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In the physical sciences, scientists developed in the 19th century the discipline of [[statistical mechanics]], where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as [[Rudolf Clausius]], most of the work had little or no randomness.<ref name="Truesdell1975page22">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=22–23|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush1967page150">{{cite journal|last1=Brush|first1=Stephen G.|title=Foundations of statistical mechanics 1845?1915|journal=Archive for History of Exact Sciences|volume=4|issue=3|year=1967|pages=150–151|issn=0003-9519|doi=10.1007/BF00412958}}</ref>

In the physical sciences, scientists developed in the 19th century the discipline of [[statistical mechanics]], where physical systems, such as containers filled with gases, can be regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as [[Rudolf Clausius]], most of the work had little or no randomness.<ref name="Truesdell1975page22">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=22–23|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush1967page150">{{cite journal|last1=Brush|first1=Stephen G.|title=Foundations of statistical mechanics 1845?1915|journal=Archive for History of Exact Sciences|volume=4|issue=3|year=1967|pages=150–151|issn=0003-9519|doi=10.1007/BF00412958}}</ref>

在物理科学中，科学家们在19世纪发展了[[统计力学]]这门学科，在这个学科中，物理系统，例如装满气体的容器，可以从数学上看作或处理为许多运动粒子的集合。尽管有些科学家试图将随机性纳入统计物理学，比如[[Rudolf Clausius]]，大部分工作几乎没有或几乎没有随机性。<ref name=“Truesdell1975page22”>{{{cite journal 124; last1=Truesdell | first 1=C.；title=早期气体动力学理论气体；journal=Archive for History of Exact科学科学史| volume=15 | issue=1 | year=1975；pages22–23 | issn=0003-9519 | doi=10.1007/bf003272232}}</ref name=“Brus1961967page150”{{ref ref name=“Brus1961967page150”[页150”[[[[[{引用journal | last1=Brush | first1=Stephen G.| title=统计力学基础1845？1915年|期刊=精确科学史档案|卷=4 |问题=3 |年=1967 |页=150–151 | issn=0003-9519 | doi=10.1007/BF00412958}</ref>

在物理科学中，科学家们在19世纪发展了[[统计力学]]这门学科，在这个学科中，物理系统，例如装满气体的容器，可以从数学上看作或处理为许多运动粒子的集合。尽管有些科学家试图将随机性纳入统计物理学，比如[[Rudolf Clausius]]，大部分工作没有或几乎没有随机性。<ref name=“Truesdell1975page22”>{cite journal | last1=Truesdell|first1=C.| title=早期气体动力学理论| journal=精确科学史档案|卷=15 |问题=1 |年份=1975 |页=22–23 | issn=0003-9519 | doi=10.1007/BF00327232}}<ref name=“Brush1967page150”>{{cite journal | last1=Brush | first1=Stephen G.| title=统计力学基础1845？1915年|期刊=精确科学史档案|卷=4 |问题=3 |年=1967 |页=150–151 | issn=0003-9519 | doi=10.1007/BF00412958}}</ref>

This changed in 1859 when [[James Clerk Maxwell]] contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he assumed the gas particles move in random directions at random velocities.<ref name="Truesdell1975page31">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=31–32|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush1958">{{cite journal|last1=Brush|first1=S.G.|title=The development of the kinetic theory of gases IV. Maxwell|journal=Annals of Science|volume=14|issue=4|year=1958|pages=243–255|issn=0003-3790|doi=10.1080/00033795800200147}}</ref> The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, [[Ludwig Boltzmann]] and [[Josiah Gibbs]], which would later have an influence on [[Albert Einstein]]'s mathematical model for [[Brownian movement]].<ref name="Brush1968page15">{{cite journal|last1=Brush|first1=Stephen G.|title=A history of random processes|journal=Archive for History of Exact Sciences|volume=5|issue=1|year=1968|pages=15–16|issn=0003-9519|doi=10.1007/BF00328110}}</ref>

This changed in 1859 when [[James Clerk Maxwell]] contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he assumed the gas particles move in random directions at random velocities.<ref name="Truesdell1975page31">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=31–32|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush1958">{{cite journal|last1=Brush|first1=S.G.|title=The development of the kinetic theory of gases IV. Maxwell|journal=Annals of Science|volume=14|issue=4|year=1958|pages=243–255|issn=0003-3790|doi=10.1080/00033795800200147}}</ref> The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, [[Ludwig Boltzmann]] and [[Josiah Gibbs]], which would later have an influence on [[Albert Einstein]]'s mathematical model for [[Brownian movement]].<ref name="Brush1968page15">{{cite journal|last1=Brush|first1=Stephen G.|title=A history of random processes|journal=Archive for History of Exact Sciences|volume=5|issue=1|year=1968|pages=15–16|issn=0003-9519|doi=10.1007/BF00328110}}</ref>

这一点在1859年发生了变化，当时[[James clark Maxwell]]对该领域做出了重大贡献，更具体地说，是对气体动力学理论的贡献，通过介绍他的工作，他假设气体粒子以随机速度随机方向移动name=“Brush1958”>{cite journal | last1=Brush | first1=S.G.| title=气体动力学理论的发展IV.麦克斯韦期刊=科学年鉴|卷=14 |问题=4 |年份=1958 |页数=243–255 | issn=0003-3790 | doi=10.1080/0003379580020147}}</ref>气体动力学理论和统计物理在下半年继续发展19世纪，克劳修斯，[[Ludwig Boltzmann]]和[[Josiah Gibbs]]主要完成的工作，后来影响到[[阿尔伯特爱因斯坦]]的[[布朗运动]][[布朗运动]]的数学模型数学模型。<ref name=“brus1961968page15”>{{cite journal | last1=Brush；first1=斯蒂芬G.；title=随机过程历史的历史；journal=journal=ArchiForHistory of EquaScience | volume=5；issue=1 | year=1968年| pages15–16 | issn=124;issn=0003-9519；doi=10.1007/BF00328110}}}}</ref</ref<在

这一点在1859年发生了变化，当时[[James clark Maxwell]]对该领域做出了重大贡献，更具体地说，是对气体动力学理论的贡献，通过介绍他的工作，他假设气体粒子以随机速度随机方向移动<ref name=“Truesdell1975page31”>{cite journal | last1=Truesdell|first1=C.| title=Early dynamic theory of gas | journal=精确科学史档案|卷=15 |问题=1 |年份=1975 | pages=31–32 | issn=0003-9519 | doi=10.1007/BF00327232}}<ref name=“Brush1958”>{cite journal | last1=Brush | first1=S.G.| title=气体动力学理论的发展IV.麦克斯韦期刊=科学年鉴|卷=14 |问题=4 |年份=1958 |页=243–255 | issn=0003-3790 | doi=10.1080/0003379580020147}}</ref>气体动力学理论和统计物理在19世纪后半叶继续发展，主要由克劳修斯，[[路德维希玻尔兹曼]]和[[约西亚吉布斯]]完成，这项工作后来对[[阿尔伯特爱因斯坦]]关于[[布朗运动]]的数学模型产生了影响。<ref name="Brush1968page15">{{cite journal|last1=Brush|first1=Stephen G.|title=A history of random processes|journal=Archive for History of Exact Sciences|volume=5|issue=1|year=1968|pages=15–16|issn=0003-9519|doi=10.1007/BF00328110}}</ref>

===Measure theory and probability theory测度论与概率论===

===Measure theory and probability theory测度论与概率论===

After the publication of Kolmogorov's book, further fundamental work on probability theory and stochastic processes was done by Khinchin and Kolmogorov as well as other mathematicians such as [[Joseph Doob]], [[William Feller]], [[Maurice Fréchet]], [[Paul Lévy (mathematician)|Paul Lévy]], [[Wolfgang Doeblin]], and [[Harald Cramér]].<ref name="Bingham2000"/><ref name="Cramer1976"/>

After the publication of Kolmogorov's book, further fundamental work on probability theory and stochastic processes was done by Khinchin and Kolmogorov as well as other mathematicians such as [[Joseph Doob]], [[William Feller]], [[Maurice Fréchet]], [[Paul Lévy (mathematician)|Paul Lévy]], [[Wolfgang Doeblin]], and [[Harald Cramér]].<ref name="Bingham2000"/><ref name="Cramer1976"/>

在科尔莫戈洛夫的书出版后，钦钦和科尔莫戈洛夫以及其他数学家如[[Joseph Doob]]、[[William Feller]]、[[Maurice Fréchet]]、[[Paul Lévy（数学家）| Paul Lévy]]、[[Wolfgang Doeblin]]等对概率论和随机过程进行了进一步的基础性工作，和[[Harald Cramér].<ref name=“Bingham2000”/><ref name=“Cramer1976”/>

在科尔莫戈洛夫的书出版后，钦钦和科尔莫戈洛夫以及其他数学家如[[Joseph Doob]]、[[William Feller]]、[[Maurice Fréchet]]、[[Paul Lévy（数学家）| Paul Lévy]]、[[Wolfgang Doeblin]]等对概率论和随机过程进行了进一步的基础性工作，和[[Harald Cramér]]。<ref name=“Bingham2000”/><ref name=“Cramer1976”/>

Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory".<ref name="Cramer1976"/> [[World War II]] greatly interrupted the development of probability theory, causing, for example, the migration of Feller from [[Sweden]] to the [[United States|United States of America]]<ref name="Cramer1976"/> and the death of Doeblin, considered now a pioneer in stochastic processes.<ref name="Lindvall1991">{{cite journal|last1=Lindvall|first1=Torgny|title=W. Doeblin, 1915-1940|journal=The Annals of Probability|volume=19|issue=3|year=1991|pages=929–934|issn=0091-1798|doi=10.1214/aop/1176990329|doi-access=free}}</ref>

Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory".<ref name="Cramer1976"/> [[World War II]] greatly interrupted the development of probability theory, causing, for example, the migration of Feller from [[Sweden]] to the [[United States|United States of America]]<ref name="Cramer1976"/> and the death of Doeblin, considered now a pioneer in stochastic processes.<ref name="Lindvall1991">{{cite journal|last1=Lindvall|first1=Torgny|title=W. Doeblin, 1915-1940|journal=The Annals of Probability|volume=19|issue=3|year=1991|pages=929–934|issn=0091-1798|doi=10.1214/aop/1176990329|doi-access=free}}</ref>