What is the fastest heat can travel? The perfect solid heat sink What dictates the conductivity of heat in an element or medium? Consider the shape, volume, area, and distance heat travels from vibrating atom to atom, not through the excitement of the an atom through transference of energy e.i.e. light radiation or the compressing of air. How heat travel along a wire, is not the same as electricity does. So, what is the scale, a particular configuration, if built, can a non-moving object move heat from a to b best?
 A: The fastest heat can move is the speed of sound. Normally phonons, excitations in the crystal lattice of a solid, will move until they scatter off some inhomogeneity such as a dislocation, foreign element or other imperfection. This means that after some distance (the mean free path) the phonon bounces in random direction, making it diffuse like normal (the distance travelled is proportional to the square root of time). But in very perfect crystals (usually 1D structures like nanowires) the transport becomes ballistic: phonons start moving freely, with $d\propto t$ instead of $d\propto \sqrt{t}$.
So clearly perfect, low-dimensional crystals work great. Is there a theoretical way of analysing this? Yes, this is transport theory in statistical mechanics (which also deals with electrical conduction: they are not that different). In particular, the transport coefficients (in this case heat conductivity) can in principle be calculated from theory using the Green-Kubo formula and its quantum versions.
