From the little I know about physics and conduction specifically, it seems to be that a system in which the heat transfer is done solely by conduction, cannot reach thermal equilibrium.
I'll preface this by stating what I think I know. Convection is the transfer of heat via atoms or molecules, meaning this is only present in fluids (and plasma I guess), as the atoms/molecules move more freely, as opposed to those in solids. Conduction is the transfer of heat via collisions of neighbouring atoms/molecules, and is thus more relevant in solids, as convection doesn't happen in solids and neighbouring atoms/molecules are further apart in fluids.
Now, it makes sense that convection would dissipate the thermal energy of a system, since the "heat-carriers" are actually moving around. With conduction though, this isn't the case. The heat carriers in a solid are to some degree stationary, and can only dissipate the energy to their neighbours.
From this minimal understanding, it seems to me that in an isolated system with conduction as the only possible method of heat transfer, thermal equilibrium cannot be reached. Instead, the atoms/molecules close to the heat source will be vibrating the strongest, and then their neighbours the second strongest, etc. I assume this is impossible, as I don't see how one could evade radiation. But, in this contrived scenario, with conduction as the only method of heat transfer, would the isolated system never reach thermal equilibrium?
Best Answer
Given enough time, the thermal energies of neighbouring atoms/molecules will eventually become equal (or as close to equal as you like) and the system will reach thermal equilibrium. This just takes a longer time if the only available method of heat transfer is conduction (which is why oven gloves work).