Understanding the minimal resources required to perform a given computation has also been a long-standing focus of research in physics. Modern work on this issue can be traced back to the work of Landauer in which he concluded that thermodynamic resources of at least $kT \ln[2]$ were needed to erase a bit on any physical system. However no work has been done before on the thermodynamic resources needed to perform more complicated computations than bit erasure. In this talk I will introduce the results of some preliminary research on this issue, focusing specifically on how the thermodynamic resources needed to implement a desired input-output function with a digital (straight-line) circuit depend on the topology of the circuit. Specifically, I will show how an analysis of the thermodynamics of digital circuits:

- Uncovers novel connections between nonequilibrium statistical physics and information theory;

- Reveals new, challenging engineering problems for how to design a circuit to have minimal thermodynamic costs;

- Allows us to extend computer science theory (specifically circuit complexity theory) to include thermodynamic costs.