Editing H-theorem (section)

=== Loschmidt's paradox ===

{{main article|Loschmidt's paradox}}

Soon after Boltzmann published his ''H'' theorem, [[Johann Josef Loschmidt]] objected that it should not be possible to deduce an irreversible process from time-symmetric dynamics and a time-symmetric formalism. If the ''H'' decreases over time in one state, then there must be a matching reversed state where ''H'' increases over time ([[Loschmidt's paradox]]). The explanation is that Boltzmann's equation is based on the assumption of "[[molecular chaos]]", i.e., that it follows from, or at least is consistent with, the underlying kinetic model that the particles be considered independent and uncorrelated. It turns out that this assumption breaks time reversal symmetry in a subtle sense, and therefore [[begs the question]]. Once the particles are allowed to collide, their velocity directions and positions in fact ''do'' become correlated (however, these correlations are encoded in an extremely complex manner). This shows that an (ongoing) assumption of independence is not consistent with the underlying particle model.

Boltzmann's reply to Loschmidt was to concede the possibility of these states, but noting that these sorts of states were so rare and unusual as to be impossible in practice. Boltzmann would go on to sharpen this notion of the "rarity" of states, resulting in his [[Boltzmann's entropy formula|entropy formula]] of 1877.