玻尔兹曼分布
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Boltzmann factor pi / pj (vertical axis) as a function of temperature T for several energy differences εi − εj.
玻尔兹曼因子 pi / pj (垂直轴) 作为温度 T 对于几个能量差异εi − εj的函数 。
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) 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:
在统计力学和数学中，波兹曼分布分布(也称为吉布斯分布[1])是一个概率分布或机率量测，它给出了一个系统处于某种状态的概率，这个概率是该状态的能量和系统温度的函数。分布情况以下列形式表示:
- [math]\displaystyle{ p_i \propto e^{-\frac{\varepsilon_i}{kT}} }[/math]
[math]\displaystyle{ p_i \propto e^{-\frac{\varepsilon_i}{kT}} }[/math]
[拉丁语]
where pi is the probability of the system being in state i, εi is the energy of that state, and a constant kT of the distribution is the product of Boltzmann's constant k and thermodynamic temperature T. The symbol [math]\displaystyle{ \propto }[/math] denotes proportionality (see 模板:Section link for the proportionality constant).
where is the probability of the system being in state , is the energy of that state, and a constant of the distribution is the product of Boltzmann's constant and thermodynamic temperature . The symbol [math]\displaystyle{ \propto }[/math] denotes proportionality (see for the proportionality constant).
pi是其中系统处于状态i的概率, εi 是该状态的能量，还有关于这个分布的一个常数kT ，它是玻耳兹曼常数k和热力学温度 T的乘积。符号 < math display = " inline" > propto </math > 表示相称性(见附录中的比例常数)。
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 .
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 .
这里的术语系统有非常广泛的含义; 它可以从单个原子到宏观系统，如天然气储罐。正因为如此，波兹曼分布可以用来解决各种各样的问题。分布表明，能量较低的状态被占据的概率总是较高的。
The ratio of probabilities of two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference:
The ratio of probabilities of two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference:
两种状态的概率之比被称为玻尔兹曼因子，其特点仅取决于两种状态的能量差:
- [math]\displaystyle{ \frac{p_i}{p_j} = e^{\frac{\varepsilon_j - \varepsilon_i}{kT}} }[/math]
[math]\displaystyle{ \frac{p_i}{p_j} = e^{\frac{\varepsilon_j - \varepsilon_i}{kT}} }[/math]
[数学][数学]
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]
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"
波兹曼分布是根据1868年他在研究统计力学气体在热平衡的时候首次提出的一个公式命名的。波尔兹曼的统计工作在他的论文《论热力学第二基本定理与热平衡条件的概率计算之间的关系》中得到了证实
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The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902.[3]:Ch.IV
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).
广义波兹曼分布是熵的统计力学定义(吉布斯熵公式 < math > s =-k { mathrm { b } sum i log pi i </math >)与熵的热力学定义(< math > ds = frac { delta q { rev }{ t } </math >)等价的充要条件。
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 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; in contrast, the latter is used to describe particle speeds in idealized gases.
波兹曼分布不应与麦克斯韦-波兹曼分布混淆。前者给出了系统处于某种状态的概率，作为该状态能量的函数; 相反，后者用于描述理想气体中的粒子速度。
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 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. It is given as
波兹曼分布是一个概率分布，它给出了某种状态的概率，作为该状态的能量和温度的函数，该分布适用于系统。它被给出为
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
[math]\displaystyle{ 《数学》 :\lt math\gt 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}}} 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 }}}}} 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}}} }[/math]
数学
</math>
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.
其中 p < sub > i 是状态 i 的概率，ε < sub > i 是状态 i 的能量，k 是波兹曼常数，t 是系统的温度，m 是感兴趣的系统所能到达的所有状态的数目。
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
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
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。它还可以给出两个状态被占领概率之间的定量关系。给出了状态 i 和状态 j 的概率比为
- [math]\displaystyle{ Q={\sum_{i=1}^{M}{e^{- {\varepsilon}_i / k T}}} \lt math\gt 《数学》 }[/math]
{\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}
{ frac { p _ i }{ p _ j } = e ^ {({ varepsilon } _ j-{ varepsilon } _ i)/k t }
</math>
数学
It results from the constraint that the probabilities of all accessible states must add up to 1.
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.
其中 p < sub > i 是状态 i 的概率，p < sub > j 状态 j 的概率，ε i 和 ε j 分别是状态 i 和 j 的能量。
The Boltzmann distribution is the distribution that maximizes the entropy
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.
波兹曼分布通常被用来描述粒子，比如原子或分子，在能量状态上的分布情况。如果我们有一个由许多粒子组成的系统，粒子处于状态 i 的概率实际上等于，如果我们从该系统中选择一个随机粒子并检查它处于什么状态，我们会发现它处于状态 i 的概率。这个概率等于状态 i 的粒子数除以系统中粒子的总数，即占据状态 i 的粒子的比例。
- [math]\displaystyle{ H(p_1,p_2,\cdots,p_M) = -\sum_{i=1}^{M} p_i\log_2 p_i }[/math]
[math]\displaystyle{ 《数学》 subject to the constraint that \lt math display="inline"\gt p_i={\frac{N_i}{N}} P _ i = { frac { n _ i }{ n } {\sum{p_i {\varepsilon}_i}} }[/math]
数学
</math> equals a particular mean energy value (which can be proven using Lagrange multipliers).
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 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.
其中 n < sub > i 是状态 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，就像我们看到的那样，等于处于 i 状态的粒子的比例。所以给出状态 i 的粒子比例，作为状态能量的函数的方程是，为了使这成为可能，必须有一些粒子处于第一个状态，才能发生跃迁。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 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 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 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]
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:
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.
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。
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.
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)
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。
- [math]\displaystyle{ 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 p_i={\frac{N_i}{N}} 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]
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:
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]
- [math]\displaystyle{ {\frac{N_i}{N}}={\frac{e^{- {\varepsilon}_i / k T}}{\sum_{j=1}^{M}{e^{- {\varepsilon}_j / k T}}}} }[/math]
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.
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
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 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 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:
- 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.
- 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.[10] 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.
- 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.
|year=1868
1868年
In mathematics
|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
|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 networks such as the Boltzmann machine, Restricted Boltzmann machine, Energy-Based models and 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
In economics
|year=1902
1902年
|title=Elementary Principles in Statistical Mechanics
统计力学的基本原理
The Boltzmann distribution can be introduced to allocate permits in emissions trading.[11][12] 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.
|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 logit model. As a discrete choice model, this is very well known in economics since Daniel McFadden made the connection to random utility maximization.
Category:Statistical mechanics
类别: 统计力学
See also
Distribution
分布
This page was moved from wikipedia:en:Boltzmann distribution. Its edit history can be viewed at 玻尔兹曼分布/edithistory
- ↑ 1.0 1.1 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 The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902. 这种分布后来在1902年由约西亚·威拉德·吉布斯进行了广泛的调查，以其现代通用形式。.
- ↑ 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
- ↑ A classic example of this is magnetic ordering. Systems of non-interacting 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.
- ↑ Park, J.-W., Kim, C. U. and Isard, W. (2012) Permit allocation in emissions trading using the Boltzmann distribution. Physica A 391: 4883–4890
- ↑ The Thorny Problem Of Fair Allocation. Technology Review blog. August 17, 2011. Cites and summarizes Park, Kim and Isard (2012).