The hot rod will give some of its heat to the cold rod, this transferal of energy will probably happen by conduction. In solids I think the heat is a measure of the kinetic energy of the crystal lattice, the more heat the more the lattice shakes, the more kinetic energy it has. Metals are good conductors of heat, silicon is not so good; if you've got metal rods I think the heat will flow quite quickly from the hot rod into the cold rod across the boundary where they touch.
If there is a temperature gradient, heat will want to flow. It's like a hill, a ball will want to roll from top to bottom in much the same way that the heat will want to flow from high temp to low temp. So when the hot touches the cold there's a sharp gradient and a lot of heat is transferred.