**7. Modeling the combinatorial multiverse as a probability space**

Now when we have specified the dynamics of the model, i.e., decided which paths (universes) can occur, it is time to attribute to each such path its probability weight so that the multiverse becomes a probability space. Following the tradition in statistical mechanics, I will frequently make use of un-normalized probabilities. This means that summing up all (un-normalized) probabilities will give the "state sum," which in general is not equal to one. To obtain the usual probabilities, one has to divide by the state sum. This may seem unnatural at first but turns out to be very practical in situations where only the relative sizes of the probabilities are needed.

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

With this setup and the random dynamics introduced earlier, each *B*-matrix contains all the information about the edges from all the states at one moment of time to the states at the next one. For example, *B*<sup>12</sup> contains the information about all edges from the single state with *S* ¼ 0 at time *t* ¼ �2 to the five states with *S*≤ 1 when *t* ¼ �1. In the same way, *B*<sup>23</sup> gives a complete description of the edges from the 5 states with *S*≤1 at time *t* ¼ �1 to the 21 states with *S*≤ 2 when *t* ¼ 0.

The number of rows and columns in the *B*-matrices are now given as follows:

For the quadratic adjacency matrix *A*, this gives the format 453 � 453. The matrices *Bk*,*k*þ<sup>1</sup> can also be described as block matrices in the following way: *B*<sup>12</sup> ¼ ð Þ 0j0101 (the first element is always a 0 and among the other four, two randomly chosen elements will be one instead of zero). For the following matrix,

Both *C*<sup>1</sup> and *C*<sup>3</sup> have rows containing only zeros, except for two randomly chosen positions where there are ones instead (these are the edges which connect to states with higher entropy one unit of time later), and *C*<sup>2</sup> is a column of zeros with two randomly chosen ones instead (these are the edges which connect to states with

where now all *D*:s and *E*:s with odd indices have rows with two randomly chosen ones and those with even indices have columns with two randomly chosen ones.

Now when we have specified the dynamics of the model, i.e., decided which paths (universes) can occur, it is time to attribute to each such path its probability weight so that the multiverse becomes a probability space. Following the tradition in statistical mechanics, I will frequently make use of un-normalized probabilities. This means that summing up all (un-normalized) probabilities will give the "state sum," which in general is not equal to one. To obtain the usual probabilities, one has to divide by the state sum. This may seem unnatural at first but turns out to be very practical in situations where only the relative sizes of the probabilities are needed.

**7. Modeling the combinatorial multiverse as a probability space**

we obtain (with certain random choices of ones as before)

lower entropy one unit of time later).

*Probability, Combinatorics and Control*

**316**

The structures of *B*<sup>34</sup> and *B*<sup>45</sup> are similar:

*B*<sup>12</sup> : 1 � 5, *B*<sup>23</sup> : 5 � 21, *B*<sup>34</sup> : 21 � 85, *B*<sup>45</sup> : 85 � 341*:* (7)

ð8Þ

ð9Þ

As for the normal phase, the choice will, to start with, be the simplest possible one: each path is either possible or not, corresponding to the probability weights 1 and 0. During the extreme phases, this assumption is no longer reasonable. Again the model will be extremely simplified, but still it is based on physical intuition and, most importantly, completely time symmetric. Assume that the only types of edges having a non-neglectable chance of occurring during the extreme phase ½ � �*m* � 1, �*m* are of the following two kinds: The first scenario is that the universe passes through the extreme phase into a state of zero entropy. The other scenario is that it passes into a state with high entropy (equal to 2*m*). Universes of one of these two types will be given the (un-normalized) probability 1 or *p*, respectively. Here *p*> 0 should be thought of as a very small number, at least when the size of the model becomes large. During the other extreme phase ½ � *m*, *m* þ 1 , near the Big Crunch, we make the completely symmetric assumption.

*Remark* 3. These assumptions may perhaps seem somewhat arbitrary. And to a certain extent, this may be so. However, they do represent the following viewpoint of what may happen at the full cosmological scale: we may think of the Big Bang and the Big Crunch as states of complete order with zero volume and entropy. Such states can very well be metastable, very much like an oversaturated gas at a temperature below the point of condensation. If no disturbance takes place, such metastable states can very well continue to exist for a substantial period of time. In particular, a low-entropy state can have a very good chance of surviving the intense but extremely short extreme phase. On the other hand, if a sufficiently large disturbance occurs, then the metastable state may almost immediately decay into a very disordered state of high entropy.

It is not my intension to further argue in favor of this viewpoint here. The main thing in this chapter is to show that completely symmetric boundary conditions at the endpoints may give rise to a broken time symmetry.

The multiverse now splits up into four different kinds of paths:


If we now denote by *NLL*, *NLH*, *NHL* and *NHH* the number of paths of the indicated kinds, then with the above assumptions we also get the corresponding probability weights for the corresponding types as

$$P\_{LL} = \mathbf{N}\_{LL}, \quad P\_{LH} = \mathbf{p} \mathbf{N}\_{LH}, \quad P\_{HL} = \mathbf{p} \mathbf{N}\_{HL}, \quad P\_{HH} = \mathbf{p}^2 \mathbf{N}\_{HH}.\tag{10}$$

We can now consider the following two types of broken time symmetry: **Definition 4.** A multiverse is said to exhibit a *weak* broken time symmetry if

$$P\_{LL} \ll P\_{LH} + P\_{HL}.\tag{11}$$

**Definition 5.** A multiverse is said to exhibit a *strong* broken time symmetry if

$$P\_{LL} + P\_{HH} \ll P\_{LH} + P\_{HL}.\tag{12}$$

Both these definitions should of course be made more precise when applied to specific models for the multiverse, e.g., by showing that the corresponding limits

$$\lim \frac{P\_{LL}}{P\_{LH} + P\_{HL}} \quad \text{and} \quad \lim \frac{P\_{LL} + P\_{HH}}{P\_{LH} + P\_{HL}} \tag{13}$$

*NLL=*ð Þ! *NLH* þ *NHL* 0 when *W* ! ∞, there is not really enough support for a firm prediction about the more precise asymptotic behavior for large *W*. Having said this, the behavior seems to be rather close to a relationship of the form *ρ* � 1*=W*. It should be possible, although perhaps not so easy, to prove exact limit theorems to confirm these kinds of predictions. The problem is that we use a large number of instances to model something much more complicated, namely, the full quantum mechanical development of the multiverse. For very special unlikely choices of these instances, the ratio *NLL=*ð Þ *NLH* þ *NHL* may behave quite differently.

**9. Can the dynamics be modified to generate a strong broken symmetry?**

However, there is one assumption which is somewhat problematic in the dynamics that we have discussed so far: the model can be said to exhibit a kind of Markov property in the sense that the probability for the entropy to go up or down at a certain step is completely independent of the prehistory of the state; it just depends on the state itself. This does not appear to be what is happening in our own universe: for instance, light emitted from (more or less) pointlike sources like stars continues to spread out concentrically for billions of years, and in this way it

harmless for the purpose of explaining time's arrow.

preserves a memory of the prehistory for a very long time.

at time *m* can be computed using the adjacency matrix *A* as

*<sup>ξ</sup>* <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*

*k*¼�*m*þ1

*ij* <sup>¼</sup> <sup>X</sup> *q*1

X *q*2

*A*<sup>2</sup>*<sup>m</sup>* � �

corresponding entropies. We can now define

and if not, then ð Þ *Sk* � *Sk*�<sup>1</sup> ð Þ¼� *Sk*þ<sup>1</sup> � *Sk* 1.

**319**

broken time symmetry.

*Combinatorial Cosmology*

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

Obviously, the above model represents an extreme simplification. But from the point of view of the author, most of the simplifications can be said to be rather

A very interesting research project is therefore to try to find better models which

I will now briefly discuss an example of such a modified model. In Section 6 it was noted that the number of paths between a state *i* at time �*m* and another state *j*

> ⋯X *<sup>q</sup>*2*m*�<sup>1</sup> *aiq*<sup>1</sup> *aq*1*q*<sup>2</sup>

This sum can now be modified by introducing various weights depending on the path. An example of such a weight can be constructed as follows: given a path *U* with vertices *v*�*<sup>m</sup>*, *v*�*m*þ1, *v*�*m*þ2, … , *vm*, we let *S*�*<sup>m</sup>*, *S*�*m*þ1, *S*�*m*þ2, … , *Sm* denote the

and note that periods of monotonic growth or decrease of the entropy will tend to make *ξ* positive, whereas switches between growth and decrease tend to make it negative. In fact, if *S* is monotonic on ½ � *k* � 1, *k* þ 1 , then ð Þ *Sk* � *Sk*�<sup>1</sup> ð Þ¼ *Sk*þ<sup>1</sup> � *Sk* 1

⋯*aq*2*m*�1*<sup>j</sup>:* (14)

ð Þ *Sk* � *Sk*�<sup>1</sup> ð Þ *Sk*þ<sup>1</sup> � *Sk* , (15)

do not exhibit this property. We can, for instance, attempt to construct models where the behavior of the entropy not only depends on the previous (or following) step but on a larger part of the prehistory (or post-history). As a particularly simple example one could let the probabilities for an increase (or decrease) of the entropy at a certain step, depend not only on the previous and following step but on the two previous (and following) steps. In fact, such dynamics would not only be more realistic but would in general also have a much better chance to exhibit a strong

equal zero when certain parameters tend to infinity in some well-defined way. However, it is worthwhile at this stage to note their implications for cosmology.

The strong broken symmetry in Definition 5 actually means that a monotonic behavior of the entropy is far more probable than a non-monotonic one. In the case of a weak broken symmetry, this is not necessarily so; it could very well be that the most probable scenario would be high entropy at both ends. Thus, this is definitely a weaker statement, but it can nevertheless be argued that it can be used to explain the time asymmetry that we observe, referring to a kind of anthropic principle: it is an obvious observational fact that we live in a universe with low entropy at at least one end. If the statement in Definition 4 is fulfilled, then clearly among such scenarios, the monotonic ones (LH and HL) are the by far most probable ones. Thus, since universes with high entropy at both ends would seem to be quite uninhabitable, one can argue that given the existence of an observer, then with almost certainty he must live in a universe with monotonic entropy.

Summing up, both limits above can be used to argue in favor of time asymmetry. Nevertheless, at least to the mind of the author, the strong broken symmetry is the preferable one. This alternative will be further studied in Section 9.
