  We investigate the breakdown of magneto-hydrodynamics at low temperature
($T$) with black holes whose extremal geometry is AdS$_2\times$R$^2$. The
breakdown is identified by the equilibration scales ($\omega_{\text{eq}},
k_{\text{eq}}$) defined as the collision point between the diffusive
hydrodynamic mode and the longest-lived non-hydrodynamic mode. We show
($\omega_{\text{eq}}, k_{\text{eq}}$) at low $T$ is determined by the diffusion
constant $D$ and the scaling dimension $\Delta(0)$ of an infra-red operator:
$\omega_{\text{eq}} = 2\pi T \Delta(0), \, k_{\text{eq}}^2 =
\omega_{\text{eq}}/D$, where $\Delta(0)=1$ in the presence of magnetic fields.
For the purpose of comparison, we have analytically shown $\Delta(0)=2$ for the
axion model independent of the translational symmetry breaking pattern
(explicit or spontaneous), which is complementary to previous numerical
results. Our results support the conjectured universal upper bound of the
energy diffusion $D \,\le\, \omega_{\text{eq}}/k_{\text{eq}}^2 \,:=\,
v_{\text{eq}}^2 \, \tau_{\text{eq}}$ where $v_{\text{eq}}:=
\omega_{\text{eq}}/k_{\text{eq}}$ and
$\tau_{\text{eq}}:=\omega_{\text{eq}}^{-1}$ are the velocity and the timescale
associated to equilibration, implying that the breakdown of hydrodynamics sets
the upper bound of the diffusion constant $D$ at low $T$.
