Couldn't resist leaving lurkerdom for this...
The Earth and sun taken together is a closed system,....
Untrue. Does all of the sun's energy reach the Earth?
...so the second law applies.
Let's assume that this is so, or at least that the 2nd Law can be applied here (which it can, with caveats). In this case, the 2nd Law says:
entropy change of sun + entropy change of earth >= 0
Given that the first term is large and positive, why cannot the second term be small and negative?
Also, any local increases in entropy are subject to the laws of probability. It is possible, for example, for the gas molecules in a container to move to one side of the container, turning the other side into a vacuum. But this is highly improbable.
Is it still "highly improbable" when a large temperature difference is maintained across the container?
On the website you reference in message 321, you say you have a BS and a PhD in Physics. For the record, so do I. But given the way you're making a pig's dinner out of first year thermo, I can only confirm that you have a lot of BS in your physics.