Histories, dynamical laws, and initial conditions −Invariance under time-reversibility and its failure in Markov processes, with application to the second law of thermodynamics and the past hypothesis

Abstract

A Markov process can be invariant under time reversal and it also can exhibit a failure of invariance that is “uniformly positive.” I show how each of these possibilities contributes to the project of deciding when a temporal sequence of states has a higher probability than its mirror image. Neither suffices, but a distinct property of the Markov process completes the project, namely the unconditional probabilities of two possible states of the system at the start of the process. The concept of forward time-translational invariance plays a role in the analysis, but I discuss backward time-translational invariance as well. I argue that the Markov framework helps clarify how the Past Hypothesis (the hypothesis that the universe began in a very low entropy state) is related to the Second Law of Thermodynamics, and how each is relevant to explaining why histories that exhibit entropy increase have higher probabilities than histories that exhibit entropy decline. I argue that the Past Hypothesis, if true, helps explain this fact about histories, but a far weaker hypothesis about the universe's initial state suffices to do so.