**3. The combinatorial multiverse**

This is not the place to try to describe all possible combinatorial models for cosmology. Rather, I have chosen to just discuss the simple case of a closed, finite universe. Many cosmologists these days support open models, and it is of course possible to apply combinatorial methods to them too. However, since such models tend to be infinite, they may be considerably more complicated from a probabilistic point of view.

To model the set of all universes in the simplest possible way, let us for each moment of time between the endpoints �*T*<sup>0</sup> and *T*<sup>0</sup> (i.e., the Big Bang and the Big Crunch) consider the finite set of all possible "states" of a universe. To make everything extremely simple, let us suppose that time is discrete in the sense that we only consider it at a finite number of points as follows:

$$\begin{array}{ccccccccc} -T\_0 & -T\_0 + 1 & & -1 & \mathbf{0} & \mathbf{1} & & T\_0 - 1 & T\_0 \\ \hline & & & & & & \mathbf{0} & \mathbf{1} & & \mathbf{0} \\ \hline & & & & & & \mathbf{0} & \mathbf{0} & \mathbf{1} \\ \end{array} \tag{1}$$

Thus, we can measure time just by counting the number of time intervals, which means that time can be viewed as integer valued. At the endpoints �*T*<sup>0</sup> and *T*0, there will just be one unique state (with zero volume), but in between, there will be many states for each *t*. All such states will be the nodes of an enormous graph, and a universe will then be just a path in this graph with the property that there is exactly one state for each moment of time. The dynamics of the model can then be specified by choosing at certain collection of edges between adjacent moments of time, say *t* and *t*+1, which correspond to those time developments which are possible. A quite schematic picture is displayed in **Figure 1**.

*Remark* 1. For the readers taking interest in the underlying physics: the word "state" is not referring to quantum states as they are usually interpreted. A better way of thinking of them is to say that they represent "distinguishable configurations." This is in fact a kind of semiclassical approximation (see Tamm [2]).

The important point here is that a given state can lead to different states in the future. This is very much what actually happens when, say, a particle decays: whether or not this happens may, according to the multiverse interpretation, lead to very different futures within a rather short time. And there is no contradiction

**Figure 1.** *One universe in the combinatorial multiverse [3].* between this and the fact that the development of the underlying wave function for the whole universe is unique.

Summarizing:

**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 possible.

**Definition 2.** A *multiverse M* is the set of all possible universes *U* in the sense of Definition 1 together with a probability measure on this set.

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.

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 nonspecified) dynamics.

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 entropy.

## **4. Entropy**

According to Boltzmann, the total entropy of a certain macro-state at a certain time is given by

$$S = k\_B \ln \Omega,\tag{2}$$

**5. The dynamics**

*Combinatorial Cosmology*

*DOI: http://dx.doi.org/10.5772/intechopen.90696*

small.

a whole.

**313**

backward in time? (compare [9]).

life-span of our multiverse up into three parts:

supposed to behave more or less as we are used to.

**6. Modeling the dynamics**

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

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

**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

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

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

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

To construct a miniature multiverse for computational purposes, one can proceed as follows: first of all, in the very small multiverses studied here, the extreme phases will only last for one single unit of time. Also, for ease of notation, let us put

The dynamics is specified by randomly choosing for each state at time *t* with entropy *S*, *K* edges to states at time *t* þ 1 with entropy *S* þ 1, and similarly *K* edges to states at time *t* � 1 with entropy *S* þ 1 (with obvious modifications at the endpoints). In this section, again to make everything as simple as possible, *K* will be set equal to 2. These random choices are in practice carried out by the random number

�*m* � 1, � *m*, � *m* þ 1, … , *m* � 1, *m*, *m* þ 1*:* (5)

*T*<sup>1</sup> ¼ *m*, so that the moments of time can in this context be denoted as

½ � �*T*0, �*T*<sup>1</sup> ∪ ½ � �*T*1, *T*<sup>1</sup> ∪ ½ � *T*1, *T*<sup>0</sup> *:* (4)

argument in both directions of time and hence we are led to the following:

or inversely

$$
\Omega = W^S, \quad \text{with} \quad W = \mathfrak{e}^{1/k\_{\beta}}, \tag{3}
$$

where Ω denotes the number of corresponding micro-states and *kB* is Boltzmann's constant.

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 simply let it be constant.

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