路德维希·玻尔兹曼 Ludwig Edward Boltzmann

路德维希·玻尔兹曼 Ludwig Edward Boltzmann
路德维希·玻尔兹曼 Ludwig Edward Boltzmann
	Ludwig Boltzmann
Born	Ludwig Eduard Boltzmann; 20, 1844; Vienna, Austrian Empire
Died	September 5, 1906 (aged 62); Tybein, Triest, Austria-Hungary
Cause of death	Suicide by hanging
Nationality	Austrian
Alma mater	University of Vienna
Known for	Boltzmann constant; Boltzmann equation; Boltzmann distribution; Detailed balance; H-theorem; Maxwell–Boltzmann distribution; Stefan–Boltzmann constant; Stefan–Boltzmann law; Maxwell-Boltzmann statistics; Boltzmann factor; Epistemological idealism;
Awards	ForMemRS (1899)
	Scientific career
Fields	Physics
Institutions	University of Graz; University of Vienna; University of Munich; University of Leipzig;
Doctoral advisor	Josef Stefan
Other academic advisors	Robert Bunsen; Leo Königsberger; Gustav Kirchhoff; Hermann von Helmholtz;
Doctoral students	Paul Ehrenfest; Philipp Frank; Gustav Herglotz; Franc Hočevar; Ignacij Klemenčič;
Other notable students	Lise Meitner; Stefan Meyer;
Signature
	150px

此词条暂由彩云小译翻译，翻译字数共2507，未经人工整理和审校，带来阅读不便，请见谅。

Ludwig Eduard Boltzmann (German pronunciation: [ˈluːtvɪg ˈbɔlt͡sman]; February 20, 1844 – September 5, 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodynamics. In 1877 he provided the current definition of entropy, [math]\displaystyle{ S = k_{\rm B} \ln \Omega \! }[/math], interpreted as a measure of statistical disorder of a system.^[2] Max Planck named the constant, $k B$ , the Boltzmann constant.^[3]

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.^[4]

统计力学是现代物理学的支柱之一。它描述了宏观观察(如温度和压力)如何与围绕平均值波动的微观参数相关。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和列表各种材料的热力学量。

Biography

[math]\displaystyle{ S = k_B \ln W }[/math]

[ math ] s = k _ b ln w [ math ]

Childhood and education

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.^[5] Boltzmann attended high school in Linz, Upper Austria. When Boltzmann was 15, his father died.^[6]

The form of the collision term assumed by Boltzmann was approximate. However, for an ideal gas the standard 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 采用的碰撞术语的形式是近似的。然而，对于理想气体，玻尔兹曼方程的标准 Chapman-Enskog 解决方案是高度准确的。只有在冲击波条件下，理想气体才有可能得到不正确的结果。

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 Maxwell's work.^[6]

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 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 Loschmidt and others over Loschmidt's paradox ultimately ended in his failure.

多年来，Boltzmann 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，他在构造碰撞术语时所作的关键假设是“分子混沌” ，这个假设破坏了时间反转对称性，这对任何可能暗示第二定律的事物都是必要的。波尔兹曼表面上的成功仅仅来自于概率假设，因此他与洛施密特和其他人就洛施密特悖论的长期争论最终以他的失败而告终。

Academic career

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, nonequilibrium statistical mechanics for dense gases and liquids focuses on the Green–Kubo relations, the fluctuation theorem, and other approaches instead.

最终，在20世纪70年代的 e.g.d。和 j. r. Dorfman 证明了系统地(幂级数)将玻尔兹曼方程扩展到高密度在数学上是不可能的。因此，稠密气体和液体的非平衡态统计力学集中在格林-库伯关系、涨落定理和其他方法上。

In 1869 at age 25, thanks to a letter of recommendation written by Stefan,^[7] 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.

Ludwig Boltzmann and co-workers in Graz, 1887: (standing, from the left) Nernst, Streintz, Arrhenius, Hiecke, (sitting, from the left) Aulinger, Ettingshausen, Boltzmann, 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: Henriette (1880), Ida (1884) and Else (1891); and a son, Arthur Ludwig (1881).^[8] Boltzmann went back to Graz to take up the chair of Experimental Physics. Among his students in Graz were Svante Arrhenius and Walther Nernst.^[9]^[10] He spent 14 happy years in Graz and it was there that he developed his statistical concept of nature.

Boltzmann's grave in the Zentralfriedhof, Vienna, with bust and entropy formula.

玻尔兹曼的坟墓[维也纳中心区弗里德霍夫，胸围和熵公式]

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.

认为热力学第二定律定律或者“熵定律”是无序定律(或者说动态有序的状态是“无限不可能的”)的观点是由于 Boltzmann 对热力学第二定律的观点。

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

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, 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). 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."

尤其是，玻尔兹曼试图把它归结为一个随机碰撞函数，或者机械粒子随机碰撞后的概率定律。继麦克斯韦尔之后，玻尔兹曼将气体分子模拟为在一个盒子里碰撞的台球，指出每次碰撞时，非平衡速度分布(以相同速度和方向运动的分子群)将变得越来越无序，导致最终的宏观均匀和最大微观无序状态或最大熵状态(宏观均匀性对应于所有场势或梯度的消失)。因此，他认为第二定律仅仅是这样一个事实的结果，即在一个机械碰撞的粒子无序状态是最有可能的。因为可能存在的无序状态比有序状态多得多，所以一个系统几乎总是处于最大无序状态——具有最多可达微观状态的宏观状态，例如平衡状态下盒子中的气体——或者向无序状态移动。玻尔兹曼总结说，一个动态有序的状态，即分子以“同样的速度和同样的方向”运动，因此是“最不可能想象的情况... ... 一个无限不可能的能量构型”

In 1894, Boltzmann succeeded his teacher Joseph Stefan as Professor of Theoretical Physics at the University of Vienna.

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.) 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. However, mathematically the odds of all the dice results not being a pair sixes is also as hard as the ones of all of them being sixes, and since statistically the data tend to balance, one in every 36 pairs of dice will tend to be a pair of sixes, and the cards -when shuffled- will sometimes present a certain temporary sequence order even if in its whole the deck was disordered.

玻尔兹曼完成了一项壮举，证明了热力学第二定律只是一个统计事实。能量的逐渐失序类似于一副最初有序的扑克牌在重复洗牌过程中的失序，就像如果洗牌次数过多，扑克牌最终会恢复到原来的秩序一样，因此，整个宇宙必定会在某一天，纯粹出于偶然，恢复到它最初开始时的状态。(当人们试图估计可能在宇宙自发发生之前就已经过去的时间线时，这种对于正在消亡的宇宙这一观点的乐观结尾就变得有些沉默了。)熵增的趋势似乎给热力学初学者带来了困难，但从概率论的角度来看，熵增的趋势是容易理解的。考虑两个普通的骰子，两个6的面朝上。摇动骰子后，发现这两个六的概率很小(1/36) ，因此可以说，骰子的随机运动(搅动) ，就像分子因热能而产生的混沌碰撞，导致较不可能的状态转变为更可能的状态。数以百万计的骰子，就像热力学计算中所涉及的数以百万计的原子一样，它们都是六的概率变得如此微小，以至于系统必须移动到一个更有可能的状态。然而，从数学上计算出所有骰子不是一对六的概率也和所有骰子都是六的概率一样困难，而且由于统计数据趋于平衡，每36对骰子中就有一对是六，而当洗牌时，有时会呈现出某种暂时的序列顺序，即使整副牌是混乱的。

Final years and death

Boltzmann spent a great deal of effort in his final years defending his theories.^[11] 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.^[12] 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.^[11] In 1903, Boltzmann, together with Gustav von Escherich and Emil Müller, founded the Austrian Mathematical Society. His students included Karl Přibram, Paul Ehrenfest and Lise Meitner.^[11]

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 Foreign Member of the Royal Society (ForMemRS) in 1899. Numerous things are named in his honour.

1885年，他成为奥地利帝国科学院的一员，1887年，他成为卡尔·弗朗岑斯大学的院长。他于1888年被选为瑞典皇家科学院院士，并于1899年被选为皇家学会的外国会员。许多事物都以他的名字命名。

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.^[13]

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.^[11]^[14] 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).^[15]^[16]^[17]^[14]

He is buried in the Viennese Zentralfriedhof. His tombstone bears the inscription of Boltzmann's entropy formula: [math]\displaystyle{ S = k \cdot \log W }[/math]^[11]

Philosophy

模板:Unreferenced section

Boltzmann's kinetic theory of gases seemed to presuppose the reality of atoms and molecules, but almost all German philosophers and many scientists like Ernst Mach and the physical chemist Wilhelm Ostwald disbelieved their existence.^[18] 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 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, 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 statistical mechanics. They are also applicable to other phenomena that do not require quantum statistics and provide insight into the meaning of temperature.

Boltzmann's 1898 I₂ molecule diagram showing atomic "sensitive region" (α, β) overlap.

Most 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 atoms and molecules, 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 theoretical constructs. Only a couple of years after Boltzmann's death, Perrin's studies of colloidal suspensions (1908–1909), based on Einstein's theoretical studies of 1905, confirmed the values of Avogadro's number and Boltzmann's constant, convincing the world that the tiny particles really exist.

To quote Planck, "The logarithmic connection between entropy and probability was first stated by L. Boltzmann in his kinetic theory of gases".^[19] This famous formula for entropy S is^[20]^[21]

[math]\displaystyle{ S = k_B \ln W }[/math]

where k_B is Boltzmann's constant, and ln is the natural logarithm. W is Wahrscheinlichkeit, a German word meaning the probability of occurrence of a macrostate^[22] or, more precisely, the number of possible microstates corresponding to the macroscopic state of a system — the number of (unobservable) "ways" in the (observable) thermodynamic state of a system that can be realized by assigning different positions and momenta to the various molecules. Boltzmann's paradigm was an ideal gas of N identical particles, of which N_i are in the ith microscopic condition (range) of position and momentum. W can be counted using the formula for permutations

[math]\displaystyle{ W = N! \prod_i \frac{1}{N_i!} }[/math]

where i ranges over all possible molecular conditions, and where [math]\displaystyle{ ! }[/math] denotes factorial. The "correction" in the denominator account for 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.

Boltzmann equation

Boltzmann's bust in the courtyard arcade of the main building, University of Vienna.

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

[math]\displaystyle{ \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 temporal and 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 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 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.

Category:1844 births

类别: 1844名出生

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 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 Loschmidt and others over Loschmidt's paradox ultimately ended in his failure.

Category:1906 deaths

分类: 1906人死亡

Category:Scientists from Vienna

类别: 来自维也纳的科学家

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, nonequilibrium statistical mechanics for dense gases and liquids focuses on the Green–Kubo relations, the fluctuation theorem, and other approaches instead.

Category:Austrian physicists

类别: 奥地利物理学家

Category:Thermodynamicists

类别: 热力学家

Second thermodynamics law as a law of disorder

Category:Fluid dynamicists

类别: 流体动力学家

Boltzmann's grave in the Zentralfriedhof, Vienna, with bust and entropy formula.

Category:Physicists who committed suicide

类别: 自杀的物理学家

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.

Category:Mathematicians who committed suicide

类别: 自杀的数学家

Category:Burials at the Vienna Central Cemetery

类别: 维也纳中央公墓的葬礼

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,^[23] 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).^[24] 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."^[25]

Category:University of Vienna alumni

类别: 维也纳大学校友

Category:Members of the Royal Swedish Academy of Sciences

类别: 瑞典皇家科学院院士

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.)^[26] 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.^[27] 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]}, 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.

Category:Corresponding Members of the St Petersburg Academy of Sciences

类别: 圣彼得堡科学院通讯员

Category:Members of the Bavarian Maximilian Order for Science and Art

分类: 美国巴伐利亚马克西米兰科学与艺术勋章（Bavarian Maximilian Order for Science and the Arts）协会会员

Awards and honours

Category:Suicides by hanging in Italy

类别: 意大利上吊自杀

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 Foreign Member of the Royal Society (ForMemRS) in 1899.^[1] Numerous things are named in his honour.

Category:Foreign Members of the Royal Society

类别: 皇家学会的外国成员

Category:Foreign associates of the National Academy of Sciences

类别: 美国国家科学院的外国合伙人

↑ ^1.0 ^1.1 "Fellows of the Royal Society". London: Royal Society. Archived from the original on 2015-03-16.
↑ Klein, Martin (1970). "Boltzmann, Ludwig". In Preece, Warren E. (in English). Encyclopædia Britannica (hard cover). 3 (Commemorative Edition for Expo 70 ed.). Chicago: William Benton. p. 893a. ISBN 0852291353.
↑ Partington, J.R. (1949), An Advanced Treatise on Physical Chemistry, vol. volume 1, Fundamental Principles, The Properties of Gases, London: Longmans, Green and Co., p. 300 {{citation}}: |volume= has extra text (help)
↑ Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons.
↑ {{cite book where k_B is Boltzmann's constant, and ln is the natural logarithm. W is Wahrscheinlichkeit, a German word meaning the probability of occurrence of a macrostate or, more precisely, the number of possible microstates corresponding to the macroscopic state of a system — the number of (unobservable) "ways" in the (observable) thermodynamic state of a system that can be realized by assigning different positions and momenta to the various molecules. Boltzmann's paradigm was an ideal gas of N identical particles, of which N_i are in the ith microscopic condition (range) of position and momentum. W can be counted using the formula for permutations 其中 k < sub > b 是 Boltzmann 常数，ln 是自然对数。W 是 Wahrscheinlichkeit，一个德语单词，意思是发生宏观状态的可能性，或者更准确地说，是对应于系统宏观状态的可能的微观状态的数量---- 一个系统的可观测的热力学状态中的“方式”的数量，可以通过给各种分子分配不同的位置和动量来实现。玻耳兹曼的范式是全同粒子的理想气体，其中 n < sub > i 处于位置和动量的微观条件。W 可以用排列的公式来计算 |title=The Scientific 100 |first1=John [math]\displaystyle{ W = N! \prod_i \frac{1}{N_i!} }[/math] W = n！1} n i数学 |last1=Simmons |first2=Lynda where i ranges over all possible molecular conditions, and where [math]\displaystyle{ ! }[/math] denotes factorial. The "correction" in the denominator account for indistinguishable particles in the same condition. 这里 i 可以覆盖所有可能的分子条件，这里 < math > ! </math > 表示阶乘。在相同条件下，分母中的“修正”解释了不可区分的粒子。 |last2=Simmons |isbn=9780806536781 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. 由于他在1877年提出物理系统的能级可以是离散的，Boltzmann 也可以被认为是量子力学的先驱之一。 |page=123 |publisher=Kensington Publishing Corp. |year=2000 Boltzmann's bust in the courtyard arcade of the main building, University of Vienna. 在维也纳大学主楼的庭院拱廊中的玻尔兹曼的半身像。 }}
↑ ^6.0 ^6.1 James 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. F 表示一定时间内单个粒子的位置和动量的分布函数(见麦克斯韦-波兹曼分布) ，f 表示一个力，m 表示一个粒子的质量，t 表示时间，v 表示粒子的平均速度。, Ioan (2004 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 integrate. David Hilbert spent years trying to solve it without any real success. 原则上，上述方程在给定适当的边界条件下，完全描述了气体粒子系综的动力学行为。这个一阶微分方程的外观看似简单，因为它可以表示任意的单粒子分布函数。作用在粒子上的力的大小直接取决于它的速度分布函数。众所周知，玻尔兹曼方程是难以整合的。大卫 · 希尔伯特花了数年时间试图解决这个问题，但没有取得任何真正的成功。). [https://archive.org/details/remarkablephysic00jame Remarkable Physicists: From Galileo to Yukawa The Boltzmann equation was developed to describe the dynamics of an ideal gas. 玻尔兹曼方程是用来描述理想气体的动力学的。]. Cambridge University Press. p. 169 This equation describes the temporal and 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. 这个方程描述了单粒子相空间中点云密度分布的位置和动量的概率分布的时间和空间变化。(参见哈密顿力学。)左边的第一项表示分布函数的显式时间变化，第二项表示空间变化，第三项描述作用在粒子上的任何力的效应。方程的右边表示碰撞的影响。. ISBN 9780521017060. https://archive.org/details/remarkablephysic00jame.
↑ Južnič, Stanislav (December 2001). "Ludwig Boltzmann in prva študentka fizike in matematike slovenskega rodu" [Ludwig Boltzmann and the First Student of Physics and Mathematics of Slovene Descent]. Kvarkadabra.net (in Slovenian) (12). Retrieved 17 February 2012.{{cite journal}}: CS1 maint: unrecognized language (link)
↑ https://www.boltzmann.com/ludwig-boltzmann/biography/
↑ Jäger, Gustav; Nabl, Josef; Meyer, Stephan (April 1999). "Three Assistants on Boltzmann". Synthese. 119 (1–2): 69–84. doi:10.1023/A:1005239104047. S2CID 30499879. Paul Ehrenfest (1880–1933) along with Nernst, Arrhenius, and Meitner must be considered among Boltzmann's most outstanding students.
↑ "Walther Hermann Nernst". Archived from the original on 2008-06-12. Walther Hermann Nernst visited lectures by Ludwig Boltzmann

↑ ^11.0 ^11.1 ^11.2 ^11.3 ^11.4 Cercignani, Carlo (1998) Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford University Press.

↑ Max Planck (1896). "Gegen die neure Energetik". Annalen der Physik. 57 (1): 72–78. Bibcode:1896AnP...293...72P. doi:10.1002/andp.18962930107.
↑ The Boltzmann Equation: Theory and Applications, E.G.D. Cohen, W. Thirring, ed., Springer Science & Business Media, 2012
↑ ^14.0 ^14.1 Nina Bausek and Stefan Washietl (February 13, 2018). "Tragic deaths in science: Ludwig Boltzmann — a mind in disorder". Paperpile. Retrieved 2020-04-26.

↑ "Eureka! Science's greatest thinkers and their key breakthroughs", Hazel Muir, p.152,

↑ Boltzmann, Ludwig (1995). "Conclusions". In Blackmore, John T.. Ludwig Boltzmann: His Later Life and Philosophy, 1900-1906. 2. Springer. pp. 206–207. ISBN 978-0-7923-3464-4. https://books.google.com/books?id=apip-Jm9WuwC&pg=PA207.
↑ Upon Boltzmann's death, Friedrich ("Fritz") Hasenöhrl became his successor in the professorial chair of physics at Vienna.
↑ Bronowski, Jacob (1974). "World Within World". The Ascent Of Man. Little Brown & Co. p. 265. ISBN 978-0-316-10930-7. https://archive.org/details/ascentofmanbron00bron.
↑ Max Planck, p. 119.
↑ 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".
↑ An alternative is the information entropy definition introduced in 1948 by Claude Shannon.[1] 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 factorials or Stirling's approximation. Similar formulas are found, however, as far back as the work of Boltzmann, and explicitly in Gibbs (see reference).
↑ Pauli, Wolfgang (1973). Statistical Mechanics. Cambridge: MIT Press. ISBN 978-0-262-66035-8. , p. 21
↑ Maxwell, J. (1871). Theory of heat. London: Longmans, Green & Co.
↑ 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)
↑ Boltzmann, L. (1974). The second law of thermodynamics. p. 20
↑ "Collier's Encyclopedia", Volume 19 Phyfe to Reni, "Physics", by David Park, p. 15
↑ "Collier's Encyclopedia", Volume 22 Sylt to Uruguay, Thermodynamics, by Leo Peters, p. 275

[frs-1] 1.0 ^1.1 "Fellows of the Royal Society". London: Royal Society. Archived from the original on 2015-03-16.

[EncycloBritan-2] Klein, Martin (1970). "Boltzmann, Ludwig". In Preece, Warren E. (in English). Encyclopædia Britannica (hard cover). 3 (Commemorative Edition for Expo 70 ed.). Chicago: William Benton. p. 893a. ISBN 0852291353.

[3] Partington, J.R. (1949), An Advanced Treatise on Physical Chemistry, vol. volume 1, Fundamental Principles, The Properties of Gases, London: Longmans, Green and Co., p. 300 {{citation}}: |volume= has extra text (help)

[gibbs-4] Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons.

[5] {{cite book where k_B is Boltzmann's constant, and ln is the natural logarithm. W is Wahrscheinlichkeit, a German word meaning the probability of occurrence of a macrostate or, more precisely, the number of possible microstates corresponding to the macroscopic state of a system — the number of (unobservable) "ways" in the (observable) thermodynamic state of a system that can be realized by assigning different positions and momenta to the various molecules. Boltzmann's paradigm was an ideal gas of N identical particles, of which N_i are in the ith microscopic condition (range) of position and momentum. W can be counted using the formula for permutations 其中 k < sub > b 是 Boltzmann 常数，ln 是自然对数。W 是 Wahrscheinlichkeit，一个德语单词，意思是发生宏观状态的可能性，或者更准确地说，是对应于系统宏观状态的可能的微观状态的数量---- 一个系统的可观测的热力学状态中的“方式”的数量，可以通过给各种分子分配不同的位置和动量来实现。玻耳兹曼的范式是全同粒子的理想气体，其中 n < sub > i 处于位置和动量的微观条件。W 可以用排列的公式来计算 |title=The Scientific 100 |first1=John [math]\displaystyle{ W = N! \prod_i \frac{1}{N_i!} }[/math] W = n！1} n i数学 |last1=Simmons |first2=Lynda where i ranges over all possible molecular conditions, and where [math]\displaystyle{ ! }[/math] denotes factorial. The "correction" in the denominator account for indistinguishable particles in the same condition. 这里 i 可以覆盖所有可能的分子条件，这里 < math > ! </math > 表示阶乘。在相同条件下，分母中的“修正”解释了不可区分的粒子。 |last2=Simmons |isbn=9780806536781 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. 由于他在1877年提出物理系统的能级可以是离散的，Boltzmann 也可以被认为是量子力学的先驱之一。 |page=123 |publisher=Kensington Publishing Corp. |year=2000 Boltzmann's bust in the courtyard arcade of the main building, University of Vienna. 在维也纳大学主楼的庭院拱廊中的玻尔兹曼的半身像。 }}

[james2004-6] 6.0 ^6.1 James 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. F 表示一定时间内单个粒子的位置和动量的分布函数(见麦克斯韦-波兹曼分布) ，f 表示一个力，m 表示一个粒子的质量，t 表示时间，v 表示粒子的平均速度。, Ioan (2004 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 integrate. David Hilbert spent years trying to solve it without any real success. 原则上，上述方程在给定适当的边界条件下，完全描述了气体粒子系综的动力学行为。这个一阶微分方程的外观看似简单，因为它可以表示任意的单粒子分布函数。作用在粒子上的力的大小直接取决于它的速度分布函数。众所周知，玻尔兹曼方程是难以整合的。大卫 · 希尔伯特花了数年时间试图解决这个问题，但没有取得任何真正的成功。). [https://archive.org/details/remarkablephysic00jame Remarkable Physicists: From Galileo to Yukawa The Boltzmann equation was developed to describe the dynamics of an ideal gas. 玻尔兹曼方程是用来描述理想气体的动力学的。]. Cambridge University Press. p. 169 This equation describes the temporal and 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. 这个方程描述了单粒子相空间中点云密度分布的位置和动量的概率分布的时间和空间变化。(参见哈密顿力学。)左边的第一项表示分布函数的显式时间变化，第二项表示空间变化，第三项描述作用在粒子上的任何力的效应。方程的右边表示碰撞的影响。. ISBN 9780521017060. https://archive.org/details/remarkablephysic00jame.

[7] Južnič, Stanislav (December 2001). "Ludwig Boltzmann in prva študentka fizike in matematike slovenskega rodu" [Ludwig Boltzmann and the First Student of Physics and Mathematics of Slovene Descent]. Kvarkadabra.net (in Slovenian) (12). Retrieved 17 February 2012.{{cite journal}}: CS1 maint: unrecognized language (link)

[8] ttps://www.boltzmann.com/ludwig-boltzmann/biography/

[springer-9] Jäger, Gustav; Nabl, Josef; Meyer, Stephan (April 1999). "Three Assistants on Boltzmann". Synthese. 119 (1–2): 69–84. doi:10.1023/A:1005239104047. S2CID 30499879. Paul Ehrenfest (1880–1933) along with Nernst, Arrhenius, and Meitner must be considered among Boltzmann's most outstanding students.

[huji-10] "Walther Hermann Nernst". Archived from the original on 2008-06-12. Walther Hermann Nernst visited lectures by Ludwig Boltzmann

[Carlo-11] 11.0 ^11.1 ^11.2 ^11.3 ^11.4 Cercignani, Carlo (1998) Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford University Press.
v
t
e
Inline tags
Attribution

{{Attribution needed}}

{{By whom}}

{{Weasel-inline}}

{{Which?}}

{{Who}}

Citation

{{Better source}}

{{Citation needed}}
{{Citation needed span}}

{{Reference necessary}}

{{Citation needed (lead)}}

{{Failed verification}}

{{Full}}

{{Primary source-inline}}

{{Request quotation}}

{{Retracted}}

{{Third-party-inline}}

Incomplete

{{Author missing}}

{{Author incomplete}}

{{Date missing}}

{{ISBN missing}}

{{Page needed}}

{{Publisher missing}}

{{Title incomplete}}

{{Year missing}}

{{Dead link}}

Content

{{Contradict-inline}}

{{Contradiction-inline}}

{{Dubious|talk page section name}}

{{Examples}}

{{Inconsistent}}

{{List fact}}

{{Lopsided}}

{{POV-statement}}

Date and place

{{Clarify timeframe}}

{{Quantify}}

{{Update after}}

{{Update-small}}

{{When}}

{{Where}}

{{Year needed}}

Wikification

{{Disambiguation needed}}

{{Pronunciation needed}}

Wording

{{Ambiguous}}

{{Awkward}}

{{Buzz}}

{{Clarify}}

{{Clarify span}}

{{Definition}}

{{Elucidate}}

{{Expand acronym}}

{{Technical-statement}}

{{Vague|optional message to be displayed on mouseover}}

{{Why?}}

Category

[12] v

[13] t

[14] e

[15] {{Attribution needed}}

[16] {{By whom}}

[17] {{Weasel-inline}}

[18] {{Which?}}

[19] {{Who}}

[20] {{Better source}}

[21] {{Citation needed}}
{{Citation needed span}}

{{Reference necessary}}

[22] {{Citation needed span}}

[23] {{Reference necessary}}

[24] {{Citation needed (lead)}}

[25] {{Failed verification}}

[26] {{Full}}

[27] {{Primary source-inline}}

[28] {{Request quotation}}

[29] {{Retracted}}

[30] {{Third-party-inline}}

[31] {{Author missing}}

[32] {{Author incomplete}}

[33] {{Date missing}}

[34] {{ISBN missing}}

[35] {{Page needed}}

[36] {{Publisher missing}}

[37] {{Title incomplete}}

[38] {{Year missing}}

[39] {{Dead link}}

[40] {{Contradict-inline}}

[41] {{Contradiction-inline}}

[42] {{Dubious|talk page section name}}

[43] {{Examples}}

[44] {{Inconsistent}}

[45] {{List fact}}

[46] {{Lopsided}}

[47] {{POV-statement}}

[48] {{Clarify timeframe}}

[49] {{Quantify}}

[50] {{Update after}}

[51] {{Update-small}}

[52] {{When}}

[53] {{Where}}

[54] {{Year needed}}

[55] {{Disambiguation needed}}

[56] {{Pronunciation needed}}

[57] {{Ambiguous}}

[58] {{Awkward}}

[59] {{Buzz}}

[60] {{Clarify}}

[61] {{Clarify span}}

[62] {{Definition}}

[63] {{Elucidate}}

[64] {{Expand acronym}}

[65] {{Technical-statement}}

[66] {{Vague|optional message to be displayed on mouseover}}

[67] {{Why?}}

[12] Max Planck (1896). "Gegen die neure Energetik". Annalen der Physik. 57 (1): 72–78. Bibcode:1896AnP...293...72P. doi:10.1002/andp.18962930107.

[13] The Boltzmann Equation: Theory and Applications, E.G.D. Cohen, W. Thirring, ed., Springer Science & Business Media, 2012

[Paperpile-14] 14.0 ^14.1 Nina Bausek and Stefan Washietl (February 13, 2018). "Tragic deaths in science: Ludwig Boltzmann — a mind in disorder". Paperpile. Retrieved 2020-04-26.

[15] "Eureka! Science's greatest thinkers and their key breakthroughs", Hazel Muir, p.152,
v
t
e
Inline tags
Attribution

{{Attribution needed}}

{{By whom}}

{{Weasel-inline}}

{{Which?}}

{{Who}}

Citation

{{Better source}}

{{Citation needed}}
{{Citation needed span}}

{{Reference necessary}}

{{Citation needed (lead)}}

{{Failed verification}}

{{Full}}

{{Primary source-inline}}

{{Request quotation}}

{{Retracted}}

{{Third-party-inline}}

Incomplete

{{Author missing}}

{{Author incomplete}}

{{Date missing}}

{{ISBN missing}}

{{Page needed}}

{{Publisher missing}}

{{Title incomplete}}

{{Year missing}}

{{Dead link}}

Content

{{Contradict-inline}}

{{Contradiction-inline}}

{{Dubious|talk page section name}}

{{Examples}}

{{Inconsistent}}

{{List fact}}

{{Lopsided}}

{{POV-statement}}

Date and place

{{Clarify timeframe}}

{{Quantify}}

{{Update after}}

{{Update-small}}

{{When}}

{{Where}}

{{Year needed}}

Wikification

{{Disambiguation needed}}

{{Pronunciation needed}}

Wording

{{Ambiguous}}

{{Awkward}}

{{Buzz}}

{{Clarify}}

{{Clarify span}}

{{Definition}}

{{Elucidate}}

{{Expand acronym}}

{{Technical-statement}}

{{Vague|optional message to be displayed on mouseover}}

{{Why?}}

Category

[72] v

[73] t

[74] e

[75] {{Attribution needed}}

[76] {{By whom}}

[77] {{Weasel-inline}}

[78] {{Which?}}

[79] {{Who}}

[80] {{Better source}}

[81] {{Citation needed}}
{{Citation needed span}}

{{Reference necessary}}

[82] {{Citation needed span}}

[83] {{Reference necessary}}

[84] {{Citation needed (lead)}}

[85] {{Failed verification}}

[86] {{Full}}

[87] {{Primary source-inline}}

[88] {{Request quotation}}

[89] {{Retracted}}

[90] {{Third-party-inline}}

[91] {{Author missing}}

[92] {{Author incomplete}}

[93] {{Date missing}}

[94] {{ISBN missing}}

[95] {{Page needed}}

[96] {{Publisher missing}}

[97] {{Title incomplete}}

[98] {{Year missing}}

[99] {{Dead link}}

[100] {{Contradict-inline}}

[101] {{Contradiction-inline}}

[102] {{Dubious|talk page section name}}

[103] {{Examples}}

[104] {{Inconsistent}}

[105] {{List fact}}

[106] {{Lopsided}}

[107] {{POV-statement}}

[108] {{Clarify timeframe}}

[109] {{Quantify}}

[110] {{Update after}}

[111] {{Update-small}}

[112] {{When}}

[113] {{Where}}

[114] {{Year needed}}

[115] {{Disambiguation needed}}

[116] {{Pronunciation needed}}

[117] {{Ambiguous}}

[118] {{Awkward}}

[119] {{Buzz}}

[120] {{Clarify}}

[121] {{Clarify span}}

[122] {{Definition}}

[123] {{Elucidate}}

[124] {{Expand acronym}}

[125] {{Technical-statement}}

[126] {{Vague|optional message to be displayed on mouseover}}

[127] {{Why?}}

[16] Boltzmann, Ludwig (1995). "Conclusions". In Blackmore, John T.. Ludwig Boltzmann: His Later Life and Philosophy, 1900-1906. 2. Springer. pp. 206–207. ISBN 978-0-7923-3464-4. https://books.google.com/books?id=apip-Jm9WuwC&pg=PA207.

[17] Upon Boltzmann's death, Friedrich ("Fritz") Hasenöhrl became his successor in the professorial chair of physics at Vienna.

[18] Bronowski, Jacob (1974). "World Within World". The Ascent Of Man. Little Brown & Co. p. 265. ISBN 978-0-316-10930-7. https://archive.org/details/ascentofmanbron00bron.

[19] Max Planck, p. 119.

[20] 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".

[21] An alternative is the information entropy definition introduced in 1948 by Claude Shannon.[1] 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 factorials or Stirling's approximation. Similar formulas are found, however, as far back as the work of Boltzmann, and explicitly in Gibbs (see reference).

[22] Pauli, Wolfgang (1973). Statistical Mechanics. Cambridge: MIT Press. ISBN 978-0-262-66035-8. , p. 21

[23] Maxwell, J. (1871). Theory of heat. London: Longmans, Green & Co.

[24] 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)

[25] Boltzmann, L. (1974). The second law of thermodynamics. p. 20

[26] "Collier's Encyclopedia", Volume 19 Phyfe to Reni, "Physics", by David Park, p. 15

[27] "Collier's Encyclopedia", Volume 22 Sylt to Uruguay, Thermodynamics, by Leo Peters, p. 275

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]