Feedback thermodynamics has spent several decades explaining how Maxwell's demon converts knowledge into work. Standard fluctuation relations, such as the Jarzynski equality and its generalizations to measurement, tacitly assume that the measurement channel has full support: the demon can obtain any outcome with nonzero probability for any state of the system. This condition breaks down for ideal error-free measurements, threshold detectors, finite-resolution sensors, and sensors that lack access to part of the phase space. Nearby in the archive sit papers on Maxwell's demons that prefer indistinguishable photons, on quantum choirs and quantum heat guards, all of which in one way or another deal with limits on the information available to the demon. This article follows the same line but takes a step toward a unified formulation for arbitrary measurement channels, including singular ones.
The author considers a stochastic thermodynamic system driven by a control parameter, with measurements taken at discrete times. For each measurement outcome, the region of phase space compatible with that outcome is identified. An indicator function is then introduced to check whether the time-reversed trajectory stays inside the compatible region. The key quantity becomes the probability that the reverse process without feedback does not leave the allowed region. From this probability, a generalized inaccessible information is constructed. The derivation of the fluctuation relation relies on a detailed equality for a fixed protocol and on the normalization of the reverse distribution. For illustration, a one-dimensional Szilard engine is used, with a Brownian particle and a movable wall, where the measurement resolution is set by the width of the interval.
A generalized integral fluctuation relation is obtained that includes an inaccessible-information term and holds for arbitrary measurement channels and feedback protocols. In the full-support limit of the channel, it reduces to the standard relation with transfer entropy, and in the error-free measurement limit, to a previously known result involving inaccessible information. The relation implies a generalized second law: the average entropy production is bounded from below not by all the information gained, but only by the accessible part of it. For the Szilard engine with finite resolution, explicit expressions are found for the average information gained, the average extracted work, and the average inaccessible information. In the high-resolution limit, the information gained grows logarithmically, as ln(L/ΔX), but the inaccessible information has the same leading divergence, so their difference, equal to the extractable work, approaches a finite value: β⟨Wext⟩ → 1 + ln 2.
The work unites two previously separate formulations of feedback fluctuation theorems into a single framework that covers intermediate cases with partial access to phase space. Unlike neighboring papers that explore specific quantum or classical realizations of the demon, this one proposes a general mathematical scaffold applicable to any measurement channel. This makes it possible to quantitatively account for constraints imposed not only by measurement precision but also by the physical realizability of the feedback protocol. The concept of inaccessible information becomes a tool for analyzing the thermodynamic efficiency of real feedback-controlled systems, from biological molecular machines to nanoelectronic devices.
A natural next step is to apply the derived relations to continuous feedback control, where the information obtained through continuous measurement may diverge. The proposed formalism makes it possible to check whether this divergence is offset by an equally divergent inaccessible information. Another direction is to compare the derived bounds with non-information-based constraints on entropy production, for instance those based on coarse-grained entropy. It would also be interesting to study biological Maxwell's demons, whose feedback mechanisms are tightly constrained by physics, in order to understand how inaccessible information affects their thermodynamic efficiency.
The results can be used in the design of micro- and nanoscale devices with feedback, where sensors have finite resolution or limited access to states. They also provide a theoretical basis for analyzing energy costs in biological systems that use information for functional purposes.
Test the predictions on experimental platforms such as colloidal particles in optical traps with finite detector resolution. Generalize the formalism to quantum feedback measurements and to multicomponent systems.
In statistical physics, it remains an open question how exactly the information constraints imposed by the physical implementation of a measuring device and an actuator affect the limits of converting information into work. This work introduces a quantitative measure of those constraints, inaccessible information, and links it to fluctuation relations, which allows the imperfections of real Maxwell's demons to be treated rigorously.
In the limit of infinitely precise measurement, the demon gains an infinite amount of information about the particle's position, yet it can extract only a finite amount of work, equal to kT(1+ln2). This happens because almost all of the extra information turns out to be inaccessible to the allowed protocol with a single movable wall.
In Stanisław Lem's "The Travels of Professor Tarantoga," Maxwell's demon serves as a metaphor for the limits of knowledge: even with complete information, the protagonist cannot break the laws of thermodynamics. This article shows that in the real world the constraints are even tighter: knowledge you cannot apply is thermodynamically useless.