Instatistical mechanics,Boltzmann's entropy formula (also known as theBoltzmann–Planck equation, not to be confused with the more generalBoltzmann equation, which is apartial differential equation) is a probability equation relating theentropy${\displaystyle S}$, also written as${\displaystyle S_{\mathrm{B}}}$, of anideal gas to themultiplicity (commonly denoted as${\displaystyle \mathrm{\Omega }}$ or${\displaystyle W}$), the number of realmicrostates corresponding to the gas'smacrostate:

${\displaystyle S=k_{\mathrm{B}}\ln W}$

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where${\displaystyle k_{\mathrm{B}}}$ is theBoltzmann constant (also written as simply${\displaystyle k}$) and equal to 1.380649 × 10⁻²³ J/K, and${\displaystyle \ln }$ is thenatural logarithm function (orlog basee, as in the image above).

In short, theBoltzmann formula shows the relationship between entropy and the number of ways theatoms ormolecules of a certain kind ofthermodynamic system can be arranged. What is important to note is that W is not all possible states of the system, but ways the system can be arranged and still have the same properties from perspective of external observer. So for example when system contains 5 particles of gas and given amount of energy distributed between them for example [1,1,2,3,4]. Energy distribution can be realized as [1,2,1,3,4] where index represent a particle, but the distribution can also be realized as [2,1,1,3,4] after swapping first two and so forth. W is measure of all possible way the distribution can be realized. When W is small for given distribution that distribution has small entropy, when W is large for given distribution it has a large entropy.

The equation was originally formulated byLudwig Boltzmann between 1872 and 1875, but later put into its current form byMax Planck in about 1900.^[2][3] To quote Planck, "thelogarithmic connection betweenentropy andprobability was first stated by L. Boltzmann in hiskinetic theory of gases".^[4]

A 'microstate' is a state specified in terms of the constituent particles of a body of matter or radiation that has been specified as a macrostate in terms of such variables asinternal energy and pressure. A macrostate is experimentally observable, with at least a finite extent inspacetime. A microstate can be instantaneous, or can be a trajectory composed of a temporal progression of instantaneous microstates. In experimental practice, such are scarcely observable. The present account concerns instantaneous microstates.

The value ofW was originally intended to be proportional to theWahrscheinlichkeit (the German word for probability) of amacroscopic state for someprobability distribution of possiblemicrostates—the collection of (unobservable microscopic single particle) "ways" in which the (observable macroscopic)thermodynamic state of a system can be realized by assigning differentpositions andmomenta to the respective molecules.

There are many instantaneous microstates that apply to a given macrostate. Boltzmann considered collections of such microstates. For a given macrostate, he called the collection of all possible instantaneous microstates of a certain kind by the namemonode, for which Gibbs' termensemble is used nowadays. For single particle instantaneous microstates, Boltzmann called the collection anergode. Subsequently, Gibbs called it amicrocanonical ensemble, and this name is widely used today, perhaps partly because Bohr was more interested in the writings of Gibbs than of Boltzmann.^[5]

Interpreted in this way, Boltzmann's formula is the most basic formula for the thermodynamicentropy. Boltzmann'sparadigm was anideal gas ofNidentical particles, of whichN_i are in thei-th microscopic condition (range) of position and momentum. For this case, the probability of each microstate of the system is equal, so it was equivalent for Boltzmann to calculate the number of microstates associated with a macrostate.W was historically misinterpreted as literally meaning the number of microstates, and that is what it usually means today.W can be counted using the formula forpermutations

${\displaystyle W=\frac{N!}{\prod _i{N}_i!}}$

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wherei ranges over all possible molecular conditions and "!" denotesfactorial. The "correction" in the denominator is due to the fact that identical particles in the same condition areindistinguishable.W is sometimes called the "thermodynamic probability" since it is aninteger greater than one, whilemathematical probabilities are alwaysnumbers between zero and one.

In Boltzmann’s 1877 paper, he clarifies molecular state counting to determine the state distribution number introducing the logarithm to simplify the equation.

Boltzmann writes:“The first task is to determine the permutation number, previously designated by𝒫, for any state distribution. Denoting by J the sum of the permutations𝒫for all possible state distributions, the quotient𝒫/J is the state distribution’s probability, henceforth denoted by W. We would first like to calculate the permutations𝒫forthe state distribution characterized by w₀ molecules withkinetic energy 0, w₁ molecules with kinetic energy ϵ, etc. …

“The most likely state distribution will be for those w₀, w₁ … values for which𝒫is a maximum or since the numerator is a constant, for which the denominator is a minimum. The values w₀, w₁ must simultaneously satisfy the two constraints (1) and (2). Since the denominator of𝒫is a product, it is easiest to determine the minimum of its logarithm, …”

Therefore, by making the denominator small, he maximizes the number of states. So to simplify the product of the factorials, he uses their natural logarithm to add them. This is the reason for the natural logarithm in Boltzmann’s entropy formula.^[6]

Boltzmann's formula applies to microstates of a system, each possible microstate of which is presumed to be equally probable.

But in thermodynamics, the universe is divided into asystem of interest, plus its surroundings; then the entropy of Boltzmann's microscopically specified system can be identified with the system entropy in classical thermodynamics. The microstates of such a thermodynamic system arenot equally probable—for example, high energy microstates are less probable than low energy microstates for a thermodynamic system kept at a fixed temperature by allowing contact with a heat bath.For thermodynamic systems where microstates of the system may not have equal probabilities, the appropriate generalization, called theGibbs entropy formula, is:

${\displaystyle S_{\mathrm{G}}=-k_{\mathrm{B}}\sum p_i\ln p_i}$

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This reduces to equation (1) if the probabilitiesp_i are all equal.

Boltzmann used a${\displaystyle \rho \ln \rho }$ formula as early as 1866.^[7] He interpretedρ as a density in phase space—without mentioning probability—but since this satisfies the axiomatic definition of a probability measure we can retrospectively interpret it as a probability anyway.Gibbs gave an explicitly probabilistic interpretation in 1878.

Boltzmann himself used an expression equivalent to (3) in his later work^[8] and recognized it as more general than equation (1). That is, equation (1) is a corollary ofequation (3)—and not vice versa. In every situation where equation (1) is valid,equation (3) is valid also—and not vice versa.

The termBoltzmann entropy is also sometimes used to indicate entropies calculated based on the approximation that the overall probability can be factored into an identical separate term for each particle—i.e., assuming each particle has an identical independent probability distribution, and ignoring interactions and correlations between the particles. This is exact for an ideal gas of identical particles that move independently apart from instantaneous collisions, and is an approximation, possibly a poor one, for other systems.^[9]

The Boltzmann entropy is obtained if one assumes one can treat all the component particles of athermodynamic system as statistically independent. The probability distribution of the system as a whole then factorises into the product ofN separate identical terms, one term for each particle; and when the summation is taken over each possible state in the 6-dimensionalphase space of asingle particle (rather than the 6N-dimensional phase space of the system as a whole), the Gibbs entropy formula

${\displaystyle S_{\mathrm{G}}=-N{k}_{\mathrm{B}}\sum _i{p}_i\ln p_i}$

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simplifies to the Boltzmann entropy${\displaystyle S_{\mathrm{B}}}$.

This reflects the original statistical entropy function introduced by Ludwig Boltzmann in 1872. For the special case of anideal gas it exactly corresponds to the properthermodynamic entropy.

For anything but the most dilute of real gases,${\displaystyle S_{\mathrm{B}}}$ leads to increasingly wrong predictions of entropies and physical behaviours, by ignoring the interactions and correlations between different molecules. Instead one must consider theensemble of states of the system as a whole, called by Boltzmann aholode, rather than single particle states.^[10] Gibbs considered several such kinds of ensembles; relevant here is the canonical one.^[9]

• History of entropy
• H theorem
• Gibbs entropy formula
• nat (unit)
• Shannon entropy
• von Neumann entropy

• Introduction to Boltzmann's Equation
• Vorlesungen über Gastheorie, Ludwig Boltzmann (1896) vol. I, J.A. Barth, Leipzig
• Vorlesungen über Gastheorie, Ludwig Boltzmann (1898) vol. II. J.A. Barth, Leipzig.

• Thermodynamic entropy
• Thermodynamic equations
• Ludwig Boltzmann

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• Short description matches Wikidata