Abstract
Maxwellian ratchets are autonomous, finite-state thermodynamic engines that implement input-output informational transformations. Previous studies of these "demons" focused on how they exploit environmental resources to generate work: They randomize ordered inputs, leveraging increased Shannon entropy to transfer energy from a thermal reservoir to a work reservoir while respecting both Liouvillian state-space dynamics and the Second Law. However, to date, correctly determining such functional thermodynamic operating regimes was restricted to a very few engines for which correlations among their information-bearing degrees of freedom could be calculated exactly and in closed form---a highly restricted set. Additionally, a key second dimension of ratchet behavior was largely ignored---ratchets do not merely change the randomness of environmental inputs, their operation constructs and deconstructs patterns. To address both dimensions, 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. In concert with the Information Processing Second Law, these methods accurately determine thermodynamic operating regimes for finite-state Maxwellian demons with arbitrary numbers of states and transitions. In addition, they facilitate analyzing structure versus randomness trade-offs that a given engine makes. The result is a greatly enhanced perspective on the information processing capabilities of information engines. As an application, we give a thorough-going analysis of the Mandal-Jarzynski ratchet, demonstrating that it has an uncountably-infinite effective state space.
Highlights
In 1867, Maxwell introduced a thought experiment designed to challenge the second law of thermodynamics [1,2], what Lord Kelvin later came to call “Maxwell’s demon.” Exploiting the fact that the second law holds only on average, i.e., the thermodynamic entropy S cannot decrease over repeated transformations, the experiment conjured an imaginary, intelligent being capable of detecting and harvesting negative entropy fluctuations to do work
Via the information processing second law [7], this allows accurate determination of the functional thermodynamics of arbitrary finite-state ratchets
We explore the structural dimension of ratchet functionality, demonstrating that the engine/eraser/dud classification does not uniquely describe ratchet information processing for a given input
Summary
In 1867, Maxwell introduced a thought experiment designed to challenge the second law of thermodynamics [1,2], what Lord Kelvin later came to call “Maxwell’s demon.” Exploiting the fact that the second law holds only on average, i.e., the thermodynamic entropy S cannot decrease over repeated transformations, the experiment conjured an imaginary, intelligent being capable of detecting and harvesting negative entropy fluctuations to do work. Fundamental progress was halted since determining thermodynamic functionality in the most general case, temporally correlated input driving a memoryful ratchet, was intractable Attempts to circumvent these problems either heavily restricted thermodynamic-controller architecture [7], invoked approximations that misclassified thermodynamic functioning, or flatly violated the second law [6]. To do this requires investigating the change in structure from the input process to the output process These structural changes were previously proved to be deeply relevant to engine thermodynamic efficiency and an engine’s ability to meet the work production bounds set by Landauer’s principle [12]. Using new methods from ergodic theory and dynamical systems that determine randomness generation and memory use, we reanalyze the original ratchet, showing that previous analyses misidentified its thermodynamic functioning This is illustrated for its operation in several distinctly correlated environments. In conjunction with the previous functional classification which can be exactly carried out, we introduce structurerandomness tradeoffs in engine operation, highlighting the multidimensional nature of ratchet information processing
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