# Boltzmann's entropy formula

In

statistical thermodynamics ,**Boltzmann's equation**is a probability equation relating theentropy "S" of an ideal gas to the quantity "W", which is the number of microstates corresponding to a givenmacrostate ::$S\; =\; k\; log\; W\; !$ (1)

where "k" is Boltzmann's constant equal to 1.38062 x 10

^{-23}joule/kelvin and "W" is the number ofmicrostate s consistent with the givenmacrostate . In short, the Boltzmann formula shows the relationship between entropy and the number of ways theatom s ormolecule s of athermodynamic system can be arranged. In 1934, Swiss physical chemistWerner Kuhn successfully derived a thermalequation of state for rubber molecules using Boltzmann's formula, which has since come to be known as the entropy model of rubber.**History**The equation was originally formulated by

Ludwig Boltzmann between 1872 to 1875, but later put into its current form byMax Planck in about 1900. [*[*] [*http://scienceworld.wolfram.com/physics/BoltzmannEquation.html Boltzmann equation*] – Eric Weisstein’s World of Physics (states the year was 1872)*cite book | author=Perrot, Pierre | title=A to Z of Thermodynamics | publisher=Oxford University Press | year=1998 | id=ISBN 0-19-856552-6 (states the year was 1875)*] To quote Planck, "thelogarithm ic connection betweenentropy andprobability was first stated by L. Boltzmann in hiskinetic theory of gases."The value of $W$, specifically, is the "Wahrscheinlichkeit", or number of possible microstates corresponding to the

macroscopic state of a system — number of (unobservable) "ways" the (observable) thermodynamic state of a system can be realized by assigning different positions and momenta to the various molecules. Boltzmann’s paradigm was anideal gas of $N$ "identical" particles, of which $N\_i$ are in the $i$-th microscopic condition (range) of position and momentum. $W$ can be counted using the formula for permutations:$W\; =\; N!;\; /\; ;\; prod\_i\; N\_i!$ (2)

where "i" ranges over all possible molecular conditions and $!$ denotes

factorial . The "correction" in the denominator is due to the fact that identical particles in the same condition are indistinguishable. $W$ is sometimes called the "thermodynamic probability" since it is aninteger greater than one, while mathematical probabilities are alwaysnumber s between zero and one.**Generalization**Boltzmann's formula applies to microstates of the universe as a whole, each possible microstate of which is presumed to be equally probable.

But in thermodynamics it is important to be able to make the approximation of dividing the universe into a system of interest, plus its surroundings; and then to be able to identify the entropy of the system with the system entropy in Classical thermodynamics. The microstates of such a thermodynamic system are "not" 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 the

Gibbs entropy , is::$S\; =\; -\; k\; sum\; p\_i\; log\; p\_i$ (3)

This reduces to equation (1) if the probabilities "p"

_{i}are all equal.Boltzmann used a $holog\; ho$ formula as early as 1866.cite journal | author=Ludwig Boltzmann | year = 1866 | title=Über die Mechanische Bedeutung des Zweiten Hauptsatzes der Wärmetheorie | journal=Wiener Berichte | volume=53 | pages=195–220] He interpreted $ho$ 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 workcite book | author=Ludwig Boltzmann | title=Vorlesungen über Gastheorie | publisher=J.A. Barth, Leipzig | year=1896 and 1898] 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.

**Boltzmann entropy excludes statistical dependencies**The term

Boltzmann 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, and may or may not be a good approximation for other systems. [*Jaynes, E. T. (1965). [*]*http://bayes.wustl.edu/etj/articles/gibbs.vs.boltzmann.pdf Gibbs vs Boltzmann entropies*] . "American Journal of Physics",**33**, 391-8.**ee also***

History of entropy

*Gibbs entropy **References****External links*** [

*http://www.chemsoc.org/exemplarchem/entries/pkirby/exemchem/Boltzmann/Boltzmann.html Introduction to Boltzmann's Equation*]

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