Jan:
I cannot locate a Arkansas Battery called Brown's battery. Do you have any other information of this battery?
In answer to the second part of your post, confederate field artillery had official names such as the Austin Arkansas Artillery, or a number such as the 9th Arkansas battery. More commonly, they were known by the name of their commander, such as Trigg's Arkansas Battery. By the way, all of the names used above were the same Arkansas battery.
If a battery changed commanders, then it was called by the name of the new commander. An example is the Clarke County Artillery orginally commanded by Capt. Franklin Roberts, fought at the battle of Shiloh became known as Wiggin's battery after Wiggins replaced Roberts.
If anything, this can only make it harder to trace a confederate battery.
Ron