Is Our Universe More Improbable Than a “Boltzmann Modern Earth” ?

Cosmologists sometimes discuss a possibility called a Boltzmann brain. A Boltzmann brain is the hypothetical possibility of a brain forming somewhere in space from an incredibly unlikely random combination of particles. Some have tried to explain the very unlikely existence of our universe by using reasoning along these lines: don't be surprised to be an observer in a universe like ours, because observers can only exist in universes like ours. But the possibility of a Boltzmann brain is sometimes presented to rebut such reasoning.

Let's consider two possibilities. In the first case, you live in a universe that is 99.9999% disorderly and chaotic, but there is just a tiny little area of space that is highly orderly, just orderly enough for your brain to exist. In the second case, you live in a universe that is orderly for vast regions stretching billions of light-years, with enough order to allow the possibility of trillions of life-bearing planets. Our reality is the second of these cases. But some cosmologists have argued that from a thermodynamic standpoint and an entropy standpoint, a "blind chance" standpoint, it is inconceivably more probable that you should find yourself as an observer under the first of these two cases.

Another possibility to consider (rather similar to a Boltzmann brain) is what we may call a “Boltzmann modern Earth.” This is the incredibly unlikely possibility that a planet the size of Earth, with all of the complexity and biology of our planet, could arise fairly suddenly from a random combination of particles.

This possibility of a "Boltzmann modern Earth" is discussed by ace cosmologist Roger Penrose in his recent scientific book Fashion Faith and Fantasy in the New Physics of the Universe. On page 316 of his book, he says, “One can make a very rough estimate of the probability that life, as it now exists on Earth, with all its detailed molecular and atomic locations and motions, came about simply by chance encounters from particles coming in from space in, let us say, six days!” Penrose then estimates that such a thing would have a probability of about 1 in 10 to the ten to the sixtieth power. That is a probability not anything like the microscopic probability of 1 in 10⁶⁰ but instead an almost infinitely smaller probability. It's the probability you would have if you started out with one tenth and then kept multiplying by one tenth a total number of times equal to a trillion trillion trillion trillion trillion times.

But then Penrose tells us that this fantastically unlikely event (a life-filled Earth like ours suddenly forming from random collisions of particles) would be far more probable than the existence of a universe as orderly as ours, saying it “would be a far 'cheaper' way of producing intelligent beings than the way in which it was actually done!” He's indicating  that the incredibly improbable sudden formation of a “Boltzmann modern Earth” would actually be much more likely than the chance of you getting a universe such as ours accidentally.

Speaking of the Second Law of Thermodynamics, Penrose states this on page 317:

The lower-entropy earlier states of the universe that initially gave rise to humanity in its earliest stages (being of lower entropy simply by virtue of the 2^nd law) must have been far more improbable (in this sense) than is the situation now. This is just the 2^nd law in action. So it must be “cheaper” (in terms of improbabilities) for the state to have come about as it is now purely by chance, than for it to have arisen from an earlier much lower entropy state – if that had come about purely by chance!

And on page 313 Penrose states that “the improbability of the universe conditions that we actually seem to find ourselves in” is roughly 1 in 10 to the 10 to the 124^th power, which is a probability almost infinitely smaller than the 1 in 10 to the 10 to the 60^th power estimate he made for the chance of a planet with all of Earth's biology appearing suddenly from random particle collisions (a “Boltzmann modern Earth” occurring). This 1 in 10 to the 10 to the 124^th power probability is the probability you would have if you started out with one tenth and then kept multiplying by one tenth a total number of times equal to ten thousand trillion trillion trillion trillion trillion trillion trillion trillion trillion trillion times.

Why such a low probability? The Second Law of Thermodynamics dictates that entropy must steadily increase. But right now the entropy of the universe is fairly low. Situations such as solar systems surrounded by vast amounts of empty space are very low entropy situations (as opposed to a universe that is a uniform sea of particles, which is high in entropy). It seems that if the universe has the low entropy it now has after 14 billion years of existence, the entropy of the universe must have been staggeringly low at the time of the Big Bang. And from a thermodynamic standpoint, such a thing seems insanely unlikely.

Penrose is one of the most well-known cosmologists around. If his statements on this topic are correct, then we have perhaps a tremendous irony. Centuries ago, people argued that our planet and its life could only have appeared if there were some higher power in the universe, on the grounds that it was too improbable that so much order could arise by chance. Now after all our advanced science, much of it done by people wishing to overturn such a conclusion, we may have discovered that the chance of this type of order existing randomly by chance (considering the history of the universe, the Big Bang, entropy and the Second Law of Thermodynamics) is not greater than was imagined long ago, but actually very much smaller.