Was an almighty God required to boot up the universe, or would a blindfolded monkey have sufficed?

What a silly question. Isn't it obvious that truly divine powers were needed to create our amazing universe capable of evolving systems of supreme complexity called 'life'? Maybe. Maybe not. It all boils down to the question: How special was the big boot, the mother of all boot-ups commonly referred to as the big bang?



Physicists have a well-defined and precise measure that describes how special, or rather how common, the state of a system is. This measure is called entropy. A system with high entropy is ordinary, and a system with low entropy is special. In a recent blog I explained that the entropy of a system is the minimum number of digits required to encode the physical state of the system. It is the number of digits required to single out what is, amongst the many possibilities that could be. If a physical system can be described in relatively few digits, a limited number of alternative states needed to be singled out, and the state of the system can be considered special. If a lot of digits are needed, the state is one out of many, and therefore rather common.

You and I are special. This should not surprise you. All life is special. Living beings are prime examples of systems of low entropy. Unfortunately, the entropy of any isolated system must increase. So the low entropy of living bodies can be maintained only by being in contact with a larger universe that generously accepts all excess entropy. To accommodate this excess entropy the universe itself needs to be in a very low entropy state. A state with hot stars in a cold dark space. Stars that provide low entropy radiation to the planets and their life forms, with the planets sending high entropy thermal radiation back into the cold universe. Low entropy in, high entropy out. The cycle that keeps you and me, and all life in the universe going.

With a universe that - after more than 13 billion years - is still in a state with low enough entropy to support life, the entropy at earlier times must have been even lower. In fact, it must have been incredibly low.


Penrose's Needle in a Haystack
In his thought provoking book The Emperor's New Mind, and more recently also in his magnum opus The Road to Reality, Wolf Prize winner Sir Roger Penrose follows these lines of reasoning, and argues that the big boot represents a truly spectacular feat. To start a low entropy universe in which life can be sustained, the operator of the universe had to select one of the relatively few acceptable boot sequences out of the virtually infinitely many sequences that can be formed using a total of 10¹²² digits.

You read that well, Penrose is not referring to the 10¹²² possible sequences consisting of 122 digits, but to the truly insane number of possible sequences consisting of 10¹²² digits. Although Penrose refrains from drawing the conclusion that an operator with divine powers must have controlled the big boot, this conclusion seems almost inevitable.

Or is it?

Let me try to convince you that the opposing hypothesis is not as far-fetched as it seems: the big boot could very well have been a random act.

Ball-'n-Boxes
Consider a ball-'n-boxes toy model. Say we have a ball that can be in one of ten boxes. Obviously, a single digit is required to describe the state of this system. The system has an entropy of one digit. Similarly, with 100 boxes being available to the ball, the entropy would be 2 digits.

In ball-'n-boxes systems a low entropy manifests itself via an encoding that requires fewer digits than expected based on the number of boxes. If 100 boxes are present, but the laws that govern the dynamics of the ball are such that for some time it is restricted to ten specific boxes, the entropy is (at least for a while) lower than two digits. With a suitable numbering of the boxes (i.e. by numbering the ten special boxes '0' - '9'), one digit would suffice to describe the state of the system. Hence, the entropy of this special ball-'n-boxes system is no more than one digit.

Penrose views the universe as a ball-'n-10^{10^122}-boxes system. Unless this system is in special state, 10¹²² digits are needed to describe it.* However, our universe is special in the sense that whilst 10^{10^122} boxes are available, the vast majority of these boxes can be ruled out as having been inaccessible since the big boot. This translates into a much reduced entropy. In fact, Penrose estimates that spectacularly few digits are needed to describe our current universe. Out of the total number of 10¹²² digits at most one in every 10²¹ digits are required to describe the current universe. And to describe the earlier universe vastly fewer numbers of digits are needed.

A total number of states of 10^{10^122} is huge but finite. This means that Penrose's arguments are applicable only to bounded (closed) universes characterized by an upper limit on the allowed entropy Whether our universe is bounded or unbounded is an open problem related to the ultimate fate of the universe. However, the observed acceleration of the expansion of the observable universe strongly suggests that our universe might indeed be unbounded. Can we define a simple ball-'n-boxes model that describes an open universe in which the entropies can grow without bound?

Fibonacci universe
The answer is that this is surprisingly simple, provided we explicitly include in the model the dynamics of the ball. Those who have followed my recent blogs will not be surprised to see the reversible Fibonacci dynamics entering the spotlight again. We have already seen that the simple Fibonacci recipe (start from two initial numbers and repetitively add the last two numbers to create the next) creates surprising chaotic behaviors. Also, using Fibonacci's dynamics the arrow of time can be understood based on the low entropy start of the universe. Here, using Fibonacci's dynamics, we delve one level deeper and try to understand why the universe started in an extremely low entropy state. In the Fibonacci model we still have one ball, but an infinite number of boxes with each box uniquely labeled with an integer. The evolution of the Fibonacci universe is described by a sequence of numbers corresponding to the integer labels of the boxes subsequently visited by the ball. If at time t-1 the ball is in the box labeled L_t-1, and at time t it is in box L_t, then at time t+1 the ball will be in box L_t+1 = L_t + L_t-1.

Suppose after 20 time steps you observe the ball to be in the box labeled 459732, and after 21 time steps in box 743862.

Is this remarkable?

Sure it is! There is no reason to expect numbers of only a few digits. Infinitely more numbers with many more digits are available. Yet you observe box labels of only six digits. Evidently, your Fibonacci toy universe can be described in surprisingly few digits, and you conclude it is in a state of very, very low entropy.

Moreover, the number of digits must have increased at every time step** so at start of the universe, the entropy must have been even lower. How much lower is not relevant for the present discussion. The important point to note is that starting from the initial pair of boxes (the boxes visited by the ball in step zero and step one) up till the boxes visited at steps 20 and 21, just six digits per time step sufficed to describe the whole dynamics. We conclude that the first 20 time steps are described by a remarkably low entropy. But this holds for any starting pair of boxes. If we start at time t = 0 and t = 1 with the ball in boxes labeled with 100-digit integers, the same entropy increasing behavior is observed. Digits get added with time no matter what starting conditions are selected, and independent of the starting conditions the history of the universe appears to be characterized by a remarkably low entropy. It is practically impossible to avoid starting entropies low compared to later entropies.

With this behavior resulting from a simple Fibonacci's dynamics, surely it will result also from more complicated reversible dynamics laws. In fact, any unbounded reversible dynamics that is mildly chaotic can be expected to result in a universe that starts with a low entropy big bang. The starting condition simply doesn't matter.

Heck, do we even need a blindfolded monkey?


Notes

* More precisely, in 90% of the cases exactly 10¹²² digits are needed, in 9% of the cases 10¹²² - 1 digits are needed, in 0.9% of the cases 10¹²² - 2 digits are needed, etc. Now, for all practical purposes, 10¹²² - 10¹⁰⁰, 10¹²² - 10³⁰, and 10¹²² - 1 are all indistinguishable from 10¹²². Therefore, with a very high degree of accuracy, the entropy of this ball-'n-10¹²²-boxes system is 10¹²² digits.

**  The Fibonacci dynamics yields box labels that tend to increase each timestep by a factor equal to the golden ratio . It follows that on average every 1/¹⁰log() = 4.785 time steps a digit gets added.