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

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 harmless for the purpose of explaining time's arrow.

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

A very interesting research project is therefore to try to find better models which 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 broken time symmetry.

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

$$\left(\mathbf{A}^{2m}\right)\_{ij} = \sum\_{q\_1} \sum\_{q\_2} \cdots \sum\_{q\_{2m-1}} a\_{iq\_1} a\_{q\_1 q\_2} \cdots a\_{q\_{2m-1} j}.\tag{14}$$

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 corresponding entropies. We can now define

$$\xi = \sum\_{k=-m+1}^{m} (\mathbb{S}\_k - \mathbb{S}\_{k-1})(\mathbb{S}\_{k+1} - \mathbb{S}\_k),\tag{15}$$

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

Given a real number *μ*≥0, we can then consider the probability measure which to each path *U* assigns the (un-normalized) probability exp f g *μξ* and replace the sum in (14) by

$$\sum\_{q\_1} \sum\_{q\_2} \cdots \sum\_{q\_{2m-1}} e^{\mu \xi} a\_{iq\_1} a\_{q\_1 q\_2} \cdots a\_{q\_{2m-1} j} \,. \tag{16}$$

With this definition, it is now again possible to compute the probability weights *PLL*, *PLH*, *PHL* and *PHH*, and we can note that for *μ* ¼ 0, these will be exactly the same as in the case without weights in Section 8. Thus, this model is really a generalization of the previous theory.

**Conjecture 1**. If *μ*>0, then we have a strong broken time symmetry in the limit *m* ! ∞ (for a suitable fixed choice of *p*, *K*, and *W* with *K* ≪*W*).
