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Random Chance

With so many miracles to choose from, where do I get off declaring Statistical Mechanics to be "the most astonishing product of human Science?" This is of course a personal opinion, but it is one shared by many physicists - perhaps even a majority. The astonishment is a result of the incredible precision with which one can predict the outcome of experiments on very complicated systems (the more complicated, the more precise!) based on the FUNDAMENTAL ASSUMPTION of STATISTICAL MECHANICS:

A system in thermal equilibrium is a priori equally likely to be found in any one of the fully-specified states accessible to it.

This seemingly trivial statement contains a couple of ringers: the word "accessible" means, for instance, that the total "internal" energy of the system - which is always written  U  - i.e. the sum of the kinetic and potential energies of all the little particles and waves that make up the big system - is fixed. There are many ways to divide up that energy, giving more to one particle and less to another, and the FUNDAMENTAL ASSUMPTION says that they are all equally likely; but in every case the energy must add up to the same U. This can obviously be very confusing, but fortunately we rarely attempt to count up the possibilities on our fingers! It is the assumption itself that is so amazing. How can anything but total ignorance result from the assumption that we know nothing at all about the minute biases a real system might have for one state over another? More emphatically, how can such an outrageous assumption lead to anything but wrong predictions? It amounts to a pronouncement that Nature runs a perfectly honest casino, in which every possible combination of the roll of the dice is actually equally likely! And yet every prediction derived from this assumption has been demonstrated to be accurate to the best precision our measurements can provide. And the consequences are numerous indeed!