1. There exists one, and only one, real, physical universe;
2. While it is possible to simulate any subset of this universe, including conscious beings, with a computer program, this program must be implemented on a physical computer, or on a virtual machine (or series of such) which is itself implemented on a physical computer;
3. The universe has a finite age and is comprised of a finite amount of matter/space/energy, but it is expanding and cooling and will continue to do so forever;
4. Some single world interpretation of quantum mechanics is correct.
My understanding is that the above assumptions, which I have deliberately chosen as being contrary to many of the ideas discussed on the Everything List, still allow for the possibility of fantastically unlikely events, such as the spontaneous formation of an exact and stable copy of our solar system from the random motion of particles in interstellar space, or from vacuum fluctuations posited by the Uncertainty Principle.
Let p(t) = probability that an event P will occur somewhere in the universe during the next year, t years from the present. The probability that P will NOT occur at some time between the present (t=0) and (t=a+1) is then given by the product:
[1-p(0)]*[1-p(1)]*[1-p(2)]...*[1-p(a)]
As a-> infinity this becomes an infinite product, representing the probability that P will NEVER occur. It is easy to see that this infinite product diverges to zero in the special case where p(t) is constant for all t; in other words, that P, however unlikely, will definitely occur at some point in the future if the probability that it occurs during a unit time period remains constant over time. The same conclusion applies if p(t) increases with increasing t: the infinite product diverges to zero, more quickly than in the case of constant p(t).
Things get more difficult, however, if p(t) decreases over time. A Google search for "infinite product" brought up some very complicated expressions for even rather simple p(t), and it is by no means obvious (to me, anyway) whether the product will converge or diverge.
Now, my question is, what happens to p(t) over time? I would have guessed that as the universe expands, chemical and nuclear reactions are less likely to occur, in the same way as chemical reaction rates are proportional to the concentration the reagents. On the other hand, it is not clear to me how more exotic processes such as spontaneous appearance of particles out of the vacuum are affected by the expansion, which after all results in "more vacuum" - doesn't it?
I'm sure the above is a gross oversimplification - I'm not a physicist - but I would welcome people's thoughts on it.
Stathis Papaioannou
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