The functional thermodynamics of finite-state Maxwellian ratchets

Autonomous Maxwellian demons exploit structured environments as a resource to generate work: They randomize ordered inputs and leverage the increased Shannon entropy to transfer energy from a thermal reservoir to a work reservoir, respecting both Liouvillian state-space dynamics and the Second Law. To date, correctly determining functional thermodynamic operating regimes was restricted to information engines for which the correlations among information-bearing degrees of freedom can be calculated exactly--a highly restricted set of engines. Although information-engine controllers are represented as finite hidden Markov chains and rate matrices, (i) no finite expression for their Shannon entropy rate exists, (ii) the set of their predictive features is generically uncountably infinite, and (iii) their effective memory--the statistical complexity--diverges. Here, we adapt recent results from dynamical-systems and ergodic theories that efficiently and accurately calculate the entropy rates and the rate of statistical complexity divergence of general hidden Markov processes. Using the Information Processing Second Law, these new methods accurately determine the thermodynamic operating regimes for finite-state Maxwellian demons with arbitrary numbers of states and transitions.