Boltzmann Bridges: the 2nd Law with Two Time Conditions

Abstract: The 2nd law of thermodynamics has provided one of the great mysteries of physics since the time of Boltzmann: why does entropy increase into the future, but not into the past? It is now the view of many physicists and philosophers that this mystery has been solved by the 'past hypothesis' that the entropy of the early universe was very low.  Arguably, we know the current entropy of the universe at least as accurately as we know its entropy 13 billion years ago. This implies that we should consider the stochastic dynamics of the universe’s entropy conditioned on those two moments, rather than just conditioned on the single moment of its inferred value 13 billion years ago. However, the two conditions that (a) the universe is out of equilibrium now and (b) that it had lower entropy at some time in the past do *not* together imply that the entropy was lower in the recent past. Rather, these conditions can easily lead to solutions to the entropy time-dependence that include "bridges," in which the 2nd law is violated. We demonstrate such entropy evolution explicitly for a random walk model and an idealized gas model, and conclude with a discussion of the role of bridges in our universe.