**5. The dynamics**

between this and the fact that the development of the underlying wave function for

**Definition 1.** A *universe U* is a chain of states (one state *Ut* for each moment of time *t*), with the property that the transition between adjacent states is always

**Definition 2.** A *multiverse M* is the set of all possible universes *U* in the sense of

As it stands, the model presented so far is too simple to generate any results. In fact, there are no observable differences at all between the states, which mean that there are no measurable variables which could be related to the (so far non-

There are of course many different variables which we can choose to enrich this structure, and which ones to choose must depend on what properties we want to explain. For explaining the second law of thermodynamics, the obvious choice is the

According to Boltzmann, the total entropy of a certain macro-state at a certain

, with *W* ¼ *e*

This formula was from the beginning derived for simple cases, like an ideal gas. Nevertheless, it does represent a kind of universal truth in statistical mechanics: the number of possible micro-states corresponding to a given macro-state grows exponentially with the entropy. Although there are many complications when one tries to consider the entropy of the universe as a whole, I will still take it as the starting point for the discussion that the entropy (at a given time *t*) is an exponential function of the total entropy as in (3). A more difficult question is if and how the constant *W* may vary with time, but for the purpose of the present paper, I will

One may of course argue that this can only be true when the universe is still quite ordered and the entropy is very far from reaching its maximum. But this is certainly what the situation is like in our universe today, and according to the computations in [10, 11], it would take an almost incredibly long time to reach such a state of maximal entropy. Thus, it will in the following be taken for granted that

where Ω denotes the number of corresponding micro-states and *kB* is

<sup>Ω</sup> <sup>¼</sup> *<sup>W</sup><sup>S</sup>*

this time is much longer than the life-span of our universe.

*S* ¼ *kB* ln Ω, (2)

<sup>1</sup>*=kB* , (3)

It may of course be said that quantum mechanics should allow for transitions between all kinds of states, although the probability for most such transitions may be extremely small. In this extremely simplified treatment, I will assume that for a given state at a given moment of time *t*, the dynamical laws will only permit transitions to a very limited number of states at the previous and next moments, which will make the probabilistic part of the investigation particularly simple. However, modifications are called for near the endpoints (the Big Bang and the Big Crunch); see Section 5.

Definition 1 together with a probability measure on this set.

the whole universe is unique.

*Probability, Combinatorics and Control*

Summarizing:

specified) dynamics.

entropy.

**4. Entropy**

time is given by

or inversely

Boltzmann's constant.

simply let it be constant.

**312**

possible.

The next step is to construct a model for the dynamics. The idea, which essentially goes back to Boltzmann (see [12]), is that any given macro-state at any given time is extremely likely to develop into a state with higher entropy at the next moment of time, simply because there are so many more states with higher entropy than with lower entropy (compare with (3)). The problem with this in the present situation, however, is that this way of thinking in fact presupposes a preferred direction of time. Otherwise, given that the dynamical laws are time symmetric, why can we not similarly argue that the entropy should also grow when we go backward in time? (compare [9]).

There have been many attempts to avoid this problem by looking for defects in the symmetries. But my conclusion here is that we must actually accept Boltzmann's argument in both directions of time and hence we are led to the following:

**Principle 1**. At every moment of time *t* and for every state with entropy *S*, there are very many "accessible states" with higher entropy, both at the previous moment of time *t* � 1 and at the next one *t* þ 1. On the other hand, the chance for finding such accessible states with lower entropy, both at times *t* � 1 and *t* þ 1, is extremely small.

This principle also implies a shift of perspective in the search for time's arrow. Rather than trying to find the reason for the asymmetry, we must concentrate on understanding why we cannot observe the symmetric structure of the multiverse as a whole.

As still one more simplification, let us assume that the entropy can only change by �1 during each unit of time. This assumption, however, has to be modified near the endpoints (BB and BC) for the following reason: it is a very important aspect of this approach to assume that physics during the first and last moments is very different from the rest of the time, since at these moments quantum phenomena can be expected to become global. To model this in a simple way, we can split the life-span of our multiverse up into three parts:

$$[-T\_0, -T\_1] \cup [-T\_1, T\_1] \cup [T\_1, T\_0].\tag{4}$$

Here the first and last parts may be called "the extreme phases," which are characterized by the property that transition between very different states can be possible. During the "normal phase" in between on the other hand, physics is supposed to behave more or less as we are used to.
