If there is no God, then persons are entirely the result of materialistic processes.
If persons are the result of materialistic processes, then persons do not have "souls."
The person who believes that we are only the result of materialistic processes is a philosophical materialist.
Free will is a problem for the philosophical materialist. It becomes difficult for the philosophical materialistic (= everything is matter and explicable in terms of material causality) to explain "free will."
One of the best contemporary examples of a philosophical materialist attempting to show that free will is compatible with determinism and indeterminism (this is called "compatibleism") is Owen Flanagan in his book The Problem of the Soul: Two Visions of Mind and How to Reconcile Them.
Flanagan writes: "The mind is the brain - mental life is realized in our brains - and it is subject to natural laws" (p. 145) This means that all mental events have a cause which is either deterministic or indeterministic. There are no uncaused mental events. Thus, no mental events are - in a Cartesian sense - purely free.
Flanagan is saying the following:
1) The Cartesian view of mental events is false.
2) There is no God
3) Therefore, because there is no God, and because Cartesian free will is logically absurd, there must be another explanation for moral agency.
I think Flanagan's book is very well-written. It serves as an excellent example of what a person does with free will if they are a philosophical materialist.
Here are some thoughts I have about Flanagan's book.
#1 - I feel he has a false dichotomy when he contrasts Cartesian disembodied mental activity with his physicalist alternative. It's true that, e.g., some Christians were influenced by Descartes' mind-body dualism. But Judeo-Christianity is precisely not mind-body dualism. Thus for Flanagan to reject Descartes is not the same as rejecting Christians who argue for the reality of persons as psycho-physical agencies and as having a soul. (USC prof. of philosophy Dallas Willard , for example.)
#2 - Flanagan's arguments for the implausibility of God are weak. For example, Flanagan dismisses the idea that God exists, using a very weak and uninformed criticism of the cosmological argument. He does not take into account the more powerful versions of this argument as presented by William Lane Craig .
#3 - Flanagan cannot get away from Cartesian language himself in putting forth his argument against Descartes. Is Flanagan's belief that Descartes is wrong causally determined? If so, why believe it to be true? He addresses this objection on pp. 149-150, which is good. But I s explanation weak and reasoning circular. Is "Flanagan" able to modify future actions? it's hard to see how he can take any credit for such modification if the mind is only the brain. I still find myself wondering how a neo-compatibilist like Flanagan can "know" that something is "true."