玻尔兹曼分布
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此article介紹的是system energy states。For particle energy levels and velocities, see Maxwell–Boltzmann distribution.
Boltzmann factor pi / pj (vertical axis) as a function of temperature T for several energy differences εi − εj.
玻尔兹曼因子 p i /p j (垂直轴)作为温度 t 的函数，几个能量差异 ε i -ε j 。
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution[1]) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form:
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Implied parentheses around the denominator kT are omitted for brevity. The normalization denominator Q (denoted by some authors by Z) is the canonical partition function
在统计力学和数学中，为了简洁起见，在分母 kT 上省略了吉布斯分布的隐含括号波兹曼分布。标准化分母 q (由一些作者用 z 表示)是标准配分函数
- [math]\displaystyle{ p_i \propto e^{-\frac{\varepsilon_i}{kT}} }[/math]
[math]\displaystyle{ 《数学》 Q={\sum_{i=1}^{M}{e^{- {\varepsilon}_i / k T}}} Q = { sum _ { i = 1} ^ { m }{ e ^ {-{ varepsilon } _ i/k t }}}}}} where {{mvar|p\lt sub\gt i\lt /sub\gt }} is the probability of the system being in state {{mvar|i}}, {{mvar|ε\lt sub\gt i\lt /sub\gt }} is the energy of that state, and a constant {{mvar|kT}} of the distribution is the product of [[Boltzmann's constant]] {{mvar|k}} and [[thermodynamic temperature]] {{mvar|T}}. The symbol \lt math display="inline"\gt \propto }[/math] denotes proportionality (see 模板:Section link for the proportionality constant).
</math>
数学
The term system here has a very wide meaning; it can range from a single atom to a macroscopic system such as a natural gas storage tank. Because of this the Boltzmann distribution can be used to solve a very wide variety of problems. The distribution shows that states with lower energy will always have a higher probability of being occupied .
It results from the constraint that the probabilities of all accessible states must add up to 1.
它是由所有可达状态的概率加起来必须等于1这一约束条件产生的。
The ratio of probabilities of two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference:
The Boltzmann distribution is the distribution that maximizes the entropy
波兹曼分布是熵最大化的分布
- [math]\displaystyle{ \frac{p_i}{p_j} = e^{\frac{\varepsilon_j - \varepsilon_i}{kT}} }[/math]
[math]\displaystyle{ H(p_1,p_2,\cdots,p_M) = -\sum_{i=1}^{M} p_i\log_2 p_i }[/math]
< math > h (p _ 1，p _ 2，cdots，p _ m) =-sum _ { i = 1} ^ { m } p _ i log _ 2 p _ i </math >
The Boltzmann distribution is named after Ludwig Boltzmann who first formulated it in 1868 during his studies of the statistical mechanics of gases in thermal equilibrium. Boltzmann's statistical work is borne out in his paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium"[2]
subject to the constraint that [math]\displaystyle{ 我们必须遵守 \lt math display = " inline" \gt 这个限制 \lt !-- {\sum{p_i {\varepsilon}_i}} {\sum{p_i {\varepsilon}_i}} It would be nice to have a citation here! The origin of the Boltzmann factor isn't entirely clear. According to some authors, Boltzmann's 1968 paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium” }[/math] equals a particular mean energy value (which can be proven using Lagrange multipliers).
</math > 等于一个特定的平均能量值(可以用拉格兰奇乘数证明)。
"Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten" is the origin but I can't find this article at the moment,
so I cannot confirm.
The partition function can be calculated if we know the energies of the states accessible to the system of interest. For atoms the partition function values can be found in the NIST Atomic Spectra Database.
如果我们知道感兴趣系统可以访问的状态的能量，我们就可以计算配分函数。对于原子来说，可以在 NIST 的原子光谱数据库中找到配分函数。
For example, this book says so, but uses suspiciously modern terminology
http://books.google.es/books?id=u13KiGlz2zcC&lpg=PA92&ots=8H1DRURdxn&pg=PA93#v=onepage&f=false
[math]\displaystyle{ 《数学》 On the other hand, Uffink's "Compendium of the foundations of classical statistical physics" does not seem to indicate quite this equation but rather that Boltzmann's 1968 distribution was the simple Maxwell–Boltzmann distribution (for a classical nonrelativistic gas), modified for particles in a potential. {\frac{N_i}{N}}={\frac{e^{- {\varepsilon}_i / k T}}{\sum_{j=1}^{M}{e^{- {\varepsilon}_j / k T}}}} { frac { n _ i }{ n } = { frac { e ^ {-{ varepsilon } _ i/k t }{ sum { j = 1}{ m }{ e ^ {-{ varepsilon } _ j/k }}}}}}} --\gt }[/math]
数学
The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902.[3]:Ch.IV
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. 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.
这个方程式对光谱学很重要。在光谱学中，我们观察原子或分子的光谱线，我们感兴趣的是从一种状态到另一种状态。为了使这成为可能，必须有一些处于第一态的粒子发生转变。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，那么在进行计算的温度下，极有可能不能观察到这种转变。一般来说，处于第一状态的分子比例越大，意味着向第二状态转变的次数越多。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，例如它是由允许的跃迁还是禁止的跃迁引起的。
The generalized Boltzmann distribution is a sufficient and necessary condition for the equivalence between the statistical mechanics definition of entropy (The Gibbs entropy formula [math]\displaystyle{ S = -k_{\mathrm{B}}\sum_i p_i \log p_i }[/math]) and the thermodynamic definition of entropy ([math]\displaystyle{ d S = \frac{\delta Q_\text{rev}}{T} }[/math], and the fundamental thermodynamic relation).[4]
The Boltzmann distribution is related to the softmax function commonly used in machine learning.
波兹曼分布学习与机器学习中常用的柔性最大激活函数学习有关。
The Boltzmann distribution should not be confused with the Maxwell–Boltzmann distribution. The former gives the probability that a system will be in a certain state as a function of that state's energy;[5] in contrast, the latter is used to describe particle speeds in idealized gases.
The distribution
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:
当考虑孤立的(或者几乎孤立的)固定组成的体系处于平衡状态时，波兹曼分布出现在《统计力学热平衡。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)也显示了波兹曼分布在不同的方面:
The Boltzmann distribution is a probability distribution that gives the probability of a certain state as a function of that state's energy and temperature of the system to which the distribution is applied.[6] It is given as
Canonical ensemble (general case)
正则系综(一般情况)
- [math]\displaystyle{ 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. 正则系综模型给出了一个封闭的固定体积系统的各种可能状态的概率，这个封闭体积系统包括一个带有热浴的热平衡。正则系综是一个玻尔兹曼概率分布。 p_i=\frac{1}{Q}} {e^{- {\varepsilon}_i / k T}=\frac{e^{- {\varepsilon}_i / k T}}{\sum_{j=1}^{M}{e^{- {\varepsilon}_j / k T}}} 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) }[/math]
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.
当感兴趣的系统是一个较小子系统的许多非交互副本的集合时，在集合中查找给定子系统状态的统计频率有时是有用的。当应用于这样一个集合时，正则系综子系统具有可分离性: 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是独立于其他子系统的，也是一个拥有属性正则系综。因此，子系统状态的期望统计频率分布具有玻耳兹曼形式。
Maxwell–Boltzmann statistics of classical gases (systems of non-interacting particles)
经典气体(非相互作用粒子系统)的 Maxwell-Boltzmann 统计
where pi is the probability of state i, εi the energy of state i, k the Boltzmann constant, T the temperature of the system and M is the number of all states accessible to the system of interest.[6][5] Implied parentheses around the denominator kT are omitted for brevity. The normalization denominator Q (denoted by some authors by Z) is the canonical partition function
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。这个预期的数分布具有玻耳兹曼形式。
- [math]\displaystyle{ 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，它们以不同的方式进行归纳，因此区分它们是有帮助的: Q={\sum_{i=1}^{M}{e^{- {\varepsilon}_i / k T}}} }[/math]
It results from the constraint that the probabilities of all accessible states must add up to 1.
The Boltzmann distribution is the distribution that maximizes the entropy
- [math]\displaystyle{ H(p_1,p_2,\cdots,p_M) = -\sum_{i=1}^{M} p_i\log_2 p_i }[/math]
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。
subject to the constraint that [math]\displaystyle{ {\sum{p_i {\varepsilon}_i}} }[/math] equals a particular mean energy value (which can be proven using Lagrange multipliers).
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 partition function can be calculated if we know the energies of the states accessible to the system of interest. For atoms the partition function values can be found in the NIST Atomic Spectra Database.[7]
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。
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
- [math]\displaystyle{ {\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T} }[/math]
where pi is the probability of state i, pj the probability of state j, and εi and εj are the energies of states i and j, respectively.
The Boltzmann distribution is often used to describe the distribution of particles, such as atoms or molecules, over energy states accessible to them. If we have a system consisting of many particles, the probability of a particle being in state i is practically the probability that, if we pick a random particle from that system and check what state it is in, we will find it is in state i. This probability is equal to the number of particles in state i divided by the total number of particles in the system, that is the fraction of particles that occupy state i.
- [math]\displaystyle{ p_i={\frac{N_i}{N}} |year=1868 1868年 }[/math]
|title=Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten
|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
where Ni 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 [5]
|journal=Wiener Berichte |volume=58 |pages=517–560
| journal = Wiener Berichte | volume = 58 | pages = 517-560
}}
}}
- [math]\displaystyle{ {\frac{N_i}{N}}={\frac{e^{- {\varepsilon}_i / k T}}{\sum_{j=1}^{M}{e^{- {\varepsilon}_j / k T}}}} |first=Josiah Willard |last=Gibbs |authorlink=Josiah Willard Gibbs 2012年10月15日 | 约西亚·威拉德·吉布斯 }[/math]
|year=1902
1902年
|title=Elementary Principles in Statistical Mechanics
统计力学的基本原理
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.[5][8] 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.[9] 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.
|title-link=Elementary Principles in Statistical Mechanics }}
| title-link = 基本原理统计力学}
The Boltzmann distribution is related to the softmax function commonly used in machine learning.
Category:Statistical mechanics
类别: 统计力学
In statistical mechanics
Distribution
分布情况
This page was moved from wikipedia:en:Boltzmann distribution. Its edit history can be viewed at 玻尔兹曼分布/edithistory
- ↑ Landau, Lev Davidovich; Lifshitz, Evgeny Mikhailovich (1980) [1976]. Statistical Physics. Course of Theoretical Physics. 5 (3 ed.). Oxford: Pergamon Press. ISBN 0-7506-3372-7. Translated by J.B. Sykes and M.J. Kearsley. See section 28
- ↑ http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf
- ↑ Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics. New York: Charles Scribner's Sons.
- ↑ Gao, Xiang; Gallicchio, Emilio; Roitberg, Adrian (2019). "The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy". The Journal of Chemical Physics. 151 (3): 034113. arXiv:1903.02121. doi:10.1063/1.5111333. PMID 31325924. S2CID 118981017.
- ↑ 5.0 5.1 5.2 5.3 Atkins, P. W. (2010) Quanta, W. H. Freeman and Company, New York
- ↑ 6.0 6.1 McQuarrie, A. (2000) Statistical Mechanics, University Science Books, California
- ↑ NIST Atomic Spectra Database Levels Form at nist.gov
- ↑ Atkins, P. W.; de Paula J. (2009) Physical Chemistry, 9th edition, Oxford University Press, Oxford, UK
- ↑ Skoog, D. A.; Holler, F. J.; Crouch, S. R. (2006) Principles of Instrumental Analysis, Brooks/Cole, Boston, MA