Demon's variational principle for informational active matter

The interplay between information, dissipation, and control is reshaping our understanding of thermodynamics in feedback-regulated systems. We develop the informational Onsager-Machlup principle, a generalized variational framework that unifies energetic, dissipative, and informational contributions within a single formalism. This framework introduces a conditioned Onsager-Machlup integral to quantify path entropy under specified memory states and enables the derivation of cumulant generating functions for arbitrary observables in systems with measurement and feedback. Our formulation is consistent with stochastic thermodynamics and information thermodynamics. Applying this principle to a minimal model of an information-driven swimmer, we obtain analytical expressions for the mean velocity and higher-order cumulants in the single-measurement case. For repeated measurements and the steady state, we derive approximate analytical expressions by using a Gaussian closure for the distribution of measured velocities. Our analytical expression shows good agreement with numerical results, except for cases of extreme drag asymmetry.