The demon stands in a room with a blindfold on and hears footsteps, but he can't see the whole room. He tries to sort molecules by speed in order to extract work from thermal motion. Classical fluctuation theorems for feedback-controlled systems assumed that the demon has access to the entire phase space, that is, he sees every molecule and every one of its velocities. In reality, sensors have finite resolution or blind spots: the demon hears only some of the footsteps, and some molecules slip past him unnoticed.
The author constructs a generalized fluctuation relation that accounts for this blindness. He introduces the concept of inaccessible information: the portion of measured information that cannot be converted into work. If the demon hears a footstep but doesn't know which side of the room it came from, then sorting molecules by that footstep is useless. The formula shows that the average entropy production is bounded from below not by all the information obtained, but only by its accessible part.
To illustrate this, a one-dimensional Szilard engine is used: a thought experiment with a Brownian particle and a moving wall. The measurement resolution is set by the width of an interval: the demon knows that the particle is somewhere within this interval, but doesn't see its exact position. At high resolution, the information gained grows logarithmically, as ln(L/ΔX), but the inaccessible information has the same leading divergence. Their difference, equal to the extractable work, tends to a finite value β⟨Wext⟩ → 1 + ln 2.
The work unites two previously separate formulations of feedback fluctuation theorems into a single framework. It applies to any measurement channel, including singular cases where some outcomes have zero probability. This makes it possible to quantitatively account for limitations imposed not only by measurement precision, but also by the physical realizability of the feedback protocol.
The blindfolded demon remains a demon, but his blindness is now measurable. Inaccessible information becomes a tool for analyzing the thermodynamic efficiency of real feedback-controlled systems, from biological molecular machines to nanoelectronic devices. Knowledge that cannot be applied is thermodynamically useless, and this uselessness has an exact mathematical measure.
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.