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 disprove the second law of thermodynamics.

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:

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:

${\displaystyle H(t)=\int _{0}^{\infty }f(E,t)\left[\log \left({\frac {f(E,t)}{\sqrt {E}}}\right)-1\right]\,dE}$

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