Non-stationary conditions

What happens with the electric charges described above when the thermoelectric circuit is removed from the external heat baths and temperature relaxes by non-stationary heat flow? Are the heat conduction equation (34) and the relation (13) still valid?

For an infinitely extended parallel metallic plate of width $2a$ the answer to these questions [4] is contained in Fig. 13. The initial surface charges turn into space charges accompanying the heat conduction process. The interior of the plate no longer is electrically neutral, but (weakly) charged!

Figure 13: Variation of temperature $(\delta T)$, electric field $(E_z)$ and electric charge density $(\rho )$ across an insulated parallel plate of thickness $2a$ with constant initial temperature gradient. Times are $t=0$ and a relatively late time $t> 0$, at which the variations are described by a single sine function.