Generalized free energy and excess/housekeeping decomposition in nonequilibrium systems: from large deviations to thermodynamic speed limits

TL;DR

This work introduces a generalized free energy $\bm{\phi}^{\star}$ for genuinely nonequilibrium, nonconservative systems via a large-deviation variational principle and decomposes entropy production into conservative (excess) and nonconservative (housekeeping) parts. The excess EPR is tied to the conservative dynamics and enables thermodynamic speed limits based on the $1$-Wasserstein distance, while the housekeeping part measures nonconservative dissipation; together they provide a robust framework for thermodynamic inference in both stochastic and deterministic settings. The approach applies broadly to MJPs, CRNs, and open systems, extends to linear response with information-geometric structure, and yields practical bounds (TURs and TSLs) that connect dynamical activity, transport distance, and dissipation. The authors demonstrate the utility of the framework on unicyclic MJPs, the Brusselator, and real metabolic networks, revealing efficiency bounds and emergent cycles, and they compare their method to steady-state Hatano–Sasa decompositions, highlighting advantages away from stationarity and for inference from short-time data. Overall, the paper provides a unifying, information-geometric and large-deviation-based perspective on nonequilibrium free energy and dissipation, with concrete implications for metabolism, oscillatory dynamics, and thermodynamic inference in driven systems.

Abstract

In genuine nonequilibrium systems that undergo continuous driving, the thermodynamic forces are nonconservative, meaning they cannot be described by any free energy potential. Nonetheless, we show that the dynamics of such systems are governed by a "generalized free energy" that is derived from a large-deviations variational principle. This variational principle also yields a decomposition of fluxes, forces, and dissipation (entropy production) into a conservative "excess" part and a nonconservative "housekeeping" part. Our decomposition is universally applicable to stochastic master equations, deterministic chemical reaction networks, and open systems. We also show that the excess entropy production obeys a thermodynamic speed limit (TSL), a fundamental thermodynamic constraint on the rate of state evolution and/or external fluxes. We demonstrate our approach on several examples, including real-world metabolic networks, where we derive fundamental dissipation bounds and uncover "futile" metabolic cycles. Our generalized free energy and decomposition are empirically accessible to thermodynamic inference in both stochastic and deterministic systems. We discuss important connections to several theoretical frameworks, including information geometry and Onsager theory, as well as previous excess/housekeeping decompositions.

Figures (10)

• Figure 1: Formalism illustrated on a two-level MJP coupled to a pair of heat baths, with distribution $\bm{p}=(p_{1},p_{2})$. Transitions occur with rates $R_{21}^{c}$ and $R_{12}^{c}$ when exchanging energy with the cold bath at inverse temperature $\beta_{c}$, and with rates $R_{21}^{h}$ and $R_{12}^{h}$ when exchanging energy with the hot bath at inverse temperature $\beta_{h}$. The four one-way transitions are characterized by the incidence matrix, forward and reverse flux vectors, and force vector.
• Figure 2: Formalism illustrated on the Brusselator CRN, a nonlinear chemical oscillator, with concentrations $\bm{c}=(c_{1},c_{2})$. There are two species $X_1,X_2$ and three reversible reactions: $\varnothing\rightleftharpoons X_{1}$ (inflow), $X_{1}\rightleftharpoons X_{2}$ (conversion), and $2X_{1}+X_{2}\rightleftharpoons3X_{1}$ (second-order autocatalysis). The six one-way reactions are characterized by the stoichiometric matrix, forward and reverse flux vectors, and thermodynamic forces.
• Figure 3: Information-geometric interpretation of excess/housekeeping decomposition. (a) The EPR $\sigma=\mathcal{D}(\bm{j}\Vert\tilde{\bm{j}})$ is the relative entropy from the forward fluxes $\bm{j}$ to the reverse fluxes $\tilde{\bm{j}}$. Excess EPR $\sigma_{\text{ex}}$ is defined by the projection of $\tilde{\bm{j}}$ onto the set of fluxes that give the actual time evolution (orange line), Eq. \ref{['eq:exdef']}. Excess/housekeeping provide an orthogonal decomposition of the EPR in flux space, Eq. \ref{['eq:pyth-1']}. (b) In the space of forces, the EPR $\sigma=\mathscr{D}(\bm{f}\Vert\bm{0})$ is the relative entropy from the actual forces $\bm{f}$ to the origin $\bm{0}$. Housekeeping EPR is defined by the (dual) projection of the forces onto the set of conservative forces (blue plane), Eq. \ref{['eq:hkdef-1']}. Excess/housekeeping provide an orthogonal decomposition of the EPR in force space, Eq. \ref{['eq:pyth']}. The two projections meet at the point corresponding to the generalized potential $\bm{\phi}^{\star}$.
• Figure 4: Large deviations interpretation of the irreversibility measure $\mathcal{L}(\bm{\phi})$. The change of state observable $\bm{\phi}$ due to reactions over time $[t,t+dt]$ is measured in $n$ independent copies, and the empirical mean change is captured by the random variable $\overline{\Delta\phi}_{n}$. Here we show schematically the probability of different outcomes $\overline{\Delta\phi}_{n}=z$. For large $n$, the probability distribution is peaked at its expectation value $z=\mathbb{E}[\Delta\phi]$. The probability that the empirical mean moves in reverse, $z=-\mathbb{E}[\Delta\phi]$, decays exponentially in $n$ as $\asymp e^{-n\mathcal{L}(\bm{\phi})\,dt}$. The generalized potential $\bm{\phi}^{\star}$ is the "most irreversible" observable, having the largest $\mathcal{L}(\bm{\phi})$, and the excess EPR $\sigma_{\text{ex}}$ is its degree of irreversibility, see Eq. \ref{['eq:eprLDP']}.
• Figure 5: Time evolution and generalized potentials for unicyclic MJP. (a): Time evolution of the probability distribution for three driving strengths ($\gamma=0,1,4$). The system has 21 microstates, and the initial distribution is concentrated on three microstates $i\in\{10,11,12\}$. (b) Our generalized potential $\bm{\phi}^{\star}$ (blue) and steady-state potential $\bm{\phi}^{\textrm{ss}}$ (dotted green) for different driving strengths, evaluated at $t=0$. (c)-(d) Generalized free energy $\bm{\phi}^{\star}$ and steady-state potential $\bm{\phi}^{\textrm{ss}}$ for different driving strengths, now evaluated at $t=10$ and $t=30$. Both potentials vanish as the system approaches the uniform steady state.