡统计力学 [[non-equilibrium statistical mechanics|nonequilibrium statistical mechanics]]'''侧重于'''格林-久保亮五关系 [[Green–Kubo relations]]'''、'''涨落定理 [[fluctuation theorem|Fluctuation theorem]]''' 和其他方法。
最后，在20世纪70年代科恩 E.G.D. Cohen和多夫曼 J. R. Dorfman证明了玻尔兹曼方程在高密度上的系统(幂级数)推广在数学上是不可能的。因此，稠密气体和液体的'''非平衡统计力学 [[non-equilibrium statistical mechanics|nonequilibrium statistical mechanics]]'''侧重于'''格林-久保亮五关系 [[Green–Kubo relations]]'''、'''涨落定理 [[fluctuation theorem|Fluctuation theorem]]''' 和其他方法。
== 玻尔兹曼：“热力学第二定律是无序定律”==
== 玻尔兹曼：“热力学第二定律是无序定律”==
[[File:Zentralfriedhof Vienna - Boltzmann.JPG|thumb|right|Boltzmann's grave in the [[Zentralfriedhof]], Vienna, with bust and entropy formula.
[[File:Zentralfriedhof Vienna - Boltzmann.JPG|thumb|right|Boltzmann's grave in the [[Zentralfriedhof]], Vienna, with bust and entropy formula.
位于维也纳中央墓园的玻尔兹曼之墓，墓碑上刻有熵公式|链接=Special:FilePath/Zentralfriedhof_Vienna_-_Boltzmann.JPG]]
位于维也纳中央墓园的玻尔兹曼之墓，墓碑上刻有熵公式|链接=Special:FilePath/Zentralfriedhof_Vienna_-_Boltzmann.JPG]]
波尔兹曼认为热力学第二定律或“熵定律”是无序定律(或动态有序状态是“无限不可能的”)。
波尔兹曼认为热力学第二定律或“熵定律”是无序定律(或动态有序状态是“无限不可能的”)。
具体来讲，玻尔兹曼试图将其简化为随机碰撞函数，或机械粒子随机碰撞后的概率定律。继麦克斯韦之后<ref>Maxwell, J. (1871). Theory of heat. London: Longmans, Green & Co.</ref> ，玻尔兹曼把气体分子模拟成一个盒子里碰撞的台球，并指出每一次碰撞非平衡态速度分布(多组分子运动的速度和方向相同)会越来越无序，并导致最终的宏观均匀和微观最无序或最大熵状态(宏观均匀的状态对应所有场势或梯度的消失)。<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>因此，他认为，热力学第二定律描述了这样一个事实结果：在一个机械碰撞粒子的世界里，无序状态是最有可能的。因为可能的无序状态比有序状态要多得多，一个系统几乎总是会处于最无序状态——具有最多可接近的微观状态的宏观状态，如处于平衡状态的盒子里的气体——或向最无序状态移动。因此，玻尔兹曼总结道，一个动态有序的状态，即分子以“相同的速度和相同的方向”运动，是“最不可思议的可能情况…… 一种无限不可能的能量配置”。<ref>Boltzmann, L. (1974). The second law of thermodynamics. p. 20</ref>
玻尔兹曼完成了证明热力学第二定律只是一个统计事实的壮举。能量的逐渐无序化类似于一叠最初有序排列的卡牌重复洗牌过程中变得无序，也正如经历了无数次洗牌而重归初始有序状态的卡牌一样，我们的宇宙最终将在某一时刻回到最初的状态。（当人们试图估计宇宙正在消亡的时间线时，这种乐观的结论就显得有些平淡了，因为时间线很可能在它自然发生之前就已经过去了。）<ref>"[[Collier's Encyclopedia]]", Volume 19 Phyfe to Reni, "Physics", by David Park, p. 15</ref>熵增加的趋势似乎对初学热力学的人们造成困难，但从概率论的观点出发就很容易理解。考虑两个普通的骰子，都是正面朝上的。摇骰子后，发现这两个6面都朝上的概率很小(1/36)；因此，我们可以说，骰子的随机运动，就像分子由于热能的混乱碰撞，会导致系统从概率较小的状态变为概率较大的状态。数以百万计的骰子，就像热力学计算中数以百万计的原子一样，它们全部为6的概率变得如此之小，以至于系统必须移动到一个更可能的状态。<ref>"Collier's Encyclopedia", Volume 22 Sylt to Uruguay, Thermodynamics, by Leo Peters, p. 275</ref>然而，数学上出现所有骰子结果都是6或者都不是6的几率都很小，由于统计数据会趋于平衡，每36对骰子会有一对6出现，而洗过的牌有时会呈现出某种临时的顺序，即使整个牌是无序的。
== 获奖经历与荣誉 ==
== 获奖经历与荣誉 ==
1885年，他成为奥地利帝国科学院的成员，1887年，他成为格拉茨大学的校长。1888年，他被选为瑞典皇家科学院成员，1899年被选为皇家学会外国成员。<ref name="frs">{{cite web|archiveurl=https://web.archive.org/web/20150316060617/https://royalsociety.org/about-us/fellowship/fellows/|archivedate=2015-03-16|url=https://royalsociety.org/about-us/fellowship/fellows/|publisher=[[Royal Society]]|location=London|title=Fellows of the Royal Society}}</ref>另外许多事物都以他的名字命名。
==另见==
==另见==
*能量学
*能量学
*玻尔兹曼大脑
*玻尔兹曼大脑