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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.

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.

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.

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==

==Physics==

Boltzmann's most important scientific contributions were in [[kinetic theory of gases|kinetic theory]], including for motivating the [[Maxwell–Boltzmann distribution]] as a description of molecular speeds in a gas. [[Maxwell–Boltzmann statistics]] and the [[Boltzmann distribution]] remain central in the foundations of [[classical mechanics|classical]] statistical mechanics. They are also applicable to other [[phenomenon|phenomena]] that do not require [[Maxwell–Boltzmann statistics#Limits of applicability|quantum statistics]] and provide insight into the meaning of [[thermodynamic temperature|temperature]].

Boltzmann's most important scientific contributions were in [[kinetic theory of gases|kinetic theory]], including for motivating the [[Maxwell–Boltzmann distribution]] as a description of molecular speeds in a gas. [[Maxwell–Boltzmann statistics]] and the [[Boltzmann distribution]] remain central in the foundations of [[classical mechanics|classical]] statistical mechanics. They are also applicable to other [[phenomenon|phenomena]] that do not require [[Maxwell–Boltzmann statistics#Limits of applicability|quantum statistics]] and provide insight into the meaning of [[thermodynamic temperature|temperature]].

[[File:Boltzmanns-molecule.jpg|225px|thumb|right|Boltzmann's 1898 I₂ molecule diagram showing atomic "sensitive region" (α, β) overlap.|链接=Special:FilePath/Boltzmanns-molecule.jpg]]

[[File:Boltzmanns-molecule.jpg|225px|thumb|right|Boltzmann's 1898 I₂ molecule diagram showing atomic "sensitive region" (α, β) overlap.|链接=Special:FilePath/Boltzmanns-molecule.jpg]]

[[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]].

[[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>

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>

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.

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.