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第205行：
第205行： + + + + + − 玻尔兹曼分布表明，能量较低的状态被占据的概率总是高于能量较高的状态被占据的概率。玻尔兹曼分布还给出两个量子态被占据的概率之间的定量关系。状态 i 和状态 j 的概率比为+ − Q={\sum_{i=1}^{M}{e^{- {\varepsilon}_i / k T}}}+ − − <math> − − 《数学》 第233行：
第234行： − + − − 第253行：
第252行： − + − N<sub>i</sub> 是位于量子态 i 的粒子数，N是系统中粒子的总数。我们可以用波兹曼分布来求出这个概率，它等于处于量子态i的粒子的比例。所以这个方程给出位于量子态 i 的粒子比例关于这个状态能量的函数。为了使这成为可能，必须有一些粒子处于第一个量子态，等待着发生跃迁。我们可以通过求第一态粒子的比例来满足这个条件。如果它可以忽略不计，那么在当前计算的温度下，则很难观察到这种跃迁。一般来说，处于第一状态的分子比例越大，意味着向第二状态跃迁的次数越多。这就产生了一条更强的谱线。然而，还有其他因素影响谱线的强度，例如它是否是被允许的跃迁还是被禁止的跃迁。 + − The Boltzmann distribution is related to the softmax function commonly used in machine learning. − 玻尔兹曼分布与机器学习中常用的归一化指数函数（Softmax 函数）有关。+ − The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states ''i'' and ''j'' is given as+ − 玻尔兹曼分布指出，能量低的量子态总是有比能量高的量子态更高的概率被粒子占据。它同时也能让我们定量地比较两个量子态分布概率的关系。+ + + − :<math>+ − + − + − </math> 第280行：
第279行： − + − + − Canonical ensemble (general case)+ − + − − − − − − The canonical ensemble gives the probabilities of the various possible states of a closed system of fixed volume, in thermal equilibrium with a heat bath. The canonical ensemble is a probability distribution with the Boltzmann form. − Statistical frequencies of subsystems' states (in a non-interacting collection)+ − − + − − − − When the system of interest is a collection of many non-interacting copies of a smaller subsystem, it is sometimes useful to find the statistical frequency of a given subsystem state, among the collection. The canonical ensemble has the property of separability when applied to such a collection: as long as the non-interacting subsystems have fixed composition, then each subsystem's state is independent of the others and is also characterized by a canonical ensemble. As a result, the expected statistical frequency distribution of subsystem states has the Boltzmann form. 第306行：
第294行： + + + + − + + 第320行：
第313行： − − * When a system is in thermodynamic equilibrium with respect to both energy exchange ''and particle exchange'', the requirement of fixed composition is relaxed and a grand canonical ensembleis obtained rather than canonical ensemble. On the other hand, if both composition and energy are fixed, then a microcanonical ensemble applies instead. − * If the subsystems within a collection ''do'' interact with each other, then the expected frequencies of subsystem states no longer follow a Boltzmann distribution, and even may not have an analytical solution. The canonical ensemble can however still be applied to the ''collective'' states of the entire system considered as a whole, provided the entire system is in thermal equilibrium. − * With ''quantum'' gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow Maxwell–Boltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the grand canonical ensemble the state-filling statistics of quantum gases are described by Fermi–Dirac statistics or Bose–Einstein statistics, depending on whether the particles are fermions or bosons, respectively. − − This equation is of great importance to [[spectroscopy]]. In spectroscopy we observe a [[spectral line]] of atoms or molecules that we are interested in going from one state to another.<ref name="Atkins, P. W. 2010"/><ref>Atkins, P. W.; de Paula J. (2009) Physical Chemistry, 9th edition, Oxford University Press, Oxford, UK</ref> In order for this to be possible, there must be some particles in the first state to undergo the transition. We may find that this condition is fulfilled by finding the fraction of particles in the first state. If it is negligible, the transition is very likely not to be observed at the temperature for which the calculation was done. In general, a larger fraction of molecules in the first state means a higher number of transitions to the second state.<ref>Skoog, D. A.; Holler, F. J.; Crouch, S. R. (2006) Principles of Instrumental Analysis, Brooks/Cole, Boston, MA</ref> This gives a stronger spectral line. However, there are other factors that influence the intensity of a spectral line, such as whether it is caused by an allowed or a [[forbidden transition]]. − − − − The Boltzmann distribution is related to the [[softmax function]] commonly used in machine learning. − − In more general mathematical settings, the Boltzmann distribution is also known as the Gibbs measure. In statistics and machine learning, it is called a log-linear model. In deep learning, the Boltzmann distribution is used in the sampling distribution of stochastic neural networks such as the Boltzmann machine, Restricted Boltzmann machine, Energy-Based models and deep Boltzmann machine. − − 在更一般的数学环境中，波兹曼分布也被称为吉布斯量度。在统计学和机器学习中，它被称为对数线性回归。在深度学习中，波兹曼分布被用于随机神经网络的抽样分布，如波茨曼机、受限玻尔兹曼机、基于能量的模型和深度波茨曼机。 − − − − == In statistical mechanics == − − {{main|Canonical ensemble|Maxwell–Boltzmann statistics}} − − − − The Boltzmann distribution can be introduced to allocate permits in emissions trading. The new allocation method using the Boltzmann distribution can describe the most probable, natural, and unbiased distribution of emissions permits among multiple countries. Simple and versatile, this new method holds potential for many economic and environmental applications. − − 可以引入欧盟波兹曼分布来分配排放交易的许可。这种新的排放量分配方法使用波兹曼分布能够描述多个国家之间最可能、最自然和最无偏见的排放量分配。这种新方法简单而通用，具有许多经济和环境应用的潜力。 − − The Boltzmann distribution appears in [[statistical mechanics]] when considering isolated (or nearly-isolated) systems of fixed composition that are in [[thermal equilibrium]] (equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble, but also some special cases (derivable from the canonical ensemble) also show the Boltzmann distribution in different aspects: − − − − The Boltzmann distribution has the same form as the multinomial logit model. As a discrete choice model, this is very well known in economics since Daniel McFadden made the connection to random utility maximization. − − 波兹曼分布与多项式 logit 模型具有相同的形式。作为一个离散选择模型，这在经济学中非常著名，因为丹尼尔 · 麦克法登提出了随机效用最大化的联系。 − − ; [[Canonical ensemble]] (general case) − − : The [[canonical ensemble]] gives the [[probabilities]] of the various possible states of a closed system of fixed volume, in thermal equilibrium with a [[heat bath]]. The canonical ensemble is a probability distribution with the Boltzmann form. − − ; Statistical frequencies of subsystems' states (in a non-interacting collection) − − : When the system of interest is a collection of many non-interacting copies of a smaller subsystem, it is sometimes useful to find the [[statistical frequency]] of a given subsystem state, among the collection. The canonical ensemble has the property of separability when applied to such a collection: as long as the non-interacting subsystems have fixed composition, then each subsystem's state is independent of the others and is also characterized by a canonical ensemble. As a result, the [[expectation value|expected]] statistical frequency distribution of subsystem states has the Boltzmann form. − − ; [[Maxwell–Boltzmann statistics]] of classical gases (systems of non-interacting particles) − − : In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space. [[Maxwell–Boltzmann statistics]] give the expected number of particles found in a given single-particle state, in a [[classical mechanics|classical]] gas of non-interacting particles at equilibrium. This expected number distribution has the Boltzmann form. − − − − Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed: − + − + + − + − − − |year=1868 − − 1868年 − |title=Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten − + + − |trans-title=Studies on the balance of living force between moving material points+ − − | 反题 = 移动物质点之间生命力平衡的研究 − − {{main|Gibbs measure|Log-linear model|Boltzmann machine}} − − |journal=Wiener Berichte |volume=58 |pages=517–560 − − | journal = Wiener Berichte | volume = 58 | pages = 517-560 − − − − }} − − }} − − In more general mathematical settings, the Boltzmann distribution is also known as the [[Gibbs measure]]. In statistics and machine learning, it is called a [[log-linear model]]. In [[deep learning]], the Boltzmann distribution is used in the sampling distribution of [[stochastic neural network]]s such as the [[Boltzmann machine]], [[Restricted Boltzmann machine]], [[Energy based model|Energy-Based models]] and [[Deep Boltzmann Machine|deep Boltzmann machine]]. − − − − |first=Josiah Willard |last=Gibbs |authorlink=Josiah Willard Gibbs − − 2012年10月15日 | 约西亚·威拉德·吉布斯 − |year=1902 − 1902年+ + − |title=Elementary Principles in Statistical Mechanics − 统计力学的基本原理 − + − − |title-link=Elementary Principles in Statistical Mechanics }} − − | title-link = 基本原理统计力学} + − − The Boltzmann distribution has the same form as the [[Multinomial logistic regression|multinomial logit]] model. As a [[discrete choice]] model, this is very well known in economics since [[Daniel McFadden]] made the connection to random utility maximization.
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在知道量子态能量的情况下，这个配分函数可以被计算。对于原子来说，配分函数的值可以在NIST Atomic Spectra Database中找到。<ref name=":1" />
在知道量子态能量的情况下，这个配分函数可以被计算。对于原子来说，配分函数的值可以在NIST Atomic Spectra Database中找到。<ref name=":1" />
The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states ''i'' and ''j'' is given as
:
The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states i and j is given as
The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states i and j is given as
玻尔兹曼分布指出，能量低的量子态总是有比能量高的量子态更高的概率被粒子占据。它同时也能让我们定量地比较两个量子态分布概率的关系。状态 i 和状态 j 的概率比为
:<math>
:<math>
{\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}
</math>
</math>
where p<sub>i</sub> is the probability of state i, p<sub>j</sub> the probability of state j, and ε<sub>i</sub> and ε<sub>j</sub> are the energies of states i and j, respectively.
where p<sub>i</sub> is the probability of state i, p<sub>j</sub> the probability of state j, and ε<sub>i</sub> and ε<sub>j</sub> are the energies of states i and j, respectively.
其中 p<sub>i</sub> 是量子态 i 的概率， p<sub>j</sub> 是量子态 j 的概率，ε i 和 ε j 分别是状态 i 和 j 的能量。
其中 p<sub>i</sub> 是量子态 i 的概率， p<sub>j</sub> 是量子态 j 的概率，ε<sub>i</sub> 和 ε<sub>j</sub> 分别是状态 i 和 j 的能量。
where N<sub>i</sub> is the number of particles in state i and N is the total number of particles in the system. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. So the equation that gives the fraction of particles in state i as a function of the energy of that state is In order for this to be possible, there must be some particles in the first state to undergo the transition. We may find that this condition is fulfilled by finding the fraction of particles in the first state. If it is negligible, the transition is very likely not to be observed at the temperature for which the calculation was done. In general, a larger fraction of molecules in the first state means a higher number of transitions to the second state. This gives a stronger spectral line. However, there are other factors that influence the intensity of a spectral line, such as whether it is caused by an allowed or a forbidden transition.
where N<sub>i</sub> is the number of particles in state i and N is the total number of particles in the system. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. So the equation that gives the fraction of particles in state i as a function of the energy of that state is
N<sub>i</sub> 是位于量子态 i 的粒子数，N是系统中粒子的总数。我们可以用波兹曼分布来求出这个概率，它等于处于量子态i的粒子的比例。所以这个方程给出位于量子态 i 的粒子比例关于这个状态能量的函数<ref name="Atkins, P. W. 2010" />。
This equation is of great importance to [[spectroscopy]]. In spectroscopy we observe a [[spectral line]] of atoms or molecules that we are interested in going from one state to another.<ref name="Atkins, P. W. 2010" /><ref name=":2">Atkins, P. W.; de Paula J. (2009) Physical Chemistry, 9th edition, Oxford University Press, Oxford, UK</ref> In order for this to be possible, there must be some particles in the first state to undergo the transition. We may find that this condition is fulfilled by finding the fraction of particles in the first state. If it is negligible, the transition is very likely not to be observed at the temperature for which the calculation was done. In general, a larger fraction of molecules in the first state means a higher number of transitions to the second state.<ref name=":3">Skoog, D. A.; Holler, F. J.; Crouch, S. R. (2006) Principles of Instrumental Analysis, Brooks/Cole, Boston, MA</ref> This gives a stronger spectral line. However, there are other factors that influence the intensity of a spectral line, such as whether it is caused by an allowed or a [[forbidden transition]]
对于光谱学来说这很重要在。在光谱中，我们期待能观察到一条原子或分子跃迁的谱线。<ref name="Atkins, P. W. 2010" /><ref name=":2" /> 在为了使这成为可能，必须有一些粒子处于第一个量子态，等待着发生跃迁。我们可以通过求第一态粒子的比例来满足这个条件。如果它可以忽略不计，那么在当前计算的温度下，则很难观察到这种跃迁。一般来说，处于第一状态的分子比例越大，意味着向第二状态跃迁的次数越多。<ref name=":3" />这就产生了一条更强的谱线。然而，还有其他因素影响谱线的强度，例如它是否是被允许的跃迁还是被禁止的跃迁。
The Boltzmann distribution is related to the [[softmax function]] commonly used in machine learning.
The Boltzmann distribution is related to the softmax function commonly used in machine learning.
玻尔兹曼分布与机器学习中常用的[[Index.php?title=Softmax function|归一化指数函数]]（Softmax 函数）有关。
== In statistical mechanics '''统计力学''' ==
{\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}
{{main|Canonical ensemble|Maxwell–Boltzmann statistics}}
:
The Boltzmann distribution appears in statistical mechanics when considering isolated (or nearly-isolated) systems of fixed composition that are in thermal equilibrium (equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble, but also some special cases (derivable from the canonical ensemble) also show the Boltzmann distribution in different aspects:
The Boltzmann distribution appears in statistical mechanics when considering isolated (or nearly-isolated) systems of fixed composition that are in thermal equilibrium (equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble, but also some special cases (derivable from the canonical ensemble) also show the Boltzmann distribution in different aspects:
当独立的(或者几乎独立的)固定体系处于统计力学热平衡状态时，玻尔兹曼分布就会出现。最普遍的情况是概率分布的正则系综，但也有一些特殊的情况(从正则系综衍生)也显示了不同形式的玻尔兹曼分布:
当独立的(或者几乎独立的)固定体系处于统计力学热平衡状态时，玻尔兹曼分布就会出现。最普遍的情况是概率分布的正则系综，但也有一些特殊的情况(从正则系综衍生)也显示了不同形式的玻尔兹曼分布:
; [[Canonical ensemble]] (general case)
; '''正则系综(一般情况)'''
;
; '''The [[canonical ensemble]] gives the [[probabilities]] of the various possible states of a closed system of fixed volume, in thermal equilibrium with a [[heat bath]]. The canonical ensemble is a probability distribution with the Boltzmann form.'''
'''正则系综(一般情况)'''
正则系综模型给出了在一个封闭固定体积，带有热浴的热平衡系统中的各种可能状态的概率。我们称正则系综为一个玻尔兹曼概率分布。
正则系综模型给出了在一个封闭固定体积，带有热浴的热平衡系统中的各种可能状态的概率。我们称正则系综为一个玻尔兹曼概率分布。
; Statistical frequencies of subsystems' states (in a non-interacting collection)
'''子系统状态的统计频率(在一个无交互的集合中)'''
'''子系统状态的统计频率(在一个无交互的集合中)'''
: When the system of interest is a collection of many non-interacting copies of a smaller subsystem, it is sometimes useful to find the [[statistical frequency]] of a given subsystem state, among the collection. The canonical ensemble has the property of separability when applied to such a collection: as long as the non-interacting subsystems have fixed composition, then each subsystem's state is independent of the others and is also characterized by a canonical ensemble. As a result, the [[expectation value|expected]] statistical frequency distribution of subsystem states has the Boltzmann form.
当系统是一个较小子系统的许多非交互副本的集合时，有时我们需要在某个子系统集合中找到某个子系统的统计频率。当应用于这样一个集合时，这个子系统的正则系综具有可分离性: 只要不相互作用的子系统的组成是固定的，那么每个子系统的状态都是独立于其他子系统的，它们也拥有自己的正则系综。因此，子系统状态的期望统计频率分布形式是玻尔兹曼分布。
当系统是一个较小子系统的许多非交互副本的集合时，有时我们需要在某个子系统集合中找到某个子系统的统计频率。当应用于这样一个集合时，这个子系统的正则系综具有可分离性: 只要不相互作用的子系统的组成是固定的，那么每个子系统的状态都是独立于其他子系统的，它们也拥有自己的正则系综。因此，子系统状态的期望统计频率分布形式是玻尔兹曼分布。
Maxwell–Boltzmann statistics of classical gases (systems of non-interacting particles)
Maxwell–Boltzmann statistics of classical gases (systems of non-interacting particles)
[[Maxwell–Boltzmann statistics]] of classical gases (systems of non-interacting particles)
'''Maxwell-Boltzmann 统计中的经典气体系统(非相互作用粒子系统)'''
'''Maxwell-Boltzmann 统计中的经典气体系统(非相互作用粒子系统)'''
: In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space. [[Maxwell–Boltzmann statistics]] give the expected number of particles found in a given single-particle state, in a [[classical mechanics|classical]] gas of non-interacting particles at equilibrium. This expected number distribution has the Boltzmann form.
In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space. Maxwell–Boltzmann statistics give the expected number of particles found in a given single-particle state, in a classical gas of non-interacting particles at equilibrium. This expected number distribution has the Boltzmann form.
In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space. Maxwell–Boltzmann statistics give the expected number of particles found in a given single-particle state, in a classical gas of non-interacting particles at equilibrium. This expected number distribution has the Boltzmann form.
在粒子系统中，许多粒子共享同一空间，并且相互之间有规律地改变位置; 它们所占据的单粒子状态空间是一个共享空间。麦克斯韦-玻尔兹曼统计给出了在一个给定的单粒子态，在一个处于平衡状态的非相互作用粒子的经典气体中所发现的粒子的预期数量。这个预期的数量分布具有玻尔兹曼分布形式。
在粒子系统中，许多粒子共享同一空间，并且相互之间有规律地改变位置; 它们所占据的单粒子状态空间是一个共享空间。[[Index.php?title=Maxwell–Boltzmann statistics|麦克斯韦-玻尔兹曼统计]]给出了在一个给定的单粒子态，在一个处于平衡状态的非相互作用粒子的经典气体中所发现的粒子的预期数量。这个预期的数量分布具有玻尔兹曼分布形式。
Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed:
虽然这些案例有很多相似之处，但是当关键假设发生变化时，我们需要对它们以不同的方式进行归纳总结，因此区分它们是有帮助的:
虽然这些案例有很多相似之处，但是当关键假设发生变化时，我们需要对它们以不同的方式进行归纳总结，因此区分它们是有帮助的:
* When a system is in thermodynamic equilibrium with respect to both energy exchange ''and particle exchange'', the requirement of fixed composition is relaxed and a [[grand canonical ensemble]] is obtained rather than canonical ensemble. On the other hand, if both composition and energy are fixed, then a [[microcanonical ensemble]] applies instead.
* When a system is in thermodynamic equilibrium with respect to both energy exchange ''and particle exchange'', the requirement of fixed composition is relaxed and a [[grand canonical ensemble]] is obtained rather than canonical ensemble. On the other hand, if both composition and energy are fixed, then a [[microcanonical ensemble]] applies instead.
* 当一个系统在能量交换和粒子交换方面都处于热力学平衡时，固定组成的要求就放宽了。这时我们就得到了一个[[Index.php?title=Microcanonical ensemble|巨正则系综]]而不是正则系综。另一方面，如果成分和能量都是固定的，则可以使用[[微正则系综Index.php?title=Microcanonical ensemble|微正则系综]]。
* If the subsystems within a collection ''do'' interact with each other, then the expected frequencies of subsystem states no longer follow a Boltzmann distribution, and even may not have an [[analytical solution]].<ref>A classic example of this is [[magnetic ordering]]. Systems of non-interacting [[spin (physics)|spins]] show [[paramagnetic]] behaviour that can be understood with a single-particle canonical ensemble (resulting in the [[Brillouin function]]). Systems of ''interacting'' spins can show much more complex behaviour such as [[ferromagnetism]] or [[antiferromagnetism]].</ref> The canonical ensemble can however still be applied to the ''collective'' states of the entire system considered as a whole, provided the entire system is isolated and in thermal equilibrium.
* If the subsystems within a collection ''do'' interact with each other, then the expected frequencies of subsystem states no longer follow a Boltzmann distribution, and even may not have an [[analytical solution]].<ref name=":4">A classic example of this is [[magnetic ordering]]. Systems of non-interacting [[spin (physics)|spins]] show [[paramagnetic]] behaviour that can be understood with a single-particle canonical ensemble (resulting in the [[Brillouin function]]). Systems of ''interacting'' spins can show much more complex behaviour such as [[ferromagnetism]] or [[antiferromagnetism]].</ref> The canonical ensemble can however still be applied to the ''collective'' states of the entire system considered as a whole, provided the entire system is isolated and in thermal equilibrium.
* 如果集合中的子系统确实相互交互，则子系统状态的预期频率不再遵循玻尔兹曼 分布，甚至可能没有解析解。<ref name=":4" />然而，如果整个系统是独立的并且处于热平衡状态，则正则系综仍然可以应用于作为一个整体考虑。
* With ''[[quantum mechanics|quantum]]'' gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow Maxwell–Boltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the grand canonical ensemble the state-filling statistics of quantum gases are described by [[Fermi–Dirac statistics]] or [[Bose–Einstein statistics]], depending on whether the particles are [[fermion]]s or [[boson]]s respectively.
* With ''[[quantum mechanics|quantum]]'' gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow Maxwell–Boltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the grand canonical ensemble the state-filling statistics of quantum gases are described by [[Fermi–Dirac statistics]] or [[Bose–Einstein statistics]], depending on whether the particles are [[fermion]]s or [[boson]]s respectively.
* 对于处于平衡状态的非相互作用粒子的量子气体，在给定的单粒子状态中发现的粒子数量并不遵循麦克斯韦-玻尔兹曼统计。并且正则系综中的量子气体没有简单的封闭表达式。在巨正则系综中，量子气体的统计分布状态由费米-狄拉克统计或玻色-爱因斯坦统计来描述，具体则取决于该粒子是费米子还是玻色子。
== In mathematics ==
== In mathematics ==
|title=Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten
In more general mathematical settings, the Boltzmann distribution is also known as the [[Gibbs measure]]. In statistics and machine learning, it is called a [[log-linear model]]. In [[deep learning]], the Boltzmann distribution is used in the sampling distribution of [[stochastic neural network]]s such as the [[Boltzmann machine]], [[Restricted Boltzmann machine]], [[Energy based model|Energy-Based models]] and [[Deep Boltzmann Machine|deep Boltzmann machine]]
In more general mathematical settings, the Boltzmann distribution is also known as the Gibbs measure. In statistics and machine learning, it is called a log-linear model. In deep learning, the Boltzmann distribution is used in the sampling distribution of stochastic neural networks such as the Boltzmann machine, Restricted Boltzmann machine, Energy-Based models and deep Boltzmann machine.
在更一般的数学环境中，玻尔兹曼分布也被称为吉布斯量度。在统计学和机器学习中，它被称为对数线性回归。在深度学习中，玻尔兹曼分布被用于随机神经网络的抽样分布，如玻尔兹曼机、受限玻尔兹曼机、基于能量的模型和深度波茨曼机。
== In economics ==
== In economics ==
The Boltzmann distribution can be introduced to allocate permits in emissions trading.<ref name="Park, J.-W. 2012">Park, J.-W., Kim, C. U. and Isard, W. (2012) Permit allocation in emissions trading using the Boltzmann distribution. Physica A 391: 4883–4890</ref><ref name=":5">[http://www.technologyreview.com/view/425051/the-thorny-problem-of-fair-allocation/ The Thorny Problem Of Fair Allocation]. ''Technology Review'' blog. August 17, 2011. Cites and summarizes Park, Kim and Isard (2012).</ref> The new allocation method using the Boltzmann distribution can describe the most probable, natural, and unbiased distribution of emissions permits among multiple countries. Simple and versatile, this new method holds potential for many economic and environmental applications.
在经济学中，玻尔兹曼分布被用来分配排放交易中的许可。<ref name="Park, J.-W. 2012" /><ref name=":5" /> 使用玻尔兹曼分布的新分配方法可以创造出在多个国家之间最可能的、自然的和无偏见的排放许可分配。这种新方法简单而通用，在经济和环保中有很大的应用潜力。
The Boltzmann distribution can be introduced to allocate permits in emissions trading.<ref name="Park, J.-W. 2012">Park, J.-W., Kim, C. U. and Isard, W. (2012) Permit allocation in emissions trading using the Boltzmann distribution. Physica A 391: 4883–4890</ref><ref>[http://www.technologyreview.com/view/425051/the-thorny-problem-of-fair-allocation/ The Thorny Problem Of Fair Allocation]. ''Technology Review'' blog. August 17, 2011. Cites and summarizes Park, Kim and Isard (2012).</ref> The new allocation method using the Boltzmann distribution can describe the most probable, natural, and unbiased distribution of emissions permits among multiple countries. Simple and versatile, this new method holds potential for many economic and environmental applications.
The Boltzmann distribution has the same form as the [[Multinomial logistic regression|multinomial logit]] model. As a [[discrete choice]] model, this is very well known in economics since [[Daniel McFadden]] made the connection to random utility maximization.
玻尔兹曼分布与多项逻辑回归模型具有相同的形式。在丹尼尔 · 麦克法登将它与随机效用最大化联系起来之后，成为了大家所熟知的一种离散选择模型。