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 $t_text{r}$.
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