  We study diffusion and butterfly velocity ($v_B$) in two holographic models,
linear axion and axion-dilaton model, with a momentum relaxation parameter
($\beta$) at finite density or chemical potential ($\mu$). Axion-dilaton model
is particularly interesting since it shows linear-$T$-resistivity, which may
have something to do with the universal bound of diffusion. At finite density,
there are two diffusion constants $D_\pm$ describing the coupled diffusion of
charge and energy. By computing $D_\pm$ exactly, we find that in the incoherent
regime ($\beta/T \gg 1,\ \beta/\mu \gg 1$) $D_+$ is identified with the charge
diffusion constant ($D_c$) and $D_-$ is identified with the energy diffusion
constant ($D_e$). In the coherent regime, at very small density, $D_\pm$ are
`maximally' mixed in the sense that $D_+(D_-)$ is identified with $D_e(D_c)$,
which is opposite to the case in the incoherent regime. In the incoherent
regime $D_e \sim C_- \hbar v_B^2 / k_B T$ where $C_- = 1/2$ or 1 so it is
universal independently of $\beta$ and $\mu$. However, $D_c \sim C_+ \hbar
v_B^2 / k_B T$ where $C_+ = 1$ or $ \beta^2/16\pi^2 T^2$ so, in general, $C_+$
may not saturate to the lower bound in the incoherent regime, which suggests
that the characteristic velocity for charge diffusion may not be the butterfly
velocity. We find that the finite density does not affect the diffusion
property at zero density in the incoherent regime.
