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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 ...
However, after sufficient time has passed, the system reaches a uniform color, a state much easier to describe and explain. Boltzmann formulated a simple relationship between entropy and the number of possible microstates of a system, which is denoted by the symbol Ω. The entropy S is proportional to the natural logarithm of this number:
The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. Coulomb interactions, is often called the Vlasov equation. This equation is more useful than the principal one above, yet still incomplete, since f cannot be solved unless the collision term in f is known.
The connection between thermodynamic entropy and information entropy is given by Boltzmann's equation, which says that S = k B ln W. If we take the base-2 logarithm of W, it will yield the average number of questions we must ask about the microstate of the physical system in order to determine its macrostate. [13]
In practice, information entropy is almost always calculated using base-2 logarithms, but this distinction amounts to nothing other than a change in units. One nat is about 1.44 shannons. For a simple compressible system that can only perform volume work, the first law of thermodynamics becomes = +.
Mathematically, the absolute entropy of any system at zero temperature is the natural log of the number of ground states times the Boltzmann constant k B = 1.38 × 10 −23 J K −1. The entropy of a perfect crystal lattice as defined by Nernst's theorem is zero provided that its ground state is unique, because ln(1) = 0.
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?
Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced k, and gave a more precise value for it (1.346 × 10 −23 J/K, about 2.5% lower than today's figure), in his derivation of the law of black-body radiation in 1900–1901. [11]