In 1872 Ludwig Boltzmann in Graz generalized Maxwell's approach for the kinetic theory of dilute gases to nonequilibrium processes, so that he could investigate the transition from nonequilibrium to equilibrium. His non-equilibrium single particle dis tribution function f : = f(x; v; t) gives the average number of molecules in a dilute gas at the position x with velocity v at time t. The temporal change of this distribution function consists of two terms, a drift term due to the motion of the molecules and a collision term due to collisions with other molecules. In the absence of an external field of force this equation, which is now called Boltzmann equation, reads:

Here J_B(ff) is the binary collision term which takes only two particle collisions into account, a good approximation for a dilute gas. A further assumption in Boltzmann's expression for the collision term is that the velocities of the colliding molecules must be uncorrelated, which was later called the assumption of "molecular chaos" by Jeans. Now Boltzmann introduced the funtional

for which he could show under very general assumptions for the intermolecular inter action that if f is a solution of the time derivative of H is always smaller than zero or at most zero:

Furthermore for an ideal gas in equilibrium he could show that the entropy S is up to a sign proportional to H. For nonequilibrium this is a generalization of the thermody namic entropy now called Boltzmann entropy

and is nothing but the second law of themodynamics for a closed system

This is Boltzmann's famous H-theorem.
The H-theorem and the Boltzmann equation met with violent objections from physicists and from mathematicians. These objections can be formulated in the form of paradoxes. The most important ones are the reversibility paradox formulated in 1876 by Boltzmann's friend Josef Loschmidt (1821-95) and the recurrence paradox formulated in 1896 by Ernst Zermelo (1871-1953).