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Thermodynamics with continuous information flow
Jordan M. Horowitz, Massimiliano Esposito
TL;DR
The paper addresses the lack of a simple thermodynamic framework for continuous information flow in interacting stochastic systems. It develops a unified stochastic-thermodynamic formalism for autonomous and nonautonomous dynamics, showing how mutual-information rates modify subsystem entropy balances while recovering nonautonomous Maxwell-demon results and analyzing autonomous steady-state flow.
Problem
Continuous information flow in autonomous systems lacks a simple thermodynamic treatment connected to the mutual-information framework used for nonautonomous protocols.
Method
The paper uses stochastic thermodynamics to formulate information transfer between two interacting systems for autonomous and nonautonomous dynamics, including cycle decomposition for autonomous steady states.
Results
The formalism shows that subsystem entropy balances are modified by information flow related to mutual-information rates and recovers accepted nonautonomous Maxwell-demon results.
Takeaways & Limitations
Information flow provides a thermodynamic bound on each subsystem and supports analysis of sensors, information engines, feedback controllers, and autonomous devices.
Abstract
from arXiv · showhide
We provide a unified thermodynamic formalism describing information transfers in autonomous as well as nonautonomous systems described by stochastic thermodynamics. We demonstrate how information is continuously generated in an auxiliary system and then transferred to a relevant system that can utilize it to fuel otherwise impossible processes. Indeed, while the joint system satisfies the second law, the entropy balance for the relevant system is modified by an information term related to the mutual information rate between the two systems. We show that many important results previously derived for nonautonomous Maxwell demons can be recovered from our formalism and use a cycle decomposition to analyze the continuous information flow in autonomous systems operating at steady-state. A model system is used to illustrate our findings.
I. INTRODUCTION
Information flow is important for feedback control, biological adaptation, and Maxwell-demon processes, but continuous coupling lacks a unified quantitative thermodynamic treatment. This paper develops a stochastic-thermodynamic framework covering both autonomous and nonautonomous information processing.
- Information transfer supports feedback control, biological adaptation, and Maxwell-demon processes across engineering, biology, and physics.
- Continuous coupling makes it difficult to separate measurement from feedback and quantify its thermodynamic influence.
- Nonautonomous Maxwell-demon analyses use mutual information to quantify measurement costs and bound feedback work, whereas autonomous continuous flow lacks a simple thermodynamic formulation.
- The paper establishes a stochastic-thermodynamic approach to information flow between interacting systems under autonomous and nonautonomous dynamics.
- The formalism addresses thermodynamic costs in sensors, information engines, and feedback controllers, and distinguishes energetic from purely informational interaction mechanisms.
II. SETUP
The setup combines two Markov systems into a bipartite joint system whose transitions alter one subsystem at a time while allowing mutual influence of transition rates. Stochastic thermodynamics assigns currents, entropy changes, and reservoir-mediated energetics to this structure.
- Systems X and Y have discrete states and independent Markov dynamics, then form a joint Markovian system with states (x, y).
- Transition rates satisfy local detailed balance, connecting graph transitions to heat exchanged with thermal or chemical reservoirs.
- The bipartite coupling lets X and Y influence each other’s transition rates without simultaneous jumps or diagonal transition mechanisms.
- Bipartite currents divide naturally into X-direction and Y-direction flows, enabling separate thermodynamic analysis of the two subsystems.
- The joint open system obeys the second law, with entropy production determined from Shannon-entropy change and entropy change in the surrounding environment.
- For a single thermal reservoir, environmental entropy flow is related to heat current by ˙Sr = −˙Q/T, while the first law connects heat, work, and internal energy.
III. BIPARTITE THERMODYNAMICS AND INFORMATION FLOW
The paper separates bipartite thermodynamics into subsystem contributions and an information flow based on changes in mutual information. This flow modifies each subsystem’s entropy balance while preserving nonnegative entropy production.
- Bipartite decomposition: The entropy flow of the joint system does not specify how energy and information move between its two subsystems.The authors therefore decompose current functionals into contributions associated with X- and Y-direction transitions.
- Information flow: Information flow is defined as the time variation of mutual information and divided into contributions from X and Y transitions.Mutual information quantifies correlations, while its derivative is written as dtI = ˙IX + ˙IY.
- Information flow: A positive ˙IX indicates that X transitions increase mutual information, whereas a negative ˙IX indicates decreasing correlations through erasure or information consumption.The latter interpretation depends on whether information is destroyed or used to extract energy.
- Subsystem second laws: Positive subsystem entropy production rates result when the joint second law is separated into X- and Y-associated pieces.The separated terms are identifiable as the entropy production rates in each subsystem.
- Subsystem second laws: Equation (11) quantifies how exchanged information modifies the entropy balances of X and Y for both autonomous and nonautonomous dynamics.From X’s perspective, hidden influence from Y can allow σX < 0, with the modification determined by ˙IX.
IV. NONAUTONOMOUS MAXWELL DEMON
The formalism recovers the thermodynamics of a nonautonomous Maxwell demon by treating measurement as information generation and feedback as information use. The information resource can reduce the engine’s entropy, while the combined memory-engine system obeys the standard second law.
- Nonautonomous Maxwell demon: A nonautonomous Maxwell-demon cycle alternates controlled measurement and feedback stages, with a memory X recording correlations with engine Y.During measurement, X is manipulated while Y is fixed; during feedback, Y evolves while X is frozen.
- Measurement: During measurement, X generates information about Y, so ˙IY = 0 and dtI = ˙IX.The systems begin uncorrelated with Iinit = 0, and measurement establishes information I.
- Feedback: During feedback, information I allows Y to reduce the entropy of itself and its environment, potentially converting heat into useful work in an isothermal process.This information functions as a resource for Y, which is consequently called an information engine.
- Composite-system resolution: When the memory and engine are treated as one composite system, the information contribution cancels and the cycle reduces to a standard thermodynamic engine.The entropy required to establish correlations during measurement supplies the energy enabling the engine’s operation.
V. AUTONOMOUS INFORMATION FLOW
At autonomous steady state, continuous information flow is decomposed into cycle currents and acts as an additional driving force alongside thermodynamic forces. Global cycles transfer energy and entropy between subsystems, while local cycles support internal flows.
- Autonomous information flow: Steady-state systems maintain constant currents that continuously exchange energy and information without external driving.The steady-state distribution is time-independent, so the mutual information itself is constant while its flow remains nonzero.
- Information processing: Positive information generation requires dissipation in the sensing subsystem, while the receiving subsystem can use the information to reduce entropy and extract energy.The information flow bounds both the energetic requirement of sensing and the benefit available to the information-consuming subsystem.
- Autonomous information flow: Information flow has the same current-times-force structure as energy dissipation, with an information force derived from conditional-probability ratios.The information force is f^y_y′(x,x′) = ln[p(y|x)/p(y|x′)].
- Cycle decomposition: Cycle decomposition separates bipartite dynamics into global cycles coupling both subsystems and local cycles confined to one subsystem.Global-cycle currents transfer energy and entropy between subsystems, whereas local cycles support internal subsystem flows.
- Cycle decomposition: Information flow occurs only on global cycles because it represents transfer between the subsystems.The cycle-current representation identifies the mesoscopic fluxes responsible for energy transfer through the network.
- Interaction regimes: When global energetic affinities vanish, subsystems can transfer energy indirectly through information flow; comparing energetic and informational affinities identifies the interaction regime.Information-dominated behavior occurs when F_X(C) ≪ F_I(C), whereas energy-dominated behavior occurs when F_X(C) ≫ F_I(C).
VI. EXAMPLE: COUPLED QUANTUM DOTS
A coupled double-quantum-dot model illustrates autonomous information engines and feedback refrigerators. Which mode occurs depends on the relative time scales of the two dots and on the interaction energy.
- Model: The example is a double quantum dot whose two single-level dots exchange electrons with leads at different temperatures and chemical potentials.The lower dot Y couples to two leads at temperature T, while the upper dot X couples to a colder lead at T_D < T.
- Cycle structure: The model has one global cycle and two local cycles, with the local Y cycles contributing to the electronic current through the lower dot.The lower-dot current is decomposed into contributions from the two local cycles.
- Information engine: When X is faster than Y, the device operates as an information engine and can pump electronic current against the bias for U ≈ 0.05–0.45.In this regime, X rapidly adapts to Y and feeds back on it; the global description identifies heat flow as the pump fuel.
- Information engine: In the ideal Maxwell-demon limit U → 0 and T_D → 0 with U/T_D finite, energetic effects disappear and the interaction becomes information dominated.The fast-controller limit also corresponds to reversible measurement and feedback in the cited model analysis.
- Feedback refrigerator: When Y is faster than X, the device acts as a feedback refrigerator that cools the upper dot’s reservoir at T_D < T.For small U, cooling is fueled by information generated in Y and consumed by X.
- Feedback refrigerator: At higher interaction energy, the heat-flow direction reverses and the refrigeration regime is lost.At U ≈ 0.14, the entropy production in X approaches equilibrium while total dissipation remains large.
VII. DISCUSSION
The discussion presents information flow as a subsystem-level refinement of the second law and situates the framework among bipartite Maxwell-demon models. It also identifies limits and extensions of the approach.
- Interpretation: Information flow bounds the thermodynamics and energetics of each subsystem, while the joint bipartite system still obeys the second law.Treating the auxiliary subsystem as unobserved gives a weaker coarse-graining than removing it entirely.
- Relation to prior work: Removing the auxiliary system completely can discard correlations needed to connect continuous-feedback descriptions with nonautonomous Maxwell-demon results.The paper argues that retaining the auxiliary subsystem preserves these relevant correlations.
- Limitation: The framework does not define how one subsystem performs work on the other because internal-energy partitioning is not unique.The paper leaves analysis of subsystem work as a direction for future research.
- Scope: Not all Maxwell-demon models are bipartite, and some can convert a low-entropy memory state into work without mutual information as the transfer medium.The authors nevertheless note that many physically and biologically relevant systems are bipartite.
- Extensions: The approach can be extended to continuous-space diffusive systems and quantum systems, including quantum feedback control and sideband cooling.A bipartite Fokker–Planck formulation is identified as a natural continuous-space generalization.
Appendix A: Cycle decomposition
The appendix constructs the cycle decomposition by selecting a fundamental oriented cycle set and decomposing transition currents into global and local cycle contributions.
- Cycle identification: A graph-theoretic procedure first identifies a fundamental set of oriented cycles for the network.Each cycle is assigned an orientation relative to every transition link.
- Current decomposition: The indicator δ^y,y′_x,x′(C) records whether a transition link lies in cycle C, with sign determined by orientation.It is +1 for matching orientation, −1 for opposite orientation, and 0 when the link is absent.
- Current decomposition: Substituting the link-level current decomposition into the entropy balance separates contributions from global cycles and local cycles C_X and C_Y.This separation is used to obtain the cycle form of the subsystem entropy-production relations.