The Sun generates heat via fusion. The heat from this reaction gets distributed around the solar system and beyond. This process of spewing heat and radiation all over the place doesn't immediately seem to be reversible. "Everyday" processes that generate and spew heat and radiation (such as burning a log in a fireplace) generally increase entropy, but these processes are molecular in nature and not nuclear, so it is not obvious to me that the analogy is apt. Does the Sun increase entropy in its neighborhood in the way that a burning log increases entropy in its neighborhood?
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Yes, the processes in the Sun result in increasing overall entropy of the universe. Entropy in neighborhood is a poorly defined quantity - the closest to this that I could think of is the local entropy - that is entropy average over a small volume, such as appears when we reduce Boltzmann equation to hydrodynamic approximation. However in case of the Sun this would probably mean averaging over a volume much larger than the Solar system.
In general, entropy production necessarily accompanies all the physical processes (happening on macroscopic scale). See, e.g.,
Entropy production in non-equilibrium systems: physical interpretation?
Maximum Principle vs. Minimum Principle in Non-equilibrium Thermodynamics
Roger V.
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