The thermodynamics of computing with finite automata

Recent breakthroughs in stochastic thermodynamics have greatly enriched our understanding of the thermodynamics of computation. To date, these analyses have concentrated on the “computation” of erasing a single bit. One of the idiosyncratic characteristics of such computation is that we know ahead of time when it will finish, and so do not need to continually observe it to tell whether it has finished - thereby avoiding the thermodynamic costs of such observation. Here we show how to analyze the thermodynamic costs of running a computer without continually observing it even though its finishing time is random, e.g., because it depends on the random input to the computer. We then use this to analyze the thermodynamics of finite automata (FA), one of the most important types of computational system. In particular, we show that due to the variability in the number of iterations it takes a given FA to finish, the Landauer cost of running that FA is a sum of novel information theoretic quantities, which we call “partial entropies”.