Inclusive Thermodynamics of Computational Machines

Abstract

We introduce a framework designed to analyze the thermodynamics of an abstractly defined logical computer like a deterministic finite automaton (DFA) or a Turing machine, without specifying any extraneous parameters (like rate matrices, Hamiltonians, etc.) of a physical process that implements the computer. Earlier investigations of how to do this were based on the continuous-time Markov chain (CTMC) formulation of stochastic thermodynamics. These investigations either assumed that there was exactly zero irreversible entropy production (EP) generated by the physical system implementing the computation, or allowed the EP to be nonzero but only considered the mismatch cost component of the EP. In addition, they only applied to a single type of computer. Our framework neither requires that EP equal zero nor restricts attention to the mismatch cost component of EP, and is designed to apply to all types of computational machines. In contrast to earlier investigations using the CTMC-based formulation, our framework is based on the inclusive Hamiltonian formulation, in which the combination of the system of interest and the baths evolve in a Hamiltonian (or unitary) dynamics. Here, we use our framework to derive an integral fluctuation theorem for computers, in which the expectation value is strictly less than 1. We also derive an exchange fluctuation theorem, and a mismatch cost formula involving first-passage times. We analyze the EP generated by a DFA, a Markov information source, and a noisy communication channel. In particular, we use the Myhill-Nerode theorem of computer science to prove that out of all DFAs which recognize the same language, the minimal complexity DFA is the one with minimal EP for all dynamics and at all iterations.