  We establish that the entropy production rate of a classically chaotic
Hamiltonian system coupled to the environment settles, after a transient, to a
meta-stable value given by the sum of positive generalized Lyapunov exponents.
A meta-stable steady state is generated in this process. This behavior also
occurs in quantum systems close to the classical limit where it leads to the
restoration of quantum-classical correspondence in chaotic systems coupled to
the environment.
