Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

188 cosmological intimations of infinityprescription of which removes the youngness paradox by giving a head start to certainbubble times.8.5.2 Infinite versus Finite Number of StatesAre the number of states accessible to the entire universe infinite or finite? In everlastingor eternal inflation, it would seem that the number is infinite, because the spacetimevolume is, but this has been disputed (e.g., Dyson et al. (2002), Banks and Fischler(2004), Bousso et al. (2006a)). Quite independent of inflation or everlasting inflation,however, an interesting set of paradoxes – going back to Boltzmann – arises if weimagine the universe to be a physical system with a finite number N of microstates (i.e.,states that completely characterize the physical system) that evolve according to a fixedHamiltonian (i.e., a fixed deterministic evolution from one microstate to the next). Inthis context, imagine two (macro) states of the universe, s i and s o with entropies S i andS o >S i (i.e., exp(S o ) different microstates are “coarse-grained” into the macrostates o that is identified by some observables that are indistinguishable for that set ofmicrostates, and likewise for S i and s i .) Here, the “i” and “0” can be considered to standfor the “initial” and “observed” states. Suppose further that the microstates in s i nearlyall evolve into microstates within s o ; thus, if the universe is in state s i at some time, itwill naturally evolve into s o later. Now let the system evolve forever. After some time,the universe will reach equilibrium (attaining entropy of roughly ln N), and thereafter(given some assumptions about ergodicity) states s i and s o , which are nonequilibriumstates, will be realized only as ultra-rare downward fluctuations in entropy.Now suppose we analyze all times at which the universe is in state s o . What arethe most typical histories of the universe just prior to this? Because, according tostatistical mechanics, the probability of fluctuating a state decreases exponentially withthe magnitude of the downward entropy fluctuation, analysis quickly reveals that ingeneral, these histories will not include the “precursor” state s i . Rather, they will almostcertainly be a direct fluctuation from the equilibrium state to s o , or (with somewhatsmaller probability) a fluctuation to a state s o ′ that is only very slightly lower entropythan s o and then evolves into s o .As one final piece of groundwork, consider some set of attributes A, which mayor may not describe a state s. Next, denote by A o some attributes that describe ouruniverse. (These might, for illustration, be “there is a cosmologist named Anthony ona planet around a star in a galaxy.”) If s o is the state our universe is in, then clearly A odescribes s o , but unless A o is so detailed as to uniquely fix the macrostate, there maybe plenty of other states also described by A o .So far so good, and now we can discuss the paradoxes. Imagine that we ask thequestion, “Why do we see a particular direction of time, as defined by an increasingentropy?” In particular, why did the universe “begin” at low entropy, and what broughtsuch a highly improbable state into being? The possibility that Boltzmann suggested(and attributed to his lab assistant Schuetz 30 ) is that the universe is generally in equilibrium,but occasionally fluctuates to a nonequilibrium state like s i , which then naturallyevolves to s o , thus accounting for the history that we see.The immediate problem, however, is the fact stated above. Supposing that we arein state s o now, the most probable precursor to our state was not s i . A much more