"If the one-asymmetry view is correct, the puzzle of the thermodynamic arrow is really the puzzle of the low entropy boundary condition. Why is entropy so low in the past ? After all, in making it unmysterious why entropy doesn't decrease in one direction, the Boltzmann measure equally makes it mysterious why it does decrease in the other - for the statistics themselves are time symmetric.

Boltzmann himself was one of the first to see the importance of this issue. In a letter to Nature in 1895, he suggests an explanation, based on an idea he attributes to `my old assistant, Dr Schuetz'. He notes that although low entropy states are very unlikely, they are very likely to occur eventually, given enough time. If the universe is very old, it will have had time to produce the kind of low entropy region we nd ourselves inhabiting simply by accident. `Assuming the universe great enough, the probability that such a small part of it as our world should be in its present state, is no longer small,' as Boltzmann puts it.

It is one thing to explain why the universe contains regions like ours, another to explain why we find ourselves in such a region. If they are so rare, isn't it more likely that we'd find ourselves somewhere else ? But Boltzmann suggests an answer to this, too. Suppose, as seems plausible, that creatures like us couldn't exist in the vast regions of near-equilibrium between such regions of low entropy. Then it's no surprise that we find ourselves in such an unlikely place."

"Faced with the deep puzzle of why the early universe had a low entropy, Boltzmann hit on the bright idea of taking advantage of the statistical nature of the Second Law. Instead of a box of gas, think of the whole universe. Imagine that it is in thermal equilibrium, the state in which the entropy is as large as possible. By construction the entropy can’t possibly increase, but it will tend to fluctuate, every so often diminishing just a bit and then returning to its maximum. We can even calculate how likely the fluctuations are; larger downward fluctuations of the entropy are much (exponentially) less likely than smaller ones. But eventually every kind of fluctuation will happen.

You can see where this is going: maybe our universe is in the midst of a fluctuation away from its typical state of equilibrium. The low entropy of the early universe, in other words, might just be a statistical accident, the kind of thing that happens every now and then."

SUMMARY

The second law of thermodynamics can be proved from the mechanical theory if one assumes that the present state of the universe, or at least that part which surrounds us, started to evolve from an improbable state and is still in a relatively improbable state. This is a reasonable assumption to make, since it enables us to explain the facts of experience, and one should not expect to be able to deduce it from anything more fundamental.
The applicability of probability theory to physical situations, which is disputed by Zermelo, cannot by rigorously proved, but the fact that one never observes those events that theoretically should be quite rare is certainly not a valid argument against the theory.

One may speculate that the universe as a whole is in thermal equilibrium and therefore dead, but there will be local deviations from equilibrium which may last for the relatively short time of a few eons. For the universe as a whole, there is no distinction between the "backwards" and "forwards" directions of time, but for the worlds on which living beings exist, and which are therefore in relatively improbable states, the direction of time will be determined by the direction of increasing entropy, proceeding from less to more probable states.