# A Combinatorial Approach to Time Asymmetry

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Preliminaries

**Definition 1.**The adjacency matrix of the (directed) graph G with nodes ${v}_{1},{v}_{2},\dots ,{v}_{m}$ is the $m\times m$-matrix $A=\left({a}_{ij}\right)$, where ${a}_{ij}=1$ if the pair ${v}_{i},{v}_{j}$ determines a (directed) edge in G and ${a}_{ij}=0$ otherwise.

**Theorem 2.**

## 3. To model the universe

**Definition 3.**A universe U is a chain of states (one state ${U}_{t}$ for each moment of time t), with the property that the transition between adjacent states is always possible.

**Definition 4.**The multiverse M is the set of all possible universes U in the sense of Definition 3.

## 4. How many universes of different types are there?

**Remark 1.**Summing up, let us note that the dynamical assumptions above are time-symmetric; the choices of the edges are to a certain extent at random, but if we consider the totality of all such graphs, this set will be preserved if we reverse the direction of the time-axis. In particular, when we consider mean values, these will be time-symmetric. Also, the dynamics reflects Boltzmann’s idea of the universe as developing from less probable states to more probable states in the sense that for any randomly chosen state there will always be several possible developments towards higher entropy but only a small chance that there will be a development towards lower entropy. It is essential to note that this is true in both directions of time.

## 5. A Model Which Includes the Big Bang and the Big Crunch

**Remark 2.**A possible physical motivation for this type of boundary conditions could look as follows; Imagine the universe at the Big Bang (and the Big Crunch) as in a state of perfect order. When the universe starts to expand, this highly ordered state may be meta-stable in the following sense: The by far most likely scenario is that the universe remains highly ordered throughout the first decisive moments of time. But there is also a small probability that some sufficiently large fluctuation will occur which instantly causes this highly ordered state to decay into a completely disordered high-entropy state. This can be compared with e.g. an over-saturated gas in thermodynamics. As already stated, all other types of behavior is assumed to have neglectable probability. Also, the behavior near the Big Crunch is completely symmetric.

## 6. The probability for high entropy at both ends

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**A schematic picture of three universes during the normal phase, as particular paths in the graph of all possible states, in a very small model with only five moments of time from $-{T}_{1}=-2$ to ${T}_{1}=2$. The red path has a monotonic behavior of the entropy whereas the blue paths represent developments with low entropy at both ends.

**Figure 3.**The ratio ${N}_{LL}/{N}_{LH}$ as a function of W for the cases ${T}_{1}=2$ (yellow) and ${T}_{1}=3$ (grey).

**Figure 4.**The ratio ${N}_{HH}/{N}_{LH}$ (left) an $\widehat{\rho}$ (right) as functions of W for the case ${T}_{1}=2$.

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Tamm, M.
A Combinatorial Approach to Time Asymmetry. *Symmetry* **2016**, *8*, 11.
https://doi.org/10.3390/sym8030011

**AMA Style**

Tamm M.
A Combinatorial Approach to Time Asymmetry. *Symmetry*. 2016; 8(3):11.
https://doi.org/10.3390/sym8030011

**Chicago/Turabian Style**

Tamm, Martin.
2016. "A Combinatorial Approach to Time Asymmetry" *Symmetry* 8, no. 3: 11.
https://doi.org/10.3390/sym8030011