Equilibrium cluster statistics of cooperative and anticooperative binding on finite one-dimensional rings

J\'er\^ome Dorignac; John Palmeri; Nils-Ole Walliser; Thomas Alfonsi

Equilibrium cluster statistics of cooperative and anticooperative binding on finite one-dimensional rings

T0 review · 0 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read Exact finite-size expressions for mean cluster numbers and size distributions are derived for a grand-canonical nearest-neighbor lattice gas on finite periodic one-dimensional rings.

desk verdict Exact cluster statistics and partition enumeration for finite 1D rings. read the letter →

arxiv 2606.18020 v1 pith:IMX2P23K submitted 2026-06-16 cond-mat.stat-mech physics.bio-ph

Thomas Alfonsi , Jérôme Dorignac , John Palmeri , Nils-Ole Walliser This is my paper · ORCID

classification cond-mat.stat-mechphysics.bio-ph

keywords latticegasclusterstatisticscooperativebindingone-dimensionalringsfinite-sizeeffectsdomainwallscorrelationfunctionsperiodicboundaryconditions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper derives exact expressions for the mean occupancy, mean domain walls, and mean clusters in a finite ring of L sites under nearest-neighbor coupling in the grand-canonical ensemble. It further obtains the mean number of clusters of each size k and both the cluster-size distribution and site-weighted cluster-size distribution from k-site correlation functions. These quantities characterize changes in spatial organization for attractive versus repulsive interactions, including pronounced finite-size and parity effects near half filling. A combinatorial reformulation based on cluster counts and sizes is introduced to reach larger L without full 2^L enumeration.

What carries the argument

Grand-canonical nearest-neighbor lattice gas on a periodic ring of L sites, together with exact k-site correlation functions and a cluster-count combinatorial enumeration whose state space scales with the number of integer partitions rather than 2^L.

What would settle it

Explicit enumeration of all microstates for L=4 or L=5 at half filling, followed by direct computation of the mean number of clusters, would match or mismatch the closed-form expressions derived from the correlation functions.

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Extended reading notes

Core claim

Using a grand-canonical formulation with nearest-neighbor coupling, exact finite-size expressions are obtained for the mean occupancy, the mean number of domain walls, and the mean number of clusters. Building on exact k-site correlation functions, expressions follow for the mean number of clusters of size k and for the cluster-size distribution together with the site-weighted cluster-size distribution. These observables show how spatial organization varies with cooperative and anticooperative interactions and exhibit finite-size and parity-dependent effects that are strongest near half filling in small systems.

Load-bearing premise

The nearest-neighbor interaction model on a finite periodic one-dimensional lattice serves as a minimal model for adsorption and binding on small ring-like substrates.

Editorial extensions

If this is right

Mean cluster number and size distributions differ qualitatively between attractive and repulsive nearest-neighbor couplings.
Parity-dependent effects appear most strongly near half filling and are visible in small rings.
The cluster-based combinatorial method reduces the effective configuration count from exponential in L to roughly exponential in sqrt(L).
Cluster observables provide information complementary to simple occupancy measures for detecting cooperativity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The derived parity effects imply that binding statistics on even-length versus odd-length rings could differ measurably even when average occupancy is similar.
The exact formulas supply reference data against which Monte Carlo or molecular-dynamics simulations of ring substrates can be validated.
Cluster-size distributions rather than occupancy alone may serve as a sharper experimental signature of interaction sign in finite biological assemblies.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

Two things stand out right away. The paper gives exact finite-size expressions for the mean number of clusters, the number of k-sized clusters, and both the cluster-size and site-weighted cluster-size distributions on periodic 1D rings. It also supplies a combinatorial formulation that groups configurations by cluster counts and sizes, cutting the enumeration cost from 2^L down to something that scales like the partition function, roughly exp(sqrt(L)).

The derivations start from the standard grand-canonical lattice gas with nearest-neighbor coupling and use the known k-site correlation functions to extract the cluster observables. That step is direct and avoids any fitting or circularity. The parity-dependent effects near half filling are called out explicitly, which is a useful finite-size detail for small rings. The combinatorial reduction is a practical addition that lets the method reach larger L than brute force allows.

The main limitation is the strict 1D nearest-neighbor setting. Results are precise benchmarks for that minimal model rather than a general theory, and the abstract gives no numerical checks against simulation, though the exact claims are in principle verifiable for small L. The claim that the model serves as a minimal description for ring-like substrates is reasonable but narrow.

This is for people who need exact finite-size cluster data in statistical mechanics of adsorption or binding on small periodic structures. Readers who run simulations or approximations would find concrete test cases here. The technical content and the new observables are solid enough to justify sending the paper to a serious referee.

Referee Report

0 major / 2 minor

Summary. The manuscript derives exact finite-size expressions for the mean occupancy, mean number of domain walls, mean number of clusters, mean number of k-sized clusters, the cluster-size distribution, and the site-weighted cluster-size distribution for a grand-canonical nearest-neighbor lattice gas on a periodic 1D ring of L sites. These are obtained from k-site correlation functions; a complementary combinatorial enumeration of configurations by cluster counts and sizes is introduced to reach larger L, with state-space size scaling as ~e^{\sqrt{L}}.

Significance. If the derivations are correct, the work supplies parameter-free exact benchmarks for finite-size and parity effects in cooperative/anticooperative 1D binding, together with cluster observables that go beyond occupancy. The combinatorial reduction and the explicit construction from transfer-matrix correlations are concrete strengths that enable direct comparison with small-ring experiments.

minor comments (2)

[Abstract] Abstract: the sentence introducing 'two complementary size statistics' does not name them; explicitly stating 'the cluster-size distribution and the site-weighted cluster-size distribution' would remove ambiguity.
The combinatorial formulation is stated to reduce the state space to integer partitions; a brief remark on how the enumeration is actually implemented (e.g., recursion or generating functions) would help readers reproduce the larger-L results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report, so we have no points requiring response or revision at this stage.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivations self-contained from standard methods

full rationale

The paper's central results consist of exact finite-size expressions for occupancy, domain walls, total clusters, k-cluster counts, and size distributions, all obtained directly from the grand-canonical partition function and k-site correlation functions of the nearest-neighbor 1D lattice gas on a periodic ring. These follow from the standard transfer-matrix solution of the model plus combinatorial grouping of microstates by cluster properties; no step reduces by construction to a fitted parameter, self-citation chain, or redefinition of inputs as outputs. The derivation chain is independent and externally verifiable against the known solvability of the 1D Ising/lattice-gas model.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Based on abstract only: no free parameters or invented entities are introduced. Relies on standard statistical mechanics assumptions for lattice gases.

assumptions (2)

domain assumption Grand-canonical ensemble applies to the lattice gas with fixed chemical potential and nearest-neighbor coupling.
Invoked in the first sentence of the abstract as the formulation used for derivations.
domain assumption Periodic boundary conditions on a finite 1D ring capture the essential physics of small ring-like substrates.
Stated as the geometry chosen for the minimal model.

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Cite this review

Pith. "Pith review of Equilibrium cluster statistics of cooperative and anticooperative binding on finite one-dimensional rings." pith.science (2026). https://pith.science/paper/IMX2P23K

@misc{pith2026260618020,
  author       = {Pith},
  title        = {Pith review of: Equilibrium cluster statistics of cooperative and anticooperative binding on finite one-dimensional rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IMX2P23K}},
  note         = {Machine review of arXiv:2606.18020}
}

abstract

We study equilibrium clustering in a finite one-dimensional lattice gas of $L$ sites with periodic boundary conditions, as a minimal model for adsorption and binding on small ring-like substrates. Using a grand-canonical formulation with nearest-neighbor coupling, we derive exact finite-size expressions for the mean occupancy, the mean number of domain walls, and the mean number of clusters. Building on exact $k$-site correlation functions, we further derive expressions for the mean number of clusters of size $k$ and for two complementary size statistics: the cluster-size distribution, and the site-weighted cluster-size distribution. These observables characterize how spatial organization changes across attractive (cooperative) and repulsive (anticooperative) interactions, and highlight finite-size and parity-dependent effects of the underlying lattice, the latter being particularly pronounced near half filling in small systems. To access larger lattices without enumerating all $2^L$ microstates, we also develop a cluster-based combinatorial formulation in which configurations are classified by cluster counts and sizes, reducing the effective state space to a set whose size scales with integer partitions, $\approx e^{\sqrt{L}}$, rather than with $\approx e^{L}$. Taken together, our results provide exact benchmarks for finite periodic systems and suggest experimentally relevant cluster observables that complement occupancy-based measures of cooperativity, with particular relevance for binding on ring-like substrates for biological assemblies.

Figures

Figures reproduced from arXiv: 2606.18020 by the authors.

**Figure 2.** FIG. 2. Parametric plot of the normalized mean number of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Contour plot and heat map of the normalized mean [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4. Parametric plot of the normalized mean number of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. (upper row) Cluster-size distribution, [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. Parametric plot of the normalized average cluster size [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: This can be characterized by studying the limit [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 7.** Figure 7: FIG. 7. Parametric plot of the normalized configuration [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 9.** Figure 9: FIG. 9. Parametric plot of the normalized mean number of [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 8.** Figure 8: FIG. 8. Contour plot and heat map of the normalized mean [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 10.** Figure 10: FIG. 10. Parametric plot of the normalized mean number of [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]

**Figure 11.** Figure 11: FIG. 11. Evolution of the equilibrium probability of occurrence of a cluster of size [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references

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In this case λ+ =u + = 1 +e µ andλ − =u − = 0, thus the partition function is simply Ξ HL = (1 +e µ)L

Hill-Langmuir The Hill-Langmuir (HL) regime represents an example of noniteracting particles, henceJ= 0. In this case λ+ =u + = 1 +e µ andλ − =u − = 0, thus the partition function is simply Ξ HL = (1 +e µ)L. In this case, the system is completely uncorrelated asξ HL = 0. The mean relative occupancy can be derived from Eq. (6): ⟨φ⟩HL = 1 1 +e −µ .(C1) Usin...
[2]

6 immediately yields⟨φ⟩ HF = 1/2, hence the title of this subsection,half-filling regime

Half-Filling AsJ+µ= 0, Eq. 6 immediately yields⟨φ⟩ HF = 1/2, hence the title of this subsection,half-filling regime. In this regime, the eigenvalues of the transfer matrix areλ ± = 1±e −J/2 andu ± = 1 2(1±e J/2)2. Thus, the partition function is Ξ HF = (1 +e −J/2)L + (1− e−J/2)L. The correlation length can be written asξ HF = 1/log(cothJ/4). Which allows ...
[3]

Thermodynamic limit As the lattice sizeLtends to infinity and when the particle density (or occupancy)⟨φ⟩is neither zero nor one, the thermodynamic limit is reached. Given that λ+ > λ −, expressions for the partition functions, the correlations, and the mean occupancy simplify as Ξ∼λ L + ,(C34) ck ∼e kµ+J(k−1) u+λk−1 + , k≪L ,(C35) and ⟨φ⟩ ∼ eµu+ λ2 + .(C...
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Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

[1] [1]

In this case λ+ =u + = 1 +e µ andλ − =u − = 0, thus the partition function is simply Ξ HL = (1 +e µ)L

Hill-Langmuir The Hill-Langmuir (HL) regime represents an example of noniteracting particles, henceJ= 0. In this case λ+ =u + = 1 +e µ andλ − =u − = 0, thus the partition function is simply Ξ HL = (1 +e µ)L. In this case, the system is completely uncorrelated asξ HL = 0. The mean relative occupancy can be derived from Eq. (6): ⟨φ⟩HL = 1 1 +e −µ .(C1) Usin...

[2] [2]

6 immediately yields⟨φ⟩ HF = 1/2, hence the title of this subsection,half-filling regime

Half-Filling AsJ+µ= 0, Eq. 6 immediately yields⟨φ⟩ HF = 1/2, hence the title of this subsection,half-filling regime. In this regime, the eigenvalues of the transfer matrix areλ ± = 1±e −J/2 andu ± = 1 2(1±e J/2)2. Thus, the partition function is Ξ HF = (1 +e −J/2)L + (1− e−J/2)L. The correlation length can be written asξ HF = 1/log(cothJ/4). Which allows ...

[3] [3]

Thermodynamic limit As the lattice sizeLtends to infinity and when the particle density (or occupancy)⟨φ⟩is neither zero nor one, the thermodynamic limit is reached. Given that λ+ > λ −, expressions for the partition functions, the correlations, and the mean occupancy simplify as Ξ∼λ L + ,(C34) ck ∼e kµ+J(k−1) u+λk−1 + , k≪L ,(C35) and ⟨φ⟩ ∼ eµu+ λ2 + .(C...

[4] [4]

clusters

K. Binder, “clusters” in the ising model, metastable states and essential singularity, Annals of physics98, 390 (1976)

1976

[5] [5]

Liu and J

Y. Liu and J. P. Dilger, Application of the one-and two- dimensional ising models to studies of cooperativity be- tween ion channels, Biophysical journal64, 26 (1993). 19

1993

[6] [6]

Yilmaz and F

M. Yilmaz and F. M. Zimmermann, Exact cluster size distribution in the one-dimensional ising model, Phys- ical Review E—Statistical, Nonlinear, and Soft Matter Physics71, 026127 (2005)

2005

[7] [7]

A. Fronczak, Cluster properties of the one-dimensional lattice gas: The microscopic meaning of grand poten- tial, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics87, 022131 (2013)

2013

[8] [8]

Ivanytskyi and V

A. Ivanytskyi and V. Chelnokov, On bimodal size distri- bution of spin clusters in the onedimensional ising model, inEPJ Web of Conferences, Vol. 182 (EDP Sciences,