  We study properties of thermal transport and quantum many-body chaos in a
lattice model with $N\to\infty$ oscillators per site, coupled by strong
anharmonic terms. We first consider a model with only optical phonons. We find
that the thermal diffusivity $D_{\rm th}$ and chaos diffusivity $D_L$ (defined
as $D_L = v_B^2/ \lambda_L$, where $v_B$ and $\lambda_L$ are the butterfly
velocity and the scrambling rate, respectively) satisfy $D_{\rm th} \approx
\gamma D_L$ with $\gamma\gtrsim 1$. At intermediate temperatures, the model
exhibits a "quantum phonon fluid" regime, where both diffusivities satisfy
$D^{-1} \propto T$, and the thermal relaxation time and inverse scrambling rate
are of the order the of Planckian timescale $\hbar/k_B T$. We then introduce
acoustic phonons to the model and study their effect on transport and chaos.
The long-wavelength acoustic modes remain long-lived even when the system is
strongly coupled, due to Goldstone's theorem. As a result, for $d=1,2$, we find
that $D_{\rm th}/D_L\to \infty$, while for $d=3$, $D_{\rm th}$ and $D_{L}$
remain comparable.
