Negative Joule heating? In general Joule-heating a.k.a. Ohmic heating due to an electromagnetic field interacting with a conducting medium leads to dissipation of energy, which is converted into mechanical energy (or is it kinetic energy of the charge-carriers?)
According to Wikipedia the corresponding power density is
$${P_d}=\mathbf{J}\cdot\mathbf{E}$$
where $\mathbf{J}$ is the current density and $\mathbf{E}$ the electric field.
Since $\mathbf{J}=\frac{\partial\mathbf{P}}{\partial t}$, where $\mathbf{P}$ is the polarization density, it looks like the instantaneous ${P_d}$ can become negative. Does this correspond to "Joule cooling" or am I getting it wrong here?
 A: To answer your question in a very broad sense, you can certainly obtain negative Joule heating ("Joule cooling"), but you'll usually be very hard-pressed to find it actually happen.
To give you an elementary example, take Ampere's law in a vacuum in terms of the current density:
$$\vec{J} = \frac{\nabla\times\vec{B}}{\mu_0} - \epsilon_0\frac{\partial\vec{E}}{\partial t}$$
If you neglect the magnetic term, you'll find that the Joule heating term is:
$$P = -\epsilon_0\left(\vec{E}\cdot\frac{\partial\vec{E}}{\partial t}\right)$$
All you'd need to do in this case is set up a scenario in which your electric field is pointing parallel to the direction in which it's changing with respect to time in order to obtain "Joule cooling". The problem is that almost all materials will try to change the electric field in them such that the $\frac{\partial\vec{E}}{\partial t}$ and $\vec{E}$ vectors are antiparallel. In the case of Ohmic materials where $\vec{J} = \sigma\vec{E}$, it's completely impossible to get Joule cooling because:
$$P = \vec{J}\cdot\vec{E} = \sigma\left(\vec{E}\cdot\vec{E}\right)$$
which is of course always positive unless you have some weird metamaterial with negative conductivity. Because polarization fields tends to oppose external fields, you'll find it's almost impossible to find an actual case of Joule cooling even though mathematically it's perfectly allowed/tractable. This is because electric fields will try to cause charges in matter to "flow" in the direction of the electric field to minimize the free energy of the system; doing the opposite would inevitably require some external work input. Just take a look at the electric part of the Lorentz force density $\vec{f}$ on a charge density $\rho$:
$$\vec{f} = \rho\vec{E}$$
This force density is pointing in the direction of the electric field, which means that charges will try to move in the electric field's direction in the absence of other non-electric forces even if the electric field was damped by material polarization.
