Font Size: 

We develop a general formalism to describe linear electrical networks in which different resistors might be at different temperatures, and that might also be subjected to external parametric driving, for example by changing in time the capacitance or inductance of the circuit elements. In particular, we study the behaviour of the heat dissipated in each resistor and discuss its proper definition in the white noise limit, clarifying some incorrect results present in the literature. Our formalism is built on top of the usual graph-theoretic description of electrical circuits employed by engineers and allows to study and design thermodynamical processes from a first principles perspective, as we illustrate with a simple example of an electrical heat pump. We also provide exact results for the low temperatures regime in which the quantum nature of the electrical noise must be taken into account. For this, we use a semiclassical approach that, due to the linearity of the system, can be shown to coincide with a fully quantum treatment of circuits for which canonical quantization is possible.