arxiv: 2604.14848 · v1 · submitted 2026-04-16 · ❄️ cond-mat.stat-mech · physics.chem-ph

Recognition: unknown

Emergence of Open Chemical Reaction Network Thermodynamics within Closed Systems

Benedikt Remlein , Massimiliano Esposito , Francesco Avanzini

Authors on Pith no claims yet

Pith reviewed 2026-05-10 09:27 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.chem-ph

keywords chemical reaction networksopen vs closed systemsnonequilibrium thermodynamicschemostatstime-scale separationabundance separationentropy productionstochastic dynamics

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The pith

Open chemical reaction network behavior with chemostats emerges as an asymptotic regime from closed finite networks under time-scale and abundance separations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Closed chemical reaction networks are finite and must relax to equilibrium, yet the standard framework for describing their nonequilibrium behavior relies on open models that introduce idealized chemostats for exchanging selected species. The paper establishes that this open behavior, including matching stochastic dynamics and thermodynamic structure, arises naturally to leading order when fast reactions serve as effective exchanges and certain species maintain diverging abundances. A sympathetic reader would care because this supplies a physically grounded origin for open descriptions without treating chemostats as external idealizations, unifying the treatment of closed and open systems. The emergence holds for arbitrary stoichiometries and recovers local detailed balance, entropy production, and free-energy balance from the closed network.

Core claim

Open-CRN behavior arises as an asymptotic regime of closed CRNs when two minimal and physically transparent conditions are met: a time-scale separation, whereby fast reactions effectively act as exchange mechanisms, and an abundance separation, whereby a subset of species behaves as chemostats with diverging chemical capacity. In this regime, both the stochastic dynamics and the thermodynamic structure—including local detailed balance, entropy production, and free-energy balance—emerge to leading order from the underlying closed CRN. Our results apply to arbitrary stoichiometries and show that chemostats need not be introduced as external idealizations, but instead arise as emergent thermod

What carries the argument

The dual conditions of time-scale separation (fast reactions acting as exchange mechanisms) and abundance separation (selected species with diverging chemical capacity acting as chemostats).

Load-bearing premise

That closed chemical reaction networks can realize sufficient time-scale and abundance separations for the asymptotic open regime to appear without higher-order corrections dominating.

What would settle it

A numerical simulation or experiment on a closed CRN with identified fast reactions and high-abundance species in which the measured entropy production or steady-state fluxes deviate from the open-CRN predictions already at leading order.

Figures

Figures reproduced from arXiv: 2604.14848 by Benedikt Remlein, Francesco Avanzini, Massimiliano Esposito.

**Figure 2.** Figure 2: FIG. 2. Hypergraph representation of the closed Brusselator (upper [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Time evolution of the average numbers of molecules [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Hypergraph representation of the closed CRN in Eqs. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Time evolution of the average entropy production rate (with [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

read the original abstract

We address a fundamental question: under which conditions do the dynamics and thermodynamics of open chemical reaction networks (CRNs), grounded on the notion of idealized chemostats that exchange selected species, emerge from underlying closed CRNs? While open CRNs provide the standard framework to describe out-of-equilibrium chemical systems, real systems are finite and ultimately relax to equilibrium, leaving the status of this description conceptually unresolved. Here we show that open-CRN behavior arises as an asymptotic regime of closed CRNs when two minimal and physically transparent conditions are met: a time-scale separation, whereby fast reactions effectively act as exchange mechanisms, and an abundance separation, whereby a subset of species behaves as chemostats with diverging chemical capacity. In this regime, both the stochastic dynamics and the thermodynamic structure \ -- including local detailed balance, entropy production, and free-energy balance \ -- emerge to leading order from the underlying closed CRN. Our results apply to arbitrary stoichiometries. They show that chemostats need not be introduced as external idealizations, but instead arise as emergent thermodynamic structures within closed systems, providing a unified and physically grounded foundation for the nonequilibrium thermodynamics of CRNs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Open CRN thermodynamics emerges as an asymptotic limit from closed systems under time-scale and abundance separations, and the derivation holds for arbitrary stoichiometries.

read the letter

The main point is that this paper derives the standard open chemical reaction network framework, including its stochastic dynamics and thermodynamics like local detailed balance, entropy production, and free-energy balance, as a leading-order asymptotic regime of closed CRNs. The two conditions are a time-scale separation where fast reactions act as effective exchanges and an abundance separation where some species behave like chemostats with diverging capacity. It works for any stoichiometry, which broadens the reach beyond special cases in earlier work. The conditions are minimal and physically transparent, which makes the unification from closed to open descriptions feel grounded rather than imposed. The asymptotic analysis avoids circularity by building directly from the closed-system properties, and the stress-test shows no internal inconsistencies or missing steps in the argument structure. A minor soft spot is whether these separations occur strongly enough in typical physical or model systems for the leading-order emergence to dominate without noticeable higher-order corrections in finite cases. The paper treats it as a regime, so that limitation is acknowledged rather than overstated. This is useful for researchers in stochastic thermodynamics and nonequilibrium chemical systems who want a foundation without external idealizations. It deserves peer review because the result is conceptually clean, the generality is a real step, and the evidence is in the derivations rather than fitting. I would bring it to a reading group to check concrete examples and would cite it when justifying open models from closed ones.

Referee Report

0 major / 3 minor

Summary. The paper claims that open chemical reaction network (CRN) dynamics and thermodynamics—including local detailed balance, entropy production, and free-energy balance—emerge to leading order from any closed CRN under two conditions: time-scale separation (fast reactions acting as effective exchange mechanisms) and abundance separation (a subset of species behaving as chemostats with diverging chemical capacity). The result is shown to hold for arbitrary stoichiometries via asymptotic analysis, without introducing chemostats as external idealizations.

Significance. If the asymptotic reduction holds, the work provides a physically grounded unification of open and closed CRN descriptions, showing that standard open-CRN thermodynamics arises naturally in finite closed systems. Credit is due for the minimal, transparent conditions, the extension to arbitrary stoichiometries, and the explicit emergence of thermodynamic structure (local detailed balance, entropy production, free-energy balance) from the underlying closed dynamics.

minor comments (3)

[Abstract] The abstract and introduction state the result holds 'to leading order' but do not specify the scaling parameters or the order of the expansion (e.g., in the time-scale ratio or abundance parameter); adding a brief statement of the asymptotic ordering would improve clarity for readers reproducing the limits.
[Derivation of stochastic dynamics] In the derivation of the emergent master equation or chemical Langevin equation, the treatment of higher-order corrections from the fast reactions is not explicitly bounded; a short remark on the conditions under which these corrections remain negligible would strengthen the claim of emergence without altering the central argument.
[Thermodynamic structure] The thermodynamic section derives entropy production and free-energy balance but could benefit from an explicit comparison table or equation showing how the closed-CRN expressions reduce to the standard open-CRN forms (e.g., the chemostat contribution to the affinity).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, accurate summary of the main results, and recommendation for minor revision. We appreciate the recognition given to the minimal conditions, the extension to arbitrary stoichiometries, and the explicit emergence of the thermodynamic structure from closed CRN dynamics.

Circularity Check

0 steps flagged

No significant circularity; asymptotic emergence from closed CRN is self-contained

full rationale

The paper derives open-CRN dynamics and thermodynamics (local detailed balance, entropy production, free-energy balance) as leading-order limits of closed CRNs under explicit time-scale separation and abundance separation with diverging chemical capacity. This is a standard singular-perturbation / asymptotic reduction applied to arbitrary stoichiometries; the target quantities are not fitted to data, not defined in terms of themselves, and not justified by self-citation chains. The two conditions are stated as minimal physical assumptions whose mathematical consequences are then proven, without renaming known results or smuggling ansatzes. The derivation therefore remains independent of its inputs and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of stochastic CRN dynamics and thermodynamic potentials together with asymptotic analysis; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Standard stochastic dynamics and thermodynamic structure of chemical reaction networks
The paper builds directly on established CRN theory for both closed and open cases.
domain assumption Existence of well-defined asymptotic limits under time-scale and abundance separations
The emergence result assumes these separations produce a leading-order regime without specifying further conditions.

pith-pipeline@v0.9.0 · 5506 in / 1401 out tokens · 31153 ms · 2026-05-10T09:27:37.372302+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

93 extracted references

[1]

Fast Time Scale As discussed in Subs. V A, Eq. (A2) describes the fast dy- namics arising from theR exc reactions which conserves the quantitiesL 𝑦 (n)andL 𝑥 (n)given in Eqs. (23) and (24), re- spectively. Hence, the zero-order probability𝑝 (0) 𝜏,𝑇 (n)can be written as in Eq. (26) and Eq. (A2) becomes d𝜏 𝜋𝜏 (n|L 𝑥,L 𝑦)= ∑︁ m∈ N (L𝑥 ,L𝑦 ) ˆ𝑊exc(n,m)𝜋 𝜏 (m|...
[2]

VI A, Eq

Slow Time Scale As discussed in Subs. VI A, Eq. (A3) becomes Eq. (35) by using the conservation of the quantitiesL 𝑦 (n)andL 𝑥 (n) given in Eqs. (23) and (24), respectively, on the fast time scale. Indeed, this implies thatÍ n∈ N (L𝑥 ,L𝑦 ) ˆ𝑊exc(n,m)=0. Then, in the long-time limit of the fast time scale, i.e.,𝜏→ ∞, implying that lim 𝜏→∞ d𝜏𝑝 (1) 𝜏,𝑇 (n)=0...
[3]

The two systems exchange energy, while the total energy of the supersystem𝐸≡𝐸 𝐴 +𝐸 𝐵 remains conserved, with𝐸 𝐴 and𝐸 𝐵 being the energies of system𝐴and𝐵, re- spectively

Thermostats Consider an isolated supersystem composed of two (weakly interacting) systems, labeled𝐴and𝐵, with constant volumes. The two systems exchange energy, while the total energy of the supersystem𝐸≡𝐸 𝐴 +𝐸 𝐵 remains conserved, with𝐸 𝐴 and𝐸 𝐵 being the energies of system𝐴and𝐵, re- spectively. The two systems are closed and do not exchange molecules: b...
[4]

The two systems exchange energy, while the total energy of the su- persystem𝐸≡𝐸 𝐴 +𝐸 𝐵 remains conserved, with𝐸 𝐴 and𝐸 𝐵 being the energies of system𝐴and𝐵, respectively

Chemostats Consider an isolated supersystem composed of a single molecular species, partitioned into two (weakly interacting) systems, labeled𝐴and𝐵, with constant volumes. The two systems exchange energy, while the total energy of the su- persystem𝐸≡𝐸 𝐴 +𝐸 𝐵 remains conserved, with𝐸 𝐴 and𝐸 𝐵 being the energies of system𝐴and𝐵, respectively. The two systems...
[5]

(21) in Eq

Contributions due to theR exc Reactions We determine here the zero-order and first-order contri- butions to⟨ ¤Σe⟩𝜏,𝑇 by using Eq. (21) in Eq. (51a). In particular, we show that both contributions vanish in the long time limit of the fast time scale independently of the large abundance- assumption. The zero-order contribution reads ⟨ ¤Σ(0) e ⟩𝜏,𝑇 = ∑︁ 𝜌e,n...
[6]

We start by using Eq

Contributions due to theR int Reactions We determine here the first-order contribution to⟨ ¤Σi⟩𝜏,𝑇 . We start by using Eq. (21) in Eq. (51b), which yields ⟨ ¤Σ(1) i ⟩𝜏,𝑇 = ∑︁ 𝜌i,n ˆ𝜔𝜌i (n)𝑝 (0) 𝜏,𝑇 (n)ln ˆ𝜔𝜌i (n)𝑝 (0) 𝜏,𝑇 (n) ˆ𝜔 −𝜌i (n+S 𝜌i )𝑝 (0) 𝜏,𝑇 (n+S 𝜌i ) (D8) where𝑝 (0) 𝜏,𝑇 (n)and𝑝 (0) 𝜏,𝑇 (n+S 𝜌i)can be written, according to Eq. (26), as 𝑝 (0) 𝜏,𝑇...
[7]

Closed CRN Dynamics.For the closed unimolecular CRN in Sec. IX, the chemical master equation (4) specializes to d𝑡 𝑝𝑡 (𝑛𝑥, 𝑛𝑦1, 𝑛𝑦2, 𝑛𝑌1, 𝑛𝑌2)=𝜀 ˆ𝜅1i (𝑛𝑥 +1)𝑝 𝑡 (𝑛𝑥 +1, 𝑛 𝑦1 −1, 𝑛 𝑦2, 𝑛𝑌1, 𝑛𝑌2) + ˆ𝜅−1i (𝑛𝑦1 +1)𝑝 𝑡 (𝑛𝑥 −1, 𝑛 𝑦1 +1, 𝑛 𝑦2, 𝑛𝑌1, 𝑛𝑌2) + ˆ𝜅2i (𝑛𝑥 +1)𝑝 𝑡 (𝑛𝑥 +1, 𝑛 𝑦1, 𝑛𝑦2 −1, 𝑛 𝑌1, 𝑛𝑌2) + ˆ𝜅−2i (𝑛𝑦2 +1)𝑝 𝑡 (𝑛𝑥 −1, 𝑛 𝑦1, 𝑛𝑦2 +1, 𝑛 𝑌1, 𝑛𝑌2) − ( ˆ𝜅...
[8]

Emergent Open CRN Dynamics.For the emergent open CRN in Sec. IX, the chemical master equation (40) specializes to d𝑇 𝑃𝑇 (𝑛𝑥 )= ˆ𝜅1i (𝑛𝑥 +1)𝑃 𝑇 (𝑛𝑥 +1) +𝑉 ˆ𝜅−1i [𝑦1]𝑃𝑇 (𝑛𝑥 −1) − ( ˆ𝜅1i𝑛𝑥 +𝑉 ˆ𝜅−1i [𝑦1])𝑃𝑇 (𝑛𝑥 ) + ˆ𝜅2i (𝑛𝑥 +1)𝑃 𝑇 (𝑛𝑥 +1) +𝑉 ˆ𝜅−2i [𝑦2]𝑃𝑇 (𝑛𝑥 −1) − ( ˆ𝜅2i𝑛𝑥 +𝑉 ˆ𝜅−2i [𝑦2])𝑃𝑇 (𝑛𝑥 ), (E6) whose solution can be written using a Poisson ansatz [82],...
[9]

(72) and featuring the steady state probability of the fast dynamics in Eq

Implications of the Abundance Separation on the Steady State Probability of the Fast Dynamics Each𝜋 (𝑦) ∞ (𝑛𝑦 |L𝜆)given in Eq. (72) and featuring the steady state probability of the fast dynamics in Eq. (71) boils down to the Poisson distribution in Eq. (74) due to the abun- dance separation. Indeed,𝑎 𝑌 (L𝜆)=O (Ω)and, there- fore, each factor(𝔠 𝑌 ) (𝑎𝑌 (L...
[10]

(78) can be approx- imated, to leading order, by{ ˜𝔠𝑦 (L𝜆)}

Implications of the Abundance Separation on the Average Reaction Rates The terms{ ˜𝔠𝑦 (L𝜆 −S 𝜆,𝜌i)}featuring the average reaction rates{⟨ ˆ𝜔𝜌i |L𝑥 −S𝑥,𝜌 i,L 𝜆 −S𝜆,𝜌i ⟩ (0) ∞ }in Eq. (78) can be approx- imated, to leading order, by{ ˜𝔠𝑦 (L𝜆)}. Indeed, according to Eq. (75), ˜𝔠𝑦 (L𝜆 −S 𝜆,𝜌i)=𝑒 {ln(𝔠 𝑦 ) −Í 𝑌∈Y (𝑦) (ln(𝔠 𝑌 ) −ln(𝑎𝑌 (L𝜆 −S𝜆,𝜌i ) ) )𝑏𝑌 ,𝑦 } , ...
[11]

Implications of the Abundance Separation on the Derivation of the Average Entropy Production Rate As mentioned in the main text, the derivation of the leading-order contribution to the the average entropy pro- duction rate follows same steps as in App. D. The only minor difference is that the term in Eq. (D10), which reads ˆ𝜔𝜌i (n)𝜋 (0) ∞ (n|L 𝑥,L 𝜆) ˆ𝜔 −...
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