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The Animate and the Inanimate William James Sidis |
CHAPTER V
THE PROBABILITIES IN THE PROBLEM
To help us towards a solution of this paradox, we must first find out what the probabilities actually lead us to conclude. We have already seen that, in a given case, the chances are even as to whether energy will run down or build up. There are also small chances of a neutral condition, in which energy remains, on the whole, at the same difference of concentration as before. But the probability of this neutrality is negligible, and we may say that the probabilities are, that in 50% of the cases the second law of thermodynamics will be obeyed, and in 50% of the cases it will be reversed. If such is the case, the universe as a whole will be neutral; that is, taking all the occurrences over all of time and space, there will be no tendency in one direction or the other.
In this reasoning we can be assured as to the probabilities in any given occasion, for we must assume all combinations of initial positions and velocities to be equally likely. Inasmuch as any event occupies a certain amount of time, let us figure on the probabilities of the positions and initial velocities at the middle of that interval. For any range of positions and velocities resulting in a combination obeying the second law of thermodynamics, we have an equal and therefore an equally probable range of positions and velocities reversing that law; namely, the identical positions with the reverse velocities. Where the positions and velocities happen to border between the two kinds of combinations, we will have a sort of neutral result, which is so improbable as to have a zero probability (though that does not make it impossible). Aside from that, the second law of thermodynamics is, on any occasion, equally probable with its reverse, and the probability of each may be taken as 50%. The probability of the second law of thermodynamics being followed on two certain occasions is, as a result, only 25%; and so on, while its probability for all occasions is almost a nullity.
The probability is, however, as a result of this 50% probability, that approximately half the events of the universe, taking all of space and time, will be in accordance with the second law of thermodynamics, while about half will tend to reverse it. The former tendency we will, for short, call the positive, while the latter tendency we will call the negative tendency. Between these two there is a bordering, or neutral, tendency, which, as a whole, neither builds energy up nor levels it down.
The universe as a whole, including all of time and space, will tend towards this neutral tendency, but this neutral tendency will simply be a compound of positive and negative tendencies at different parts of space and time tending to cancel one another. Taking definite portions of space and time, the chances are that there will be some sort of preponderance of tendency in one direction or the other, the preponderance being greater the smaller the section of space and time that we take into consideration. We may, therefore, assume that, in the part of space and time under our observation (which, we know is very limited) the preponderance is towards the positive tendency. We may suppose that there are other parts of space, and other periods of time, when the preponderance will be in the reverse direction.
But even where the preponderance is toward the positive tendency; it still remains merely a preponderance, and instances of the negative tendency would be almost certain to occur it is true that the probabilities are that, in such a part of space and time, instances of the negative tendency will occur to a very limited amount; but, all the same, they will occur.
The probabilities of the situation, then, are as follows: the whole universe, including all of space and time, will tend to have as much of the positive as of the negative tendency, with a certain amount of the neutral tendency. At a particular moment of time the probabilities are that there will also be about as much of one tendency as of the other, but that in some sections of space there will be a preponderance towards the positive tendency, while in other sections of space the preponderance will be the reverse; about half of space falling under one heading, and about half of space falling under the other. In each of those portions of space there will be instances of events opposed to the prevailing tendency, presumably in certain material objects. The same applies to a more limited extent if we take one section of space with respect to the different moments of time.