Demon's variational principle for informational active matter

TL;DR

This work develops the informational Onsager-Machlup principle (IOMP), a demon's variational framework that unifies energetic, dissipative, and informational contributions in stochastic systems under measurement and feedback. By introducing the conditioned Onsager-Machlup integral (OMI) and its mutual variant, the authors derive a variational route to the cumulant generating function for observables and apply it to the information swimmer, obtaining analytical expressions for the mean velocity and higher cumulants under single and multiple measurements. The key contributions include a rigorous CGF construction for information-driven dynamics, demonstration that information-based feedback can sustain persistent motion in dissipative environments, and a foundation for designing informational engines operating far from equilibrium. The framework connects to generalized variational principles (GOMP) and offers a versatile toolkit for analyzing informational active matter in colloidal, biological, and synthetic systems where memory and information exchange are central.

Abstract

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. Applying this principle to a minimal model of an information-driven swimmer, where feedback adaptively modulates viscous drag based on velocity measurements, we obtain analytical expressions for the mean velocity and higher-order cumulants. Here, we show that information-based feedback can sustain persistent motion even in dissipative environments, establishing a theoretical foundation for informational active matter and providing a systematic route for designing feedback-powered engines operating far from equilibrium.

Figures (6)

• Figure 1: Examples of informational active matter. Information-fueled particle models, such as the information swimmer Huang20Hou24 (top left), exhibit directional motion by rectifying random Brownian motion through measurement and feedback. Beyond single-particle behavior, ensembles of information swimmers (top right) have been investigated and shown to display characteristic pattern formation VanSaders23. Particles that exploit both information and intrinsic activity Ito15, such as bacteria performing chemotaxis Berg72Wadhams04Celani10 (bottom left), also represent important examples of informational active matter. Externally controlled active particles can also function as informational active engines Paneru22Saha23Malgaretti22Cocconi23Cocconi24Rafeek24Schuttler25 (bottom right).
• Figure 2: Schematic flowchart of the informational Onsager-Machlup principle (IOMP) [see Eqs.(\ref{['Eq:IOMVP0']})-(\ref{['Eq:MOM']})]. The stochastic system, subjected to thermal fluctuations, is characterized by the free energy $F$. If the system is out of equilibrium, the active power $\dot{W}$ can also be incorporated into the IOMP framework. The system is in contact with a heat bath at temperature $T$, and its energy dissipation is quantified by the dissipation function $\Phi$. Importantly, the system interacts with a memory (demon), and it is quantified by the amount of information, represented here by the mutual Onsager-Machlup integral (OMI) $M$ [see Eq.(\ref{['mutualOMItext']})]. These four quantities, free energy $F$, active power $\dot{W}$, dissipation $\Phi$, and information $M$, are systematically incorporated into the conditioned OMI $O$. By maximizing the modified OMI with respect to an observable $A$, one obtains the cumulant generating function (CGF) $K_A$.
• Figure 3: (a) Schematic illustration of the information swimmer. A particle of mass $m$ undergoes Brownian motion in a one-dimensional space at temperature $T$. At each measurement interval $\tau$, the particle velocity $v(t)$ is recorded, and its sign is stored in a binary memory $y = \pm 1$. The drag coefficient switches between $\zeta_+$ and $\zeta_-$ depending on the memory state. (b) Sample trajectory of the particle velocity $v$ and memory state $y$ obtained from Langevin simulations [see Eqs. (\ref{['Eq:Langevin']}) and (\ref{['Eq:FDT']})], where the parameters are $\gamma_\pm = \zeta_\pm/m = \gamma(1 \mp \delta)$ with $\delta = 0.3$ and $\gamma\tau = 1$. Red and blue circles represent the measured velocities $V_0, V_1, V_2, \dots$, together with their signs, while the red and blue shaded regions indicate the corresponding memory states.
• Figure 4: (a) First cumulant [see Eq. (\ref{['Eq:Result-V1']})], (b) second cumulant [see Eq. (\ref{['Eq:Result-V1second']})], (c) third cumulant [see Eq. (\ref{['Eq:Result-VVV1']})], and (d) fourth cumulant [see Eq. (\ref{['Eq:Result-VVVV1']})] of $V_1$ for the single measurement case as functions of the dimensionless measurement time $\gamma \tau$. Here, $\gamma$ is defined by $\gamma_\pm = \gamma(1 \mp \delta)$, and the drag asymmetry is varied as $\delta=0.3$ (black), $0.6$ (red), and $0.9$ (blue). The thermal velocity $V_T= \sqrt{k_{\rm B}T/m}$ is used to scale the velocity.
• Figure 5: First cumulant $\langle V_N \rangle_\mathrm{c}$ for the multiple measurements case [see Eq. (\ref{['Eq:Result-VN']})] as functions of the dimensionless measurement time $\gamma \tau$. The drag asymmetry is chosen as (a) $\delta = 0.3$, (b) $\delta = 0.6$, and (c) $\delta = 0.9$. The black and red lines are the results for $N=2$ and $N=4$ measurements, respectively, while the blue lines are the steady-state velocity $\langle V_\infty \rangle_{\rm c}$ [see Eq. (\ref{['Eq:Result-Vss']})]. The dotted lines are single-measurement velocity $\langle V_1 \rangle_\mathrm{c}$ [see Eq. (\ref{['Eq:Result-V1']}) and Fig. \ref{['Fig:V-result']}(a)]. (d) The solid lines are the steady-state velocity $\langle V_\infty \rangle_{\rm c}$ [see Eq. (\ref{['Eq:Result-Vss']})] as functions of $\gamma \tau$ when $\delta$ is varied. The symbols are the result of numerical simulation of the Langevin equation in Eqs. (\ref{['Eq:Langevin']}) and (\ref{['Eq:FDT']}) with $N = 2\times 10^3$ measurements. The details of the simulation conditions are provided in the text.