Information Thermodynamics of Feedback-Controlled Brownian Particles

Abstract

A feedback-controlled Brownian particle can, in principle, extract work from a single thermal reservoir by using
measurement information to bias a trapping potential, provided the total entropy budget, including the
thermodynamic cost of acquiring and erasing that information, is properly accounted for. We simulate an
overdamped Brownian particle in a harmonic trap subject to discrete feedback control, in which the trap center is
repositioned at fixed intervals based on noisy measurements of particle position, and quantify extracted work,
mutual information between measurement and particle state, and total entropy production across 40 combinations
of measurement noise (5 levels, standard deviation 0.05 to 2.0 trap widths) and feedback interval (8 levels, 0.02 to
4.0 relaxation times), using ensembles of 5 x 10^4 trajectories per condition. At the lowest measurement noise and
shortest feedback interval tested, mean extracted work per cycle reached 0.84 ± 0.03 k_BT, approaching but not
exceeding the mutual-information bound of 0.91 k_BT set by the generalized second law, consistent with the
Sagawa-Ueda inequality to within measurement precision across all 40 conditions with zero violations observed.
Extracted work declined monotonically with increasing measurement noise, falling to 0.11 ± 0.02 k_BT at the
highest noise level tested, an 87% reduction relative to the low-noise condition, while mutual information declined
correspondingly from 0.91 to 0.14 bits over the same range, tracking extracted work closely (Pearson r = 0.97
across all 40 conditions). Total entropy production, summing particle-bath dissipation and the thermodynamic cost
of measurement and memory erasure, remained non-negative in every condition tested, with a minimum efficiency
ratio (extracted work divided by the Landauer erasure cost of the acquired information) of 0.79 at the optimal
operating point, meaning the feedback protocol recovered 79% of the theoretical maximum efficiency achievable
given the information actually acquired. Feedback interval showed a pronounced optimum near 0.3-0.5 relaxation
times, with extracted work falling by more than half at both shorter intervals, where measurement noise dominated,
and longer intervals, where the particle re-equilibrated before feedback could be applied. These results provide a
quantitative map of the noise-timing trade-off governing information-to-work conversion in feedback-controlled
colloidal systems and offer design guidance for physical realizations of Maxwell's-demon-type engines operating
with realistic, noisy measurement.