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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.

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.

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，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(1880年)、艾达(1884年)、爱尔莎(1891年)、亚瑟·路德维希(1881年)。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·阿伦尼乌斯和瓦尔特·能斯特。

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，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(1880年)、艾达(1884年)、爱尔莎(1891年)、亚瑟·路德维希(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·阿伦尼乌斯和瓦尔特·能斯特。<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>

他在格拉茨度过了快乐的14年，正是在那里，他发展了他的自然界的统计概念。

他在格拉茨度过了快乐的14年，正是在那里，他发展了他的自然界的统计概念。

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]].

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]].

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，但几乎所有的德国哲学家和许多科学家，如恩斯特·马赫(Ernst Mach)和物理化学家威廉·奥斯特瓦尔德(Wilhelm Ostwald)，都不相信它们的存在。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，但几乎所有的德国哲学家和许多科学家，如恩斯特·马赫(Ernst Mach)和物理化学家威廉·奥斯特瓦尔德(Wilhelm Ostwald)，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。

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.

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.

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.

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.

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”。也就是著名的熵公式：<math> S = k_B \ln W </math>，其中''k_B'' 是玻尔兹曼常数，''W'' 代表德文中宏观状态出现的概率，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_B'' 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。

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 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.