Boltzmann distribution

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In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
or
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gen ...
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: :$p_i \propto e^$ 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 the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
and
thermodynamic temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wo ...
. The symbol $\propto$ denotes proportionality (see for the proportionality constant). The term ''system'' here has a very wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom to a macroscopic system such as a natural gas storage tank. Therefore 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: :$\frac = e^$ The Boltzmann distribution is named after
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...
who first formulated it in 1868 during his studies of the statistical mechanics of gases in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
. 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 distribution was later investigated extensively, in its modern generic form, by
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
in 1902. The Boltzmann distribution should not be confused with the
Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and used ...
or Maxwell-Boltzmann statistics. The Boltzmann distribution gives the probability that a system will be in a certain ''state'' as a function of that state's energy,Atkins, P. W. (2010) Quanta, W. H. Freeman and Company, New York while the Maxwell-Boltzmann distributions give the probabilities of particle ''speeds'' or ''energies'' in ideal gases. The distribution of energies in a one-dimensional gas however, does follow the Boltzmann distribution.

# The distribution

The Boltzmann distribution is a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
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 :$p_i=\frac\right\} \left\{e^\left\{- \left\{\varepsilon\right\}_i / \left(k T\right)\right\}=\frac\left\{e^\left\{- \left\{\varepsilon\right\}_i / \left(k T\right)\left\{\sum_\left\{j=1\right\}^\left\{M\right\}\left\{e^\left\{- \left\{\varepsilon\right\}_j / \left(k T\right)\right\}$ where ''pi'' is the probability of state ''i'', ''εi'' the energy of state ''i'', ''k'' the Boltzmann constant, ''T'' the absolute temperature of the system and ''M'' is the number of all states accessible to the system of interest. The normalization denominator ''Q'' (denoted by some authors by ''Z'') is the canonical partition function :$Q=\left\{\sum_\left\{i=1\right\}^\left\{M\right\}\left\{e^\left\{- \left\{\varepsilon\right\}_i / \left(k T\right)\right\}$ 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 Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
:$H\left(p_1,p_2,\cdots,p_M\right) = -\sum_\left\{i=1\right\}^\left\{M\right\} p_i\log_2 p_i$ subject to the normalization constraint and the constraint that $\sum {p_i {\varepsilon}_i}$ equals a particular mean energy value (which can be proven using Lagrange multipliers). 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 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 :$\left\{\frac\left\{p_i\right\}\left\{p_j=e^\left\{\left(\left\{\varepsilon\right\}_j-\left\{\varepsilon\right\}_i\right) / \left(k T\right)\right\}$ 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 corresponding ratio of populations of energy levels must also take their degeneracies into account. The Boltzmann distribution is often used to describe the distribution of particles, such as atoms or molecules, over bound 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''. :$p_i=\left\{\frac\left\{N_i\right\}\left\{N$ 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 :$\left\{\frac\left\{N_i\right\}\left\{N=\left\{\frac\left\{e^\left\{- \left\{\varepsilon\right\}_i / \left(k T\right)\left\{\sum_\left\{j=1\right\}^\left\{M\right\}\left\{e^\left\{- \left\{\varepsilon\right\}_j / \left(k T\right)$ This equation is of great importance to spectroscopy. In spectroscopy we observe a spectral line of atoms or molecules undergoing transitions 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 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 softmax function commonly used in machine learning is related to the Boltzmann distribution: :$\left(p_1, \ldots, p_M\right) = \operatorname\left\{softmax\right\}\left(- \left\{\varepsilon\right\}_1 / \left(k T\right), \ldots, - \left\{\varepsilon\right\}_M / \left(k T\right)\right)$

# Generalized Boltzmann distribution

Distribution of the form :
fundamental thermodynamic relation In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentiall ...
where state functions are described by ensemble average.

# In statistical mechanics

The Boltzmann distribution appears in statistical mechanics when considering closed systems of fixed composition that are in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
(equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble. Some special cases (derivable from the canonical ensemble) show the Boltzmann distribution in different aspects: ; Canonical ensemble (general case) : The canonical ensemble gives the
probabilities Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
of the various possible states of a closed system of fixed volume, in thermal equilibrium with a
heat bath In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is a ...
. The canonical ensemble has a state 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 In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the particle density i ...
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 In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the particle density i ...
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 in thermal equilibrium. * With ''
quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity ( physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizat ...
'' 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 In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic e ...
, depending on whether the particles are
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s or
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer sp ...
s, respectively.

# In mathematics

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 Deep learning (also known as deep structured learning) is part of a broader family of machine learning methods based on artificial neural networks with representation learning. Learning can be supervised, semi-supervised or unsupervised. D ...
, 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 deep learning, the Boltzmann machine is considered to be one of the unsupervised learning models. In the design of Boltzmann machine in deep learning , as the number of nodes are increased the difficulty of implementing in real time applications becomes critical, so a different type of architecture named Restricted Boltzmann machine is introduced.

# In economics

The Boltzmann distribution can be introduced to allocate permits in
emissions trading Emissions trading is a market-based approach to controlling pollution by providing economic incentives for reducing the emissions of pollutants. The concept is also known as cap and trade (CAT) or emissions trading scheme (ETS). Carbon emissio ...
.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 new allocation method using the Boltzmann distribution can describe the most probable, natural, and unbiased distribution of emissions permits among multiple countries. The Boltzmann distribution has the same form as the
multinomial logit In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the prob ...
model. As a
discrete choice In economics, discrete choice models, or qualitative choice models, describe, explain, and predict choices between two or more discrete alternatives, such as entering or not entering the labor market, or choosing between modes of transport. Such ...
model, this is very well known in economics since Daniel McFadden made the connection to random utility maximization.