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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 higher terms have singularities. The problem of developing mathematically the limiting processes, which lead from the atomistic view (represented by Boltzmann's equation) to the laws of motion of continua, is an important part of Hilbert's sixth problem. [21]
The H-theorem is a natural consequence of the kinetic equation derived by Boltzmann that has come to be known as Boltzmann's equation. The H-theorem has led to considerable discussion about its actual implications, [6] with major themes being: What is entropy? In what sense does Boltzmann's quantity H correspond to the thermodynamic entropy?
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: = ,
Ludwig Boltzmann defined entropy as a measure of the number of possible microscopic states (microstates) of a system in thermodynamic equilibrium, consistent with its macroscopic thermodynamic properties, which constitute the macrostate of the system. A useful illustration is the example of a sample of gas contained in a container.
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.
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's grave in the Zentralfriedhof, Vienna, with bust and entropy formula. In statistical mechanics, the entropy S of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of W , the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy E ...