Excess & Housekeeping Demons in Nonequilibrium Systems

• Excess and housekeeping demons are conceptual constructs that distinguish transient reorganization costs from persistent dissipation in nonequilibrium systems.
• They are mathematically defined via geometric splittings and orthogonal projections, enhancing our understanding of fluctuation theorems and thermodynamic trade-offs.
• This framework applies to diverse settings—stochastic, chemical, and quantum systems—offering practical insights into energy flows and information budgets.

Excess and housekeeping demons are conceptual and mathematical constructs used in modern nonequilibrium thermodynamics and information theory to rigorously distinguish two fundamentally different modes of entropy production and information flow in driven, stochastic, and quantum systems. The excess demon quantifies the dissipative costs associated with transient reorganization or relaxation toward a new steady state or marginal distribution, while the housekeeping demon accounts for the persistent, irreducible dissipation required to sustain nonequilibrium currents or maintain steady correlations. These decompositions generalize the paradigm of Maxwell's demon, but with technical precision and wide applicability across classical, quantum, and informational contexts. Their formal definition emerges from geometric splittings of currents and forces—often as orthogonal projections in the relevant Hilbert, Riemannian, or operator-valued spaces—and is supported by universal fluctuation theorems and uncertainty relations.

1. Formal Definitions across Stochastic and Quantum Systems

The decomposition into excess and housekeeping components arises in Langevin systems, Markov jump networks, chemical reaction models, quantum master equations, and information-flow frameworks.

• Stochastic Langevin dynamics: For a system with state $x \in \mathbb{R}^d$ evolving under $\dot x = F(x) + \sqrt{2D(x)}\,\xi(t)$, with symmetric positive-definite diffusion matrix $D(x)$ and noise $\xi(t)$, one defines the current velocity $v(x,t) = F(x) - D(x) \nabla\ln P(x,t)$ and decomposes it into

$$v(x) = v_{hk}(x) + v_{ex}(x) = v_{hk}(x) + D(x)\nabla\phi(x),$$

where $v_{ex}$ is gradient (excess) and $v_{hk}$ is solenoidal (housekeeping), orthogonal under the metric $M(x) = D(x)^{-1}$. The total entropy production rate splits:

$\Sigma_{tot} = \Sigma_{ex} + \Sigma_{hk}$

where $\Sigma_{ex}$ is associated with relaxation (internal reorganization) and $\Sigma_{hk}$ with maintaining steady currents and detailed-balance violation (Sireci et al., 8 Jul 2025, Dechant et al., 2022, Yeo et al., 2015, Lahiri et al., 2013).

• Discrete stochastic networks: In a master equation $\dot p_i = \sum_j (p_j w_{ji} - p_i w_{ij})$ with edges $e = (i \to j)$, the thermodynamic force $F_e = \ln(p_i w_{ij} / p_j w_{ji})$ admits a geometric split via the Onsager metric ($L = \mathrm{diag}(\ell_e)$, $\ell_e = J_e / F_e$) into conservative (excess) and cyclic (housekeeping) components:

$$F = F^* + (F - F^*),\quad \Sigma^{ex} = \|F^*\|_L^2,\quad \Sigma^{hk} = \|F - F^*\|_L^2,$$

with $J^{rel} = L F^*$, $J^{cyc} = L (F - F^*)$ (Yoshimura et al., 2022, Maekawa et al., 26 Sep 2025).

• Quantum open systems: For Markovian Lindblad dynamics, the entropy production rate is operator-valued:

$$\sigma(\rho) = \mathrm{Tr}[\mathbf{J}(\rho)^\dagger \mathbf{F}(\rho)] = \|\mathbf{F}(\rho)\|^2_{\Gamma \otimes \rho},$$

with $\mathbf{F} = \mathbf{F}_c + \mathbf{F}_{nc}$ (conservative/excess and nonconservative/housekeeping force operators), yielding

$$\sigma_{ex} = \|\mathbf{F}_c(\rho)\|^2,\quad \sigma_{hk} = \|\mathbf{F}_{nc}(\rho)\|^2$$

(Yoshimura et al., 2024).

• Information-flow decomposition: In bipartite Markov or Langevin systems, the information flow from subsystem $Y$ to $X$ ($I_X$) is split as

$I_X = I_X^{ex} + I_X^{hk}$

where the excess part ($I_X^{ex}$) is associated with conservative forces driving change in mutual information, while the housekeeping part ($I_X^{hk}$) consists of cyclic modes that sustain correlations but do not alter total mutual information (Maekawa et al., 26 Sep 2025, Ito et al., 28 Dec 2025).

2. Geometric Structure and Orthogonal Projections

The decomposition is universally geometric, derived as orthogonal projections onto gradient (excess) and solenoidal/cyclic (housekeeping) subspaces endowed with problem-specific metrics:

• Langevin/Hilbert space: Riemannian geometry with metric $M(x) = D(x)^{-1}$ supports a Hodge decomposition. Excess dissipation corresponds to gradient flows (curl-free), housekeeping to divergence-free (in stationary measure) flows sustaining NESS symmetry breaking (Sireci et al., 8 Jul 2025, Dechant et al., 2022).
• Discrete network: Projections in $L$-metric space (Onsager coefficients) split force vectors; the cycle basis expansion of housekeeping modes generalizes Schnakenberg decomposition beyond the steady state, valid for multistable, oscillatory, and chaotic regimes (Yoshimura et al., 2022).
• Quantum case: The inner product in force-operator space ($\langle A, B \rangle_{\Gamma \otimes \rho}$) ensures the excess/housekeeping forces are orthogonal in the superoperator structure (Yoshimura et al., 2024).

3. Physical Interpretation: Demonic Taxonomies

The excess and housekeeping "demons" generalize the Maxwell/Szilard framework, but with explicit energetic and informational budgets:

• Excess demon: Responsible for driving the system through probability-space reorganizations—transient relaxation, learning, forgetting, expansion, erasure—quantified by excess dissipation or excess information-flow. Its activity vanishes in NESS or steady regimes. In feedback or engine cycles (Szilard/Maxwell), excess demons are idealized constructs that neglect the energetic cost of control (Kish et al., 2011, Lahiri et al., 2013, Yeo et al., 2015).
• Housekeeping demon: Perpetually maintains steady non-equilibrium currents—cyclic transport, broken detailed balance, persistent correlations—via irreducible entropy and energy production. This demon's cost is unavoidable and reflects all active control, decision, and synchronization operations. In all physical completions of Szilard or Maxwell engines, the total energy for control and signal generation far exceeds the extractable work (Kish et al., 2011).

Demons are not literal entities but mappings between components of entropy/information budgets and dynamical processes.

4. Universal Fluctuation Theorems and Thermodynamic Trade-Offs

Both excess and housekeeping parts obey exact integral fluctuation theorems, granting statistical symmetry to their dynamics:

• Hatano–Sasa theorem for excess: $\langle e^{-\Delta S_{ex}} \rangle = 1$, $\langle Q_{ex} \rangle \ge 0$ (Sireci et al., 8 Jul 2025, Lahiri et al., 2013, Yeo et al., 2015).
• Housekeeping heat/entropy theorems: Subject to parity or velocity-symmetry constraints, $\langle e^{-\Delta S_{hk}} \rangle = 1$, $\langle Q_{hk} \rangle \ge 0$ (Yeo et al., 2015, Lahiri et al., 2013).
• Thermodynamic uncertainty relations: Excess and housekeeping dissipation furnish tighter bounds on the fluctuations and precision of currents, with TUR and dissipation-speed limits often sharper than the classical total entropy TUR (Dechant et al., 2022, Yoshimura et al., 2022, Yoshimura et al., 2024, Ito et al., 28 Dec 2025).
• Quantum diffusion bounds: In open quantum systems, quantum diffusivity constrains the achievable rate of state change, with excess dissipation encoding the minimal cost of transformation (Yoshimura et al., 2024).

5. Applications, Examples, and Symmetry Breaking

Excess and housekeeping demons elucidate symmetry breaking, energy flow, and correlation maintenance in varied physical settings:

• Driven ring particle: A constant torque produces nonzero housekeeping current and broken parity; enhanced dissipation is purely housekeeping (Sireci et al., 8 Jul 2025).
• Brownian gyrator: Anisotropic temperature coupling yields both solenoidal housekeeping currents (parity breaking) and excess flows measuring relative entropy to uncoupled equilibrium (Sireci et al., 8 Jul 2025).
• Bistable potential under shear: Excess velocity acquires divergence with noise, signaling stabilization of excited states (Sireci et al., 8 Jul 2025).
• Molecular oscillator synchronization: Excess dissipation quantifies free-energy transduction into synchronization; housekeeping term corresponds to ATP-driven persistent cycles (Sireci et al., 8 Jul 2025).
• Markov-jump networks: Fundamental cycles in the probability graph maintain housekeeping information flow sustaining correlations, while excess modes change mutual information by relaxation (Maekawa et al., 26 Sep 2025).
• Underdamped/NES systems: Decomposition into excess and housekeeping heat clarifies which part is available for work extraction and which constitutes irreducible loss, guiding feedback and control strategies (Lahiri et al., 2013).

6. Information-thermodynamic and Optimal-Transport Perspectives

Recent work embeds excess and housekeeping demons within the framework of optimal transport and information theory:

• Information-flow decomposition: The excess demon is realized as the optimal transport speed (2-Wasserstein distance per unit time) between marginal distributions, setting sharp speed limits for learning or reorganization in subsystems (Ito et al., 28 Dec 2025).
• Subsystem trade-offs: Speed-limit and TUR inequalities for marginal distributions generalize second-law constraints and quantify the energetic costs and information budgeting for subsystem dynamics in coupled systems (Ito et al., 28 Dec 2025, Maekawa et al., 26 Sep 2025).
• Conditional Fisher information bounds: Both excess and housekeeping flows are universally governed by conditional Fisher information metrics, imposing sharp lower bounds on subsystem dissipation rates (Ito et al., 28 Dec 2025).

7. Impact, Limitations, and Classification in Engine/Demon Traditions

A rigorous split into excess and housekeeping modes has profound implications for thermodynamic engines, feedback devices, and Maxwell demon analogs:

• Szilard/Maxwell engine cycles: Once physical control cost (housekeeping) and information acquisition/generation are accounted, the Second Law remains intact, with “excess demon” analyses erroneously suggesting its violation only by neglecting necessary control and error costs (Kish et al., 2011).
• Classification scheme: Housekeeping and excess demons represent distinct archetypes: the housekeeping demon upholds cyclic currents and structure at persistent energetic expense, while the excess demon governs irreversible transitions. Both must be considered to accurately assess total thermodynamic budgets and trade-offs, for example in molecular machines, computational feedback, and dissipative systems (Kish et al., 2011, Yoshimura et al., 2022, Sireci et al., 8 Jul 2025).

In conclusion, excess and housekeeping demons provide a unified, geometric, and operationally meaningful decomposition of dissipation, symmetry breaking, and information flow in nonequilibrium systems. Their precise mathematical structure underpins advances in thermodynamic speed limits, uncertainty relations, and the classification of cost and efficiency in physical and informational engines across classical, quantum, and network settings.