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Concept
H-theorem
Summary
In classical statistical mechanics, the H-theorem, introduced by Ludwig Boltzmann in 1872, describes the tendency to decrease in the quantity H (defined below) in a nearly-ideal gas of molecules. As this quantity H was meant to represent the entropy of thermodynamics, the H-theorem was an early demonstration of the power of statistical mechanics as it claimed to derive the second law of thermodynamics—a statement about fundamentally irreversible processes—from reversible microscopic mechanics. It is thought to prove the second law of thermodynamics, albeit under the assumption of low-entropy initial conditions.
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, with major themes being:
What is entropy? In what sense does Boltzmann's quantity H correspond to the thermodynamic entropy?
Are the assumptions (especially the assumption of molecular chaos) behind Boltzmann's equation too strong? When are these assumptions violated?
Boltzmann in his original publication writes the symbol E (as in entropy) for its statistical function. Years later, Samuel Hawksley Burbury, one of the critics of the theorem, wrote the function with the symbol H, a notation that was subsequently adopted by Boltzmann when referring to his "H-theorem". The notation has led to some confusion regarding the name of the theorem. Even though the statement is usually referred to as the "Aitch theorem", sometimes it is instead called the "Eta theorem", as the capital Greek letter Eta (Η) is undistinguishable from the capital version of Latin letter h (H). Discussions have been raised on how the symbol should be understood, but it remains unclear due to the lack of written sources from the time of the theorem. Studies of the typography and the work of J.W. Gibbs seem to favour the interpretation of H as Eta.
The H value is determined from the function f(E, t) dE, which is the energy distribution function of molecules at time t.
Official source
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