Let $A$ be an integral domain which is a finitely generated algebra over an algebraically closed field $k$. Let $\phi :A \to A^{[1]}$ be a $k$-algebra homomorphism and let us write $\phi_t : A \to A[t]$ to emphasize the indeterminate $t$. $\phi$ is called an exponential map if the following two conditions are satisfied:

(i) $\epsilon_0 \phi_t : A \to A$ is the identity map, where $\epsilon_0 : A[t] \to A$ is the evaluation map at $t=0$.

(ii) $\phi_{s} \phi_t=\phi_{s+t}$, where $\phi_s$ is extended to a homomorphism $A[t] \to A[t,s]$ by setting $\phi_s(t)=t$.

Can someone please refer me to some book or lecture note or paper explaining the bijective correspondence between these exponential maps and algebraic actions of $\mathbb G_a=(k,+)$ on $X$ , the affine variety corresponding to $A$ ?

So far I have only been able to locate the bijective correspondence between locally nilpotent derivations on finitely generated domain over algebraically closed field of characteristic zero and the $\mathbb G_a$ actions on affine variety corresponding to the affine $k$-domain, in Gene Freudenberg's book : Algebraic theory Locally nilpotent derivations. But I haven't been able to find explicit detail concerning the correspondence between $\mathbb G_a$-actions and exponential maps (that there is a correspondence is mentioned,for example, in Gupta, N. Invent. math. (2014) 195: 279-288, On the cancellation problem for the affine space $\mathbb A^3$ in characteristic $p$