The classical thermodynamic doesn't resolve the arrow of time problem, but confuses it even further: according to the 2nd law of thermodynamic the future is more deterministic than the past; however, we seem to remember past rather than the future.
However, there have been some interesting development in the last 5 years or so, relating some I would say philosophical issues of quantum mechanics with the arrow of time of statistical physics. An easy review is available here, and a more detailed one in the original article it references. There are many very interesting philosophical question arising from that work, a possible resolution of the arrow of time problem being one of them.
EDIT: an explanation for the 1st paragraph above. Here's what I meant:
The 2nd law of thermodynamics precludes one from knowing some things from the past. If you mixed hot water and cold water you cannot later tell from measuring the temperature of the mix how cold was cold water and how hot was the hot one. However, the future remains deterministic: you can predict the temperature of the mix by measuring the ingredients before mixing them. In a matter of speaking, the law of thermodynamics allow more knowledge of the future from the present state than knowledge of the past, also based solely on the present state.
The human experience is kind of opposite: in our present state we remember the past, but we cannot directly "remember" the future. We can predict certain things, with varying degrees of certainty, but almost never with the same clarity as we remember the past.
Thus we have the Arrow of Time enigma: the basic laws of mechanics are deterministic; the laws of thermodynamics (which is based in small scale on the laws of mechanics) are suddenly non-deterministic in "hiding past, not future" sort of way; and the human experience, as well as the desire for the existence of free will, calls for non-determinism in "hiding future, not the past" sort of way.
The "Quantum Arrow of Time" article I quoted attempts to resolve this problem.