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Boltzmann's entropy formula—carved on his gravestone. [1]In statistical mechanics, Boltzmann's entropy formula (also known as the Boltzmann–Planck equation, not to be confused with the more general Boltzmann equation, which is a partial differential equation) is a probability equation relating the entropy, also written as , of an ideal gas to the multiplicity (commonly denoted as or ), the ...
The statistical entropy perspective was introduced in 1870 by Austrian physicist Ludwig Boltzmann, who established a new field of physics that provided the descriptive linkage between the macroscopic observation of nature and the microscopic view based on the rigorous treatment of large ensembles of microscopic states that constitute ...
Boltzmann's grave in the Zentralfriedhof, Vienna, with bust and entropy formula. The defining expression for entropy in the theory of statistical mechanics established by Ludwig Boltzmann and J. Willard Gibbs in the 1870s, is of the form: = ,
Entropy can also be defined for any Markov processes with reversible dynamics and the detailed balance property. In Boltzmann's 1896 Lectures on Gas Theory, he showed that this expression gives a measure of entropy for systems of atoms and molecules in the gas phase, thus providing a measure for the entropy of classical thermodynamics.
Later, Boltzmann, in efforts to develop a kinetic theory for the behavior of a gas, applied the laws of probability to Maxwell's and Clausius' molecular interpretation of entropy so as to begin to interpret entropy in terms of order and disorder.
When these probabilities are substituted into the above expression for the Gibbs entropy (or equivalently k B times the Shannon entropy), Boltzmann's equation results. In information theoretic terms, the information entropy of a system is the amount of "missing" information needed to determine a microstate, given the macrostate.
Boltzmann in his original publication writes the symbol E (as in entropy) for its statistical function. [1] Years later, Samuel Hawksley Burbury, one of the critics of the theorem, [7] wrote the function with the symbol H, [8] a notation that was subsequently adopted by Boltzmann when referring to his "H-theorem". [9]
An alternative to Boltzmann's formula for entropy, above, is the information entropy definition introduced in 1948 by Claude Shannon. [35] Shannon's definition was intended for use in communication theory but is applicable in all areas.