The thermodynamics of computing with circuits

As Landauer showed, any physical process that implements a given computation must generate a minimal amount of heat. Common engineered systems implement computations using circuits, as do many biological systems (e.g., gene regulatory networks). The topology of such circuits introduces additional constraints on the physical system implementing the computation. One might expect that this increases the minimal amount of heat needed to implement that computation, beyond the minimal amount needed if there are no constraints on how the computation can be implemented. We derive exact equations for the minimal amount of heat that is generated by any physical process that implements a given computation using a specified circuit. We also quantify how that minimal amount of heat compares to the minimal amount when the constraint of implementing the computation with the specified circuit is removed. These results provide a rich, new set of optimization problems that must be addressed by any designer of a circuit, if they wish to minimize thermodynamic costs.