  The dynamics of chaotic systems are, by definition, exponentially sensitive
to initial conditions and may appear rather random. In this work, we explore
relations between the chaotic dynamics of an observable and the dynamics of
information (entropy) contained in this observable, focussing on a disordered
metal coupled to a dissipative, e.g. phononic, bath. The chaotic dynamics is
characterised by Lyapunov exponents $\lambda$, the rates of growth of
out-of-time order correlators (OTOCs), quantities of the form $\langle[\hat
A(t),\hat B(0)]^{2}\rangle\propto\exp(2\lambda t)$, where $\hat A$ and $\hat B$
are the operators of, e.g., the total current of electrons in a metallic
quantum dot. We demonstrate that the Lyapunov exponent $\lambda$ matches the
rate of decay of information stored in the observable $\langle \hat
A(t)\rangle$ after applying a small perturbation with a small classical
uncertainty. This relation suggests a way to measure Lyapunov exponents in
experiment. We compute both the Lyapunov exponent and the rate of decay of
information microscopically in a disordered metal in the presence of a bosonic
bath, which may, in particular, represent interactions in the system. For a
sufficiently short range of the correlations in the bath, the exponent has the
form $\lambda=\lambda_{0}-1/\tau$, where $\lambda_{0}$ is the
(temperature-independent) Lyapunov exponent in the absence of the bath and
$1/\tau$ is the inelastic scattering rate. Our results demonstrate also the
existence of a transition between chaotic and non-chaotic behaviour at
$\lambda_{0}=1/\tau$, which may be triggered, e.g., by changing the temperature
of the bath.
