Boltzmann Bridges

It is often stated that the second law of thermodynamics follows from the condition that at some given time in the past the entropy was lower than it is now. Formally, this condition is the statement that $E[S(t)|S(t_0)]$, the expected entropy of the universe at the current time $t$ conditioned on its value $S(t_0)$ at a time $t_0$ in the past, is an increasing function of $t$. We point out that in general this is incorrect. The epistemic axioms underlying probability theory say that we should condition expectations on all that we know, and on nothing that we do not know. Arguably, we know the value of the universe's entropy at the present time $t$ at least as well as its value at a time in the past, $t_0$. However, as we show here, conditioning expected entropy on its value at two times rather than one radically changes its dynamics, resulting in a unexpected, very rich structure. For example, the expectation value conditioned on two times can have a maximum at an intermediate time between $t_0$ and $t$, i.e., in our past. Moreover, it can have a negative rather than positive time derivative at the present. In such "Boltzmann bridge" situations, the second law would not hold at the present time. We illustrate and investigate these phenomena for a random walk model and an idealized gas model, and briefly discuss the role of Boltzmann bridges in our universe.