The Unlikely Universe - Part 1

The work of a curious fellow

This page is on a topic that I have wondered about. I would appreciate any feedback that you might be able to provide. Especially errors in concept or calculation. Please send an email to if you would care to comment.

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Earth's suitability for life - a long string of lucky breaks?...

I finished up the "Predictability Principle" page with the idea of exploring the chain of events that resulted in planet Earth being so congenial to life, in particular to human life. As far as I know there is no one who can make a thorough exploration of that chain beginning with the big bang and tracing events up to today. I do wonder about a few links in that chain where it seems to me the the whole thing might easily have come apart.

I think that the development of humans on earth depended on a fairly short list of requirements in addition to the reasonable predictability we have already covered:

1. the existence of appropriate laws of nature
2. the existence on earth of the atoms found in the body
3. the existence of abundant water in the liquid state
4. the initiation of first life
5. the existence of the moon for the tides
6. the existence of oxygen for respiration
7. the existence of low entropy food
My guess is that the probability of meeting each of these requirements ranges from highly unlikely to downright miraculous. One approach to this topic would be to take on these requirements one at a time and try to make some estimates of their probabilities. Then we could multiply the probabilities together and make some judgment about whether or not we arrived on this world by chance. In fact I may try follow this approach eventually but it would be be too bulky a project for inclusion in the "Life on Earth" series. It turns out that there is a shortcut. All of the requirements in my list depend on a single event that is itself the most improbable of them all - so improbable in fact that the odds against its happening by chance is the largest number I have ever seen applied to a physical quantity. If the probability of every one of my listed requirements were dead certainties it would make no difference in the overall probability (very close to zero) that the universe as we find it was a chance occurrence.

My challenge now is to make this extreme improbability seem reasonable to folks who have not spent a lot of time studying the underlying principles. To remove any suspense around what I am talking about, it is the very special nature of the big bang itself. To get at this special nature we will need to back up a bit and establish some concepts we haven't dealt with in any detail. This may require an additional page or two but I will take on the first few of these concepts right here. I am indebted to Professor Roger Penrose for this shortcut. His book "The Road to Reality" provided the line of reasoning I am going to try to summarize here for your consideration.

There is a law of nature that seems to be universal, applying to all sorts of situations. It was initially formulated with respect to thermodynamics, the study of heat flow. In that context this "second law of thermodynamics" states that heat flows from hot objects to cold objects with a consequent decrease in the hot object's temperature and increase in the cold object's temperature. The temperature changes reduce the distinction between the hot and cold objects, increasing the entropy of the system. It is the tendency for entropy to increase over time that is the universal aspect of the second law. This tendency is so ubiquitous that it has been taken to define the direction of time's flow from past to future.

The second law is intimately connected with the state of the system under consideration. In its broadest sense the state of a system at any instant is the information required to reproduce that system as it exists in that instant. The set of numbers defining the state of a system, for systems made of objects the size of a molecule or bigger, is the position and momentum of all the moving parts at the same instant. Applying the laws of Nature to this initial state, allows us to calculate a future state. The accuracy of this prediction of the future depends on the precision with which the initial state was known and the correctness of our calculations.

Let's consider a box full of 1023 nitrogen molecules. The set of numbers identifying the position and momentum of each and every molecule in the box at a particular time is called a microstate of the system. If I move a single molecule to another location or alter its momentum the new set of numbers is another microstate. With so many molecules and so much random thermal motion among them we may consider every possible microstate equally probable, including the microstates where all the molecules reside in one half of the container.

Why then do we not observe all the molecules in one half of the container for however brief a period of time? Here we need to introduce the notion of a macrostate. Macrostates are collections of microstates that are indistinguishable from one another. If I take two molecules and swap their positions and momenta, since all the molecules are identical the two microstates must be in the same macrostate.

The situation where all the gas is in one half of the container is a macrostate with a huge number of microstates in it. The situation where the gas is uniformly distributed throughout the container is a macrostate with enormously more microstates in it. With all microstates being equally likely, the probability of the occurrence of a particular macrostate is equal to its number of microstates divided by the total number of possible microstates. That makes the probability of the half empty container macrostate so close to zero that the difference may be neglected. The justification for the previous statement may be found on the "Arithmetic of Chance" page

Next we need to pin down the notion of entropy a bit. We have said that as the macrostate in which a system finds itself become more disordered, the system entropy increases. In fact the numerical value of the entropy of a system is proportional to the logarithm of the number of microstates in the macrostate of the system. Now recall the notion of phase space that appeared on the "Economy and Extravagance" page. The state of the system at any instance was represented by a single point in phase space. In the context of a single moving part, dynamical system, "state", "macrostate" and "microstate" were identical. In the present context of systems of many independent particles, macrostates do not represent a single point in phase space but occupy a volume containing all their indistinguishable microstates. The entropy then of each macrostate is a number proportional to the logarithm of the volume of phase space occupied by that macrostate.

If we could somehow lay our hands on the total entropy available to the universe and the entropy of the universe at about the time of the big bang we could use those entropies to calculate the relative volumes of the phase space each entropy represents and arrive at the probability that just by chance the universe started out the way it did. That is the path I am going to try to take on the page that follows to quantify the likelihood of this universe unfolding in the way it is just by chance.

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