Do stars tend to fuse all hydrogen even when the mass is small? Suppose a brown dwarf requires only one more hydrogen atom to become massive enough to start fusion, what will happen if we add a hydrogen atom to it?


It fuses until the mass is dropped and then fusion stops suddenly
It starts to fuse until all hydrogen use up


Which one is correct?
The difference between a brown dwarf and a star is not a sharp boundary. A brown dwarf is simply a ball of gas where the (small) fusion rate is incapable of providing a significant fraction of the luminosity (which is mainly provided by gravitational contraction).

A star will contract and reach a minimum luminosity, whilst a brown dwarf's luminosity will monotonically decrease throughout its life. The core becomes degenerate and it is this that provides the pressure that supports a brown dwarf, even though it becomes colder.

If you add some mass to a brown dwarf, then the result depends when you add it. If you add it early during the strong contraction phase then you might get a low-mass star. If it is added after the brown dwarf is degenerate, then it could contract further, but do so without increasing the temperature enough to ignite hydrogen strongly. This is an interesting problem that deserves a model simulation!

If fusion can start to any degree, then convective mixing in these very low mass objects will very gradually (trillions of years), turn nearly all their H to He.