"It's hard to put ourselves in the shoes of a late-nineteenth-century thermodynamicist. Those folks felt that the inability of entropy to decrease in a closed system was not just a good idea; it was a Law. The idea that entropy would "probably" increase wasn't any more palatable than a suggestion that energy would "probably" be conserved would have been. In reality, the numbers are just so overwhelming that the probabilistic reasoning of statistical mechanics might as well be absolute, for all intents and purposes. But Boltzmann wanted to prove something more definite than that.... In 18702, Boltzmann (twenty-eight years old at the time) published a paper in which he purported to show that entropy would always increase or remain constant - a result called the "H-Theorem", which has been the subject of countless debates ever since."

"The next step belongs to Boltzmann, in whose work the connection between statistical mechanics and the second law of thermodynamics first becomes absolutely explicit. Like Maxwell, Boltzmann considered the effects of collisions on the distribution of velocities of the molecules of a gas. In effect, his approach was to partition the available velocities into a series of tiny intervals, and to consider the effect of collisions on the number of molecules whose velocities fell in each of these intervals. He was able to argue that no matter what the initial distribution of velocities - that is, no matter how the particles were initially assigned to the available velocity intervals - the effect of collisions was to make the distribution of velocities approach the distribution which Maxwell had characterized a few years earlier.

Boltmann's approach was to define a quantity, originally denoted E, with respect to which he was able to argue (1) that E takes its minimum possible value when the gas has the Maxwell distribution, and (2) that when the gas does not already have the Maxwell distribution, the effect of collisions is to ensure that E always decreases with time. Together these results seemed to imply not only that a gas will not depart from the Maxwell distribution, but also that any sample of gas will tend to that distribution, if not already there. These are the defining characteristics of the notion of equilibrium, however, and Boltzmann's result therefore provides powerful confirmation that the Maxwell distribution does indeed characterize the condition of a gas in thermal equilibrium, as Maxwell himself had suggested. (E later came to be called H, and the result is now known as Boltzmann's H-theorem.)"
pp 25-26