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20世纪40年代末，'''斯坦尼斯拉夫·乌拉姆 Stanislaw Ulam'''在洛斯阿拉莫斯国家实验室研究核武器项目时，发明了现代版的马尔可夫链蒙特卡罗方法。在乌拉姆的突破之后，约翰·冯·诺伊曼立即意识到了它的重要性。冯·诺伊曼为ENIAC（人类第一台电子数字积分计算机）编写了程序来进行蒙特卡罗计算。1946年，洛斯阿拉莫斯的核武器物理学家正在研究中子在可裂变材料中的扩散。尽管拥有大部分必要的数据，例如中子在与原子核碰撞之前在物质中的平均运行距离，以及碰撞后中子可能释放出多少能量，但洛斯阿拉莫斯的物理学家们无法用传统的、确定性的数学方法解决这个问题。此时乌拉姆建议使用随机实验。他后来回忆当初灵感产生过程:

20世纪40年代末，'''斯坦尼斯拉夫·乌拉姆 Stanislaw Ulam'''在洛斯阿拉莫斯国家实验室研究核武器项目时，发明了现代版的马尔可夫链蒙特卡罗方法。在乌拉姆的突破之后，约翰·冯·诺伊曼立即意识到了它的重要性。冯·诺伊曼为ENIAC（人类第一台电子数字积分计算机）编写了程序来进行蒙特卡罗计算。1946年，洛斯阿拉莫斯的核武器物理学家正在研究中子在可裂变材料中的扩散。尽管拥有大部分必要的数据，例如中子在与原子核碰撞之前在物质中的平均运行距离，以及碰撞后中子可能释放出多少能量，但洛斯阿拉莫斯的物理学家们无法用传统的、确定性的数学方法解决这个问题。此时乌拉姆建议使用随机实验。他后来回忆当初灵感产生过程:

The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by a question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations. Later [in 1946], I described the idea to John von Neumann, and we began to plan actual calculations.我最初的想法和尝试蒙特卡洛法是在1946年，当时我正从疾病中康复，时常玩单人纸牌游戏。那时我会思考这样一个问题：一盘52张的加菲尔德纸牌成功出牌的几率有多大?在花了大量时间尝试通过纯粹的组合计算来估计它们之后，我想知道是否有一种比“抽象思维”更实际的方法，可能不是将它展开100次，然后简单地观察和计算成功的游戏数量。在快速计算机新时代开始时，这已经是可以想象的了，我立刻想到了中子扩散和其他数学物理的问题，以及更一般的情形—如何将由某些微分方程描述的过程转换成可解释为一系列随机操作的等价形式。后来(1946年)，我向约翰·冯·诺伊曼描述了这个想法，然后我们开始计划实际的计算。

The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by a question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations. Later [in 1946], I described the idea to John von Neumann, and we began to plan actual calculations.

 

我最初的想法和尝试蒙特卡洛法是在1946年，当时我正从疾病中康复，时常玩单人纸牌游戏。那时我会思考这样一个问题：一盘52张的加菲尔德纸牌成功出牌的几率有多大?在花了大量时间尝试通过纯粹的组合计算来估计它们之后，我想知道是否有一种比“抽象思维”更实际的方法，可能不是将它展开100次，然后简单地观察和计算成功的游戏数量。在快速计算机新时代开始时，这已经是可以想象的了，我立刻想到了中子扩散和其他数学物理的问题，以及更一般的情形—如何将由某些微分方程描述的过程转换成可解释为一系列随机操作的等价形式。后来(1946年)，我向约翰·冯·诺伊曼描述了这个想法，然后我们开始计划实际的计算。

{{quote|The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by a question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a [[Canfield (solitaire)|Canfield solitaire]] laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations. Later [in 1946], I described the idea to [[John von Neumann]], and we began to plan actual calculations.{{sfn|Eckhardt|1987}}}}

{{quote|The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by a question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a [[Canfield (solitaire)|Canfield solitaire]] laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations. Later [in 1946], I described the idea to [[John von Neumann]], and we began to plan actual calculations.{{sfn|Eckhardt|1987}}}}

The theory of more sophisticated mean field type particle Monte Carlo methods had certainly started by the mid-1960s, with the work of Henry P. McKean Jr. on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics. We also quote an earlier pioneering article by Theodore E. Harris and Herman Kahn, published in 1951, using mean field genetic-type Monte Carlo methods for estimating particle transmission energies. Mean field genetic type Monte Carlo methodologies are also used as heuristic natural search algorithms (a.k.a. metaheuristic) in evolutionary computing. The origins of these mean field computational techniques can be traced to 1950 and 1954 with the work of Alan Turing on genetic type mutation-selection learning machines and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey.

The theory of more sophisticated mean field type particle Monte Carlo methods had certainly started by the mid-1960s, with the work of Henry P. McKean Jr. on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics. We also quote an earlier pioneering article by Theodore E. Harris and Herman Kahn, published in 1951, using mean field genetic-type Monte Carlo methods for estimating particle transmission energies. Mean field genetic type Monte Carlo methodologies are also used as heuristic natural search algorithms (a.k.a. metaheuristic) in evolutionary computing. The origins of these mean field computational techniques can be traced to 1950 and 1954 with the work of Alan Turing on genetic type mutation-selection learning machines and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey.

更复杂的平均场型粒子蒙特卡罗方法的理论产生于20世纪60年代中期，最初来自于'''小亨利·麦基恩 Henry P. McKean Jr.''' 研究流体力学中出现的一类非线性抛物型偏微分方程的马尔可夫解释。我们还引用Theodore E. Harris和Herman Kahn在1951年发表的一篇较早的开创性文章，使用平均场遗传型蒙特卡罗方法来估计粒子传输能量平均场遗传型蒙特卡罗方法在进化计算中也被用作启发式自然搜索算法(又称元启发式)。这些平均场计算技术的起源可以追溯到1950年和1954年，当时艾伦·图灵(Alan Turing)在基因类型突变-选择学习机器[24]上的工作，以及新泽西州普林斯顿高等研究院(Institute for Advanced Study)的尼尔斯·阿尔·巴里里利(Nils Aall Barricelli)的文章。

更复杂的平均场型粒子蒙特卡罗方法的理论产生于20世纪60年代中期，最初来自于'''小亨利·麦基恩 Henry P. McKean Jr.''' 研究流体力学中出现的一类非线性抛物型偏微分方程的马尔可夫解释。'''西奥多·爱德华·哈里斯 Theodore E. Harris'''和'''赫曼·卡恩 Herman Kahn'''在1951年发表了一篇开创性文章，使用平均场遗传型蒙特卡罗方法来估计粒子传输能量。这一方法在演化计算中也被用作启发式自然搜索算法(又称元启发式)。这些平均场计算技术的起源可以追溯到1950年和1954年，当时'''艾伦·图灵 Alan Turing'''在基因类型突变-选择学习机器上的工作，以及来自新泽西州普林斯顿高等研究院的'''尼尔斯·阿尔·巴里里利 Nils Aall Barricelli'''的文章。

[[Quantum Monte Carlo]], and more specifically [[Diffusion Monte Carlo|diffusion Monte Carlo methods]] can also be interpreted as a mean field particle Monte Carlo approximation of [[Richard Feynman|Feynman]]–[[Mark Kac|Kac]] path integrals.<ref name="dp04">{{cite book |last = Del Moral |first = Pierre|title = Feynman–Kac formulae. Genealogical and interacting particle approximations |year = 2004 |publisher = Springer |quote = Series: Probability and Applications |url = https://www.springer.com/mathematics/probability/book/978-0-387-20268-6 |page = 575 |isbn = 9780387202686|series = Probability and Its Applications}}</ref><ref name="dmm002">{{cite book

[[Quantum Monte Carlo]], and more specifically [[Diffusion Monte Carlo|diffusion Monte Carlo methods]] can also be interpreted as a mean field particle Monte Carlo approximation of [[Richard Feynman|Feynman]]–[[Mark Kac|Kac]] path integrals.<ref name="dp04">{{cite book |last = Del Moral |first = Pierre|title = Feynman–Kac formulae. Genealogical and interacting particle approximations |year = 2004 |publisher = Springer |quote = Series: Probability and Applications |url = https://www.springer.com/mathematics/probability/book/978-0-387-20268-6 |page = 575 |isbn = 9780387202686|series = Probability and Its Applications}}</ref><ref name="dmm002">{{cite book

  }}</ref><ref name="dmm00m">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Miclo|first2 = Laurent|title = A Moran particle system approximation of Feynman–Kac formulae.|journal = Stochastic Processes and Their Applications |year = 2000|volume = 86|issue = 2|pages = 193–216|doi = 10.1016/S0304-4149(99)00094-0|doi-access = free}}</ref><ref name="dm-esaim03">{{cite journal|last1 = Del Moral|first1 = Pierre|title = Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups|journal = ESAIM Probability & Statistics|date = 2003|volume = 7|pages = 171–208|url = http://journals.cambridge.org/download.php?file=%2FPSS%2FPSS7%2FS1292810003000016a.pdf&code=a0dbaa7ffca871126dc05fe2f918880a|doi = 10.1051/ps:2003001|doi-access = free}}</ref><ref name="caffarel1">{{cite journal|last1 = Assaraf|first1 = Roland|last2 = Caffarel|first2 = Michel|last3 = Khelif|first3 = Anatole|title = Diffusion Monte Carlo Methods with a fixed number of walkers|journal = Phys. Rev. E|url = http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf|date = 2000|volume = 61|issue = 4|pages = 4566–4575|doi = 10.1103/physreve.61.4566|pmid = 11088257|bibcode = 2000PhRvE..61.4566A|url-status = dead|archiveurl = https://web.archive.org/web/20141107015724/http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf|archivedate = 2014-11-07 }}</ref><ref name="caffarel2">{{cite journal|last1 = Caffarel|first1 = Michel|last2 = Ceperley|first2 = David |last3 = Kalos|first3 = Malvin|title = Comment on Feynman–Kac Path-Integral Calculation of the Ground-State Energies of Atoms|journal = Phys. Rev. Lett.|date = 1993|volume = 71|issue = 13|doi = 10.1103/physrevlett.71.2159|bibcode = 1993PhRvL..71.2159C|pages=2159|pmid=10054598}}</ref><ref name="h84">{{cite journal |last = Hetherington|first = Jack, H.|title = Observations on the statistical iteration of matrices|journal = Phys. Rev. A |date = 1984|volume = 30|issue = 2713|doi = 10.1103/PhysRevA.30.2713|pages = 2713–2719|bibcode = 1984PhRvA..30.2713H}}</ref> The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and [[Robert D. Richtmyer|Robert Richtmyer]] who developed in 1948 a mean field particle interpretation of neutron-chain reactions,<ref>{{cite journal|last1 = Fermi|first1 = Enrique|last2 = Richtmyer|first2 = Robert, D.|title = Note on census-taking in Monte Carlo calculations|journal = LAM|date = 1948|volume = 805|issue = A|url = http://scienze-como.uninsubria.it/bressanini/montecarlo-history/fermi-1948.pdf|quote = Declassified report Los Alamos Archive}}</ref> but the first heuristic-like and genetic type particle algorithm (a.k.a. Resampled or Reconfiguration Monte Carlo methods) for estimating ground state energies of quantum systems (in reduced matrix models) is due to Jack H. Hetherington in 1984<ref name="h84" /> In molecular chemistry, the use of genetic heuristic-like particle methodologies (a.k.a. pruning and enrichment strategies) can be traced back to 1955 with the seminal work of [[Marshall Rosenbluth|Marshall N. Rosenbluth]] and [[Arianna W. Rosenbluth]].<ref name=":0">{{cite journal |last1 = Rosenbluth|first1 = Marshall, N.|last2 = Rosenbluth|first2 = Arianna, W.|title = Monte-Carlo calculations of the average extension of macromolecular chains|journal = J. Chem. Phys.|date = 1955|volume = 23|issue = 2|pages = 356–359|bibcode = 1955JChPh..23..356R|doi = 10.1063/1.1741967 |s2cid = 89611599|url = https://semanticscholar.org/paper/1570c85ba9aca1cb413ada31e215e0917c3ccba7}}</ref>

  }}</ref><ref name="dmm00m">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Miclo|first2 = Laurent|title = A Moran particle system approximation of Feynman–Kac formulae.|journal = Stochastic Processes and Their Applications |year = 2000|volume = 86|issue = 2|pages = 193–216|doi = 10.1016/S0304-4149(99)00094-0|doi-access = free}}</ref><ref name="dm-esaim03">{{cite journal|last1 = Del Moral|first1 = Pierre|title = Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups|journal = ESAIM Probability & Statistics|date = 2003|volume = 7|pages = 171–208|url = http://journals.cambridge.org/download.php?file=%2FPSS%2FPSS7%2FS1292810003000016a.pdf&code=a0dbaa7ffca871126dc05fe2f918880a|doi = 10.1051/ps:2003001|doi-access = free}}</ref><ref name="caffarel1">{{cite journal|last1 = Assaraf|first1 = Roland|last2 = Caffarel|first2 = Michel|last3 = Khelif|first3 = Anatole|title = Diffusion Monte Carlo Methods with a fixed number of walkers|journal = Phys. Rev. E|url = http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf|date = 2000|volume = 61|issue = 4|pages = 4566–4575|doi = 10.1103/physreve.61.4566|pmid = 11088257|bibcode = 2000PhRvE..61.4566A|url-status = dead|archiveurl = https://web.archive.org/web/20141107015724/http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf|archivedate = 2014-11-07 }}</ref><ref name="caffarel2">{{cite journal|last1 = Caffarel|first1 = Michel|last2 = Ceperley|first2 = David |last3 = Kalos|first3 = Malvin|title = Comment on Feynman–Kac Path-Integral Calculation of the Ground-State Energies of Atoms|journal = Phys. Rev. Lett.|date = 1993|volume = 71|issue = 13|doi = 10.1103/physrevlett.71.2159|bibcode = 1993PhRvL..71.2159C|pages=2159|pmid=10054598}}</ref><ref name="h84">{{cite journal |last = Hetherington|first = Jack, H.|title = Observations on the statistical iteration of matrices|journal = Phys. Rev. A |date = 1984|volume = 30|issue = 2713|doi = 10.1103/PhysRevA.30.2713|pages = 2713–2719|bibcode = 1984PhRvA..30.2713H}}</ref> The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and [[Robert D. Richtmyer|Robert Richtmyer]] who developed in 1948 a mean field particle interpretation of neutron-chain reactions,<ref>{{cite journal|last1 = Fermi|first1 = Enrique|last2 = Richtmyer|first2 = Robert, D.|title = Note on census-taking in Monte Carlo calculations|journal = LAM|date = 1948|volume = 805|issue = A|url = http://scienze-como.uninsubria.it/bressanini/montecarlo-history/fermi-1948.pdf|quote = Declassified report Los Alamos Archive}}</ref> but the first heuristic-like and genetic type particle algorithm (a.k.a. Resampled or Reconfiguration Monte Carlo methods) for estimating ground state energies of quantum systems (in reduced matrix models) is due to Jack H. Hetherington in 1984<ref name="h84" /> In molecular chemistry, the use of genetic heuristic-like particle methodologies (a.k.a. pruning and enrichment strategies) can be traced back to 1955 with the seminal work of [[Marshall Rosenbluth|Marshall N. Rosenbluth]] and [[Arianna W. Rosenbluth]].<ref name=":0">{{cite journal |last1 = Rosenbluth|first1 = Marshall, N.|last2 = Rosenbluth|first2 = Arianna, W.|title = Monte-Carlo calculations of the average extension of macromolecular chains|journal = J. Chem. Phys.|date = 1955|volume = 23|issue = 2|pages = 356–359|bibcode = 1955JChPh..23..356R|doi = 10.1063/1.1741967 |s2cid = 89611599|url = https://semanticscholar.org/paper/1570c85ba9aca1cb413ada31e215e0917c3ccba7}}</ref>

Kalos and Whitlock point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. "Indeed, the same computer code can be viewed simultaneously as a 'natural simulation' or as a solution of the equations by natural sampling."

Kalos and Whitlock point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. "Indeed, the same computer code can be viewed simultaneously as a 'natural simulation' or as a solution of the equations by natural sampling."