# 玻尔兹曼分布

Boltzmann factor pi / pj (vertical axis) as a function of temperature T for several energy differences εi − εj.

Boltzmann factor pi / pj (vertical axis) as a function of temperature T for several energy differences εi − εj.

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:

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:

$p_i \propto e^{-\frac{\varepsilon_i}{kT}}$

$p_i \propto e^{-\frac{\varepsilon_i}{kT}}$

[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 $\propto$ 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 $\propto$ denotes proportionality (see for the proportionality constant).

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 .

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:

$\frac{p_i}{p_j} = e^{\frac{\varepsilon_j - \varepsilon_i}{kT}}$

$\frac{p_i}{p_j} = e^{\frac{\varepsilon_j - \varepsilon_i}{kT}}$

[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"

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"

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The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902.: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 $S = -k_{\mathrm{B}}\sum_i p_i \log p_i$) and the thermodynamic definition of entropy ($d S = \frac{\delta Q_\text{rev}}{T}$, and the fundamental thermodynamic relation).

The generalized Boltzmann distribution is a sufficient and necessary condition for the equivalence between the statistical mechanics definition of entropy (The Gibbs entropy formula $S = -k_{\mathrm{B}}\sum_i p_i \log p_i$) and the thermodynamic definition of entropy ($d S = \frac{\delta Q_\text{rev}}{T}$, and the fundamental thermodynamic relation).

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.

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.

## 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

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

$《数学》 :\lt math\gt p_i=\frac{1}{Q}} {e^{- {\varepsilon}_i / k T}=\frac{e^{- {\varepsilon}_i / k T}}{\sum_{j=1}^{M}{e^{- {\varepsilon}_j / k T}}} P _ i = frac {1}{ q }{ e ^ {-{ varepsilon } _ i/k t } = frac { e ^ {-{ varepsilon } _ i/k t }{ sum { j = 1} ^ { m }{ e ^ {-{ varepsilon } _ j/k t }}}}} p_i=\frac{1}{Q}} {e^{- {\varepsilon}_i / k T}=\frac{e^{- {\varepsilon}_i / k T}}{\sum_{j=1}^{M}{e^{- {\varepsilon}_j / k T}}}$

[/itex]

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.

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. 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

$Q={\sum_{i=1}^{M}{e^{- {\varepsilon}_i / k T}}} \lt math\gt 《数学》$

{\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}

{ frac { p _ i }{ p _ j } = e ^ {({ varepsilon } _ j-{ varepsilon } _ i)/k t }

[/itex]

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.

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.

$H(p_1,p_2,\cdots,p_M) = -\sum_{i=1}^{M} p_i\log_2 p_i$

$《数学》 subject to the constraint that \lt math display="inline"\gt p_i={\frac{N_i}{N}} P _ i = { frac { n _ i }{ n } {\sum{p_i {\varepsilon}_i}}$

[/itex] 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.

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.

The Boltzmann distribution is related to the softmax function commonly used in machine learning.

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

${\frac{p_i}{p_j}}=e^{({\varepsilon}_j-{\varepsilon}_i) / k T}$

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:

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)


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.


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)


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.


$Maxwell–Boltzmann statistics of classical gases (systems of non-interacting particles) 经典气体(非相互作用粒子系统)的 Maxwell-Boltzmann 统计 p_i={\frac{N_i}{N}} In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space. Maxwell–Boltzmann statistics give the expected number of particles found in a given single-particle state, in a classical gas of non-interacting particles at equilibrium. This expected number distribution has the Boltzmann form. 在粒子系统中，许多粒子共享同一空间，并且相互之间有规律地改变位置; 它们所占据的单粒子状态空间是一个共享空间。麦克斯韦-玻尔兹曼统计给出了在一个给定的单粒子态，在一个处于平衡状态的非相互作用粒子的经典气体中所发现的粒子的预期数量。这个预期的数分布具有玻耳兹曼形式。$

Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed:

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 

${\frac{N_i}{N}}={\frac{e^{- {\varepsilon}_i / k T}}{\sum_{j=1}^{M}{e^{- {\varepsilon}_j / k T}}}}$

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.

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.

## 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.

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.

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. 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

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.

|title-link=Elementary Principles in Statistical Mechanics }}

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

Distribution

This page was moved from wikipedia:en:Boltzmann distribution. Its edit history can be viewed at 玻尔兹曼分布/edithistory

1. Landau, Lev Davidovich; Lifshitz, Evgeny Mikhailovich (1980) . 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
2. http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf
3. 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。.
4. 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. Unknown parameter |s2cid= ignored (help)
5. Atkins, P. W. (2010) Quanta, W. H. Freeman and Company, New York
6. McQuarrie, A. (2000) Statistical Mechanics, University Science Books, California
7. NIST Atomic Spectra Database Levels Form at nist.gov
8. Atkins, P. W.; de Paula J. (2009) Physical Chemistry, 9th edition, Oxford University Press, Oxford, UK
9. Skoog, D. A.; Holler, F. J.; Crouch, S. R. (2006) Principles of Instrumental Analysis, Brooks/Cole, Boston, MA
10. 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.
11. Park, J.-W., Kim, C. U. and Isard, W. (2012) Permit allocation in emissions trading using the Boltzmann distribution. Physica A 391: 4883–4890
12. The Thorny Problem Of Fair Allocation. Technology Review blog. August 17, 2011. Cites and summarizes Park, Kim and Isard (2012).