  The entropy production rate for an open quantum system with a classically
chaotic limit has been previously argued to be independent of $\hbar$ and $D$,
the parameter denoting coupling to the environment, and to be equal to the sum
of generalized Lyapunov exponents, with these results applying in the
near-classical regime. We present results for a specific system going well
beyond earlier work, considering how these dynamics are altered for the Duffing
problem by changing $\hbar,D$ and show that the entropy dynamics have a
transition from classical to quantum behavior that scales, at least for a
finite time, as a function of $\hbar^2/D$.
