H-theorem

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. H   = d e f   ∫ P ( ln ⁡ P ) d 3 v = ⟨ ln ⁡ P ⟩ {displaystyle displaystyle H {stackrel {mathrm {def} }{=}} int {P({ln P}),d^{3}v}=leftlangle ln P ight angle } 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: 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. The value f(E, t) dE is the number of molecules that have kinetic energy between E and E + dE. H itself is defined as For an isolated ideal gas (with fixed total energy and fixed total number of particles), the function H is at a minimum when the particles have a Maxwell–Boltzmann distribution; if the molecules of the ideal gas are distributed in some other way (say, all having the same kinetic energy), then the value of H will be higher. Boltzmann's H-theorem, described in the next section, shows that when collisions between molecules are allowed, such distributions are unstable and tend to irreversibly seek towards the minimum value of H (towards the Maxwell–Boltzmann distribution). (Note on notation: Boltzmann originally used the letter E for quantity H; most of the literature after Boltzmann uses the letter H as here. Boltzmann also used the symbol x to refer to the kinetic energy of a particle.)