We describe a minimal model of an autonomous Maxwell demon, a device that delivers work by rectifying thermal fluctuations while simultaneously writing information to a memory register. We solve exactly for the steady-state behavior of our model, and we construct its phase diagram. We find that our device can also act as a "Landauer eraser", using externally supplied work to remove information from the memory register. By exposing an explicit, transparent mechanism of operation, our model offers a simple paradigm for investigating the thermodynamics of information processing by small systems.Landauer's principle | nonequilibrium statistical mechanics A system in thermal equilibrium undergoes random microscopic fluctuations, and it is tempting to speculate that an ingeniously designed device could deliver useful work by rectifying these fluctuations. The suspicion that this would violate the second law of thermodynamics has inspired nearly 150 years of provocative thought experiments (1-5), leading to discussions of the thermodynamic implications of information processing (6-12). Although both Maxwell (1) and Szilard (3) famously took the rectifying agent to be an intelligent being, later analyses have explored the feasibility of a fully mechanical "demon". There has emerged a kind of consensus, based largely on the works of Landauer (6) and Bennett (7,8), and independently Penrose (13), according to which a mechanical demon can indeed deliver work by rectifying fluctuations, but in doing so it gathers information that must be written to physical memory. The eventual erasure of this information carries a thermodynamic cost, no less than k B T ln 2 per bit (Landauer's principle), which eliminates any gains obtained from the rectification of fluctuations.The past few years have seen increased interest in the thermodynamics of information processing (14-19). Discussions of Maxwell's demon, Landauer's principle and related topics arise in contexts such as quantum information theory (20), the synthesis of artificial nanoscale machines (21), feedback control in microscopic systems (22-33), and single-photon cooling of atoms (34). Experiments have been performed with the explicit aim of testing theoretical predictions (27), including Landauer's principle (35). Moreover the consensus or "favored explanation" (36) described above is widely but not universally accepted, as suspicions persist that it assigns an unwarranted thermodynamic significance to random data (12,(36)(37)(38)(39).In spite of this attention, the field still lacks a tangible example or model of a device that converts heat into work at the expense of writing information. Discussions are often framed around general principles rather than a particular instance, and the demon is typically described in generic terms, as a system capable of performing microscopic feedback control, but otherwise unspecified. In this paper we propose an explicit, solvable model of a system that behaves as a Maxwell demon. Our device, which extracts energy from a single thermal r...

We introduce a family of Maxwellian Demons for which correlations among information bearing degrees of freedom can be calculated exactly and in compact analytical form. This allows one to precisely determine Demon functional thermodynamic operating regimes, when previous methods either misclassify or simply fail due to approximations they invoke. This reveals that these Demons are more functional than previous candidates. They too behave either as engines, lifting a mass against gravity by extracting energy from a single heat reservoir, or as Landauer erasers, consuming external work to remove information from a sequence of binary symbols by decreasing their individual uncertainty. Going beyond these, our Demon exhibits a new functionality that erases bits not by simply decreasing individual-symbol uncertainty, but by increasing inter-bit correlations (that is, by adding temporal order) while increasing single-symbol uncertainty. In all cases, but especially in the new erasure regime, exactly accounting for informational correlations leads to tight bounds on Demon performance, expressed as a refined Second Law of thermodynamics that relies on the Kolmogorov-Sinai entropy for dynamical processes and not on changes purely in system configurational entropy, as previously employed. We rigorously derive the refined Second Law under minimal assumptions and so it applies quite broadly-for Demons with and without memory and input sequences that are correlated or not. We note that general Maxwellian Demons readily violate previously proposed, alternative such bounds, while the current bound still holds. As such, it broadly describes the minimal energetic cost of any computation by a thermodynamic system. 'intelligence' is necessary; a frictionless trapdoor connected to a spring acting as a valve, for example, cannot achieve the same feat [10].Maxwell's Demon posed a fundamental challenge. Either such a Demon could not exist, even in principle, or the Second Law itself needed modification. A glimmer of a resolution came with Szilard's reformulation of Maxwell's Demon in terms of measurement and feedback-control of a single-molecule engine. Critically, Szilard emphasized hitherto-neglected information-theoretic aspects of the Demon's operations [11]. Later, through the works of Landauer, Penrose, and Bennett, it was recognized that the Demon's operation necessarily accumulated information and, for a repeating thermodynamic cycle, erasing this information has an entropic cost that ultimately compensates for the total amount of negative entropy production leveraged by the Demon to extract work [12][13][14]. In other words, with intelligence and information-processing capabilities, the Demon merely shifts the entropy burden temporarily to an information reservoir, such as its memory. The cost is repaid whenever the information reservoir becomes full and needs to be reset. This resolution is concisely summarized in Landauer's principle [15]: the Demon's erasure of one bit of information at temperature T K requires at least k T ...

We describe a simple and solvable model of a device that -like the "neat-fingered being" in Maxwell's famous thought experiment -transfers energy from a cold system to a hot system by rectifying thermal fluctuations. In order to accomplish this task, our device requires a memory register to which it can write information: the increase in the Shannon entropy of the memory compensates the decrease in the thermodynamic entropy arising from the flow of heat against a thermal gradient. We construct the nonequilibrium phase diagram for this device, and find that it can alternatively act as an eraser of information. We discuss our model in the context of the second law of thermodynamics.

Active biological systems reside far from equilibrium, dissipating heat even in their steady state, thus requiring an extension of conventional equilibrium thermodynamics and statistical mechanics. In this Letter, we have extended the emerging framework of stochastic thermodynamics to active matter. In particular, for the active Ornstein-Uhlenbeck model, we have provided consistent definitions of thermodynamic quantities such as work, energy, heat, entropy, and entropy production at the level of single, stochastic trajectories and derived related fluctuation relations. We have developed a generalization of the Clausius inequality, which is valid even in the presence of the non-Hamiltonian dynamics underlying active matter systems. We have illustrated our results with explicit numerical studies.

Information processing typically occurs via the composition of modular units, such as the universal logic gates found in discrete computation circuits. The benefit of modular information processing, in contrast to globally integrated information processing, is that complex computations are more easily and flexibly implemented via a series of simpler, localized information processing operations that only control and change local degrees of freedom. We show that, despite these benefits, there are unavoidable thermodynamic costs to modularity-costs that arise directly from the operation of localized processing and that go beyond Landauer's bound on the work required to erase information. Localized operations are unable to leverage global correlations, which are a thermodynamic fuel. We quantify the minimum irretrievable dissipation of modular computations in terms of the difference between the change in global nonequilibrium free energy, which captures these global correlations, and the local (marginal) change in nonequilibrium free energy, which bounds modular work production. This modularity dissipation is proportional to the amount of additional work required to perform a computational task modularly, measuring a structural energy cost. It determines the thermodynamic efficiency of different modular implementations of the same computation, and so it has immediate consequences for the architecture of physically embedded transducers, known as information ratchets. Constructively, we show how to circumvent modularity dissipation by designing internal ratchet states that capture the information reservoir's global correlations and patterns. Thus, there are routes to thermodynamic efficiency that circumvent globally integrated protocols and instead reduce modularity dissipation to optimize the architecture of computations composed of a series of localized operations.