Loschmidt's Paradox

Charles University is the oldest university in central and eastern Europe, founded in 1348, so more than twice as old as William and Mary or Yale. It has a host of distinguished alumni, including several Nobel Prize winners, famous historical figures, Novelists Karel KapekČapek and Franz Kafka as well as Nicola Tesla, Ivana Trump and Lubos Motl. Einstein and Mach appear on the faculty list. (Though the place never entered my consciousness before the last named alum beat the crap out of me for dissing the joint).

Another student there was Johann Josef Loschmidt, who was the first to determine the number of molecules in a cm^3 of air, and made other contributions to Chemistry. He was a friend of Boltzmann's and contributed to the clarification of the foundations of statistical mechanics, notably through posing Loschmidt's Paradox, which is deeply connected to the Boltzmann Brain problem. (This is kind of a second indignity at the hands of history, since not only should Avogodro's number probably been named after Loschmidt, but Boltzmann's brain probably should be too.)

From the Wikipedia article:

Any process that happens regularly in the forward direction of time but rarely or never in the opposite direction, such as entropy increasing in an isolated system, defines what physicists call an arrow of time in nature. This term only refers to an observation of an asymmetry in time, it is not meant to suggest an explanation for such asymmetries. Loschmidt's paradox is equivalent to the question of how it is possible that there could be a thermodynamic arrow of time given time-symmetric fundamental laws, since time-symmetry implies that for any process compatible with these fundamental laws, a reversed version that looked exactly like a film of the first process played backwards would be equally compatible with the same fundamental laws, and would even be equally probable if one were to pick the system's initial state randomly from the phase space of all possible states for that system.

And:

Another way of dealing with Loschmidt's paradox is to see the second law as an expression of a set of boundary conditions, in which our universe's time coordinate has a low-entropy endpoint: the Big Bang. From this point of view, the arrow of time is determined entirely by the direction that leads to the Big Bang, and a hypothetical universe with a maximum-entropy Big Bang would have no arrow of time. The theory of cosmic inflation tries to give reason why the early universe had such a low entropy.