System-independent lower bounds on entropy production incurred by running a computer program

Abstract: Mismatch cost (MMC) is a universally applicable lower bound on both the entropy production (EP) of any fixed physical process across a given time interval. In this work we use MMC for the first time to lower-bound the amount of work dissipated by running a high-level or assembler-level computer program. To begin, we extend previous results concerning MMC to prove that it scales at least linearly with the total heat flow in the worst case over initial distributions. This establishes that - in contrast to results like the thermodynamic speed limit theorem or thermodynamic uncertainty relations - it is often a substantial fraction of the work dissipated on macroscopic scales. We also prove that the MMC lower bound over a given time interval never decreases if the time interval is subdivided into a sequence of sub-intervals, and that the bound often increases. Armed with these results, we then introduce a general framework for computing the minimal EP (i.e., the MMC) associated with running any computer program on any physical system that implements a modern digital computer. Crucially, this framework holds completely independently of the microscopic physical details of that system. Next we apply this general framework to compare lower bounds on the EP incurred by running two canonical sorting algorithms, bubble sort and Bucket sort. This enables us to investigate how thermodynamic cost depends on features like input size and structure (e.g., with or without repeated entries). Finally, we extend the framework to programs that call subroutines.