  A typical strategy of realizing an adiabatic change of a many-particle system
is to vary parameters very slowly on a time scale $t_\text{r}$ much larger than
intrinsic equilibration time scales. In the ideal case of adiabatic state
preparation, $t_\text{r} \to \infty$, the entropy production vanishes. In
systems with conservation laws, the approach to the adiabatic limit is hampered
by hydrodynamic long-time tails, arising from the algebraically slow relaxation
of hydrodynamic fluctuations. We argue that the entropy production $\Delta S$
of a diffusive system at finite temperature in one or two dimensions is
governed by hydrodynamic modes resulting in $\Delta S \sim 1/\sqrt{t_\text{r}}$
in $d=1$ and $\Delta S \sim \ln(t_\text{r})/t_\text{r}$ in $d=2$. In higher
dimensions, entropy production is instead dominated by other high-energy modes
with $\Delta S \sim 1/t_\text{r}$. In order to verify the analytic prediction,
we simulate the non-equilibrium dynamics of a classical two-component gas with
point-like particles in one spatial dimension and examine the total entropy
production as a function of $t_\text{r}$.
