“…We consider three geometries as typical examples, figure 1, which are a rectangle, a nanowire [18,19,[23][24][25][26][49][50][51] of constant width w, and a bow tie (or wedge) constriction [20,21,[51][52][53] of varying width down to a minimum w. We compare the temperature profiles for devices with the same macroscopic dimensions and characteristic parameters, for either a fixed applied potential difference or a fixed current. The nanowire and bow tie are excellent representative examples because they exhibit markedly different behaviour: the electrical resistances of the nanowire and bow tie have different dependences on the width w, and this means that Joule heating is independent of w for the nanowire (for small w and (4). Light grey indicates two leads at temperature T 0 with boundary condition (5), dark grey indicates insulating regions with boundary condition (6).…”