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arxiv: 2410.06778 · v1 · submitted 2024-10-09 · 🧮 math.PR · math.CO

On Interactions for Large Scale Interacting Systems

Kenichi Bannai , Jun Koriki , Makiko Sasada , Hidetada Wachi , Shuji Yamamoto This is my paper

Pith reviewed 2026-05-23 19:12 UTC · model grok-4.3

classification 🧮 math.PR math.CO
keywords interactionsconserved quantitiesequivalence classesseparable interactionswedge sumsbox productshydrodynamic limitslarge-scale interacting systems
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The pith

Equivalence classes of separable interactions for systems with two, three, or four local states number one, two, and five respectively.

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

The paper introduces interactions as symmetric directed graphs that represent possible state transitions on adjacent sites in large-scale particle systems. It defines an equivalence relation on these interactions based on their spaces of conserved quantities to group those with similar macroscopic properties. For cases where local states number two, three, or four, it determines there are one, two, and five equivalence classes of separable interactions. The work also constructs new interactions via wedge sums and box products and shows these operations preserve the irreducibly quantified condition needed for hydrodynamic limits. This provides a systematic way to build and classify interactions for studying macroscopic phenomena from microscopic rules.

Core claim

When the set of local states consists of two, three or four elements, the number of equivalence classes of separable interactions are respectively one, two and five. The irreducibly quantified condition for interactions is preserved by wedge sums and box products.

What carries the argument

The equivalence relation on interactions defined using the space of conserved quantities, which classifies interactions reflecting their macroscopic properties.

If this is right

  • Interactions underlying exclusion processes, multi-species processes, and lattice-gas processes can be classified by their conserved quantities.
  • Wedge sums and box products provide methods to construct new interactions from existing ones.
  • The irreducibly quantified condition, important for hydrodynamic limits, remains intact under these constructions.
  • Abundant examples are generated suitable for considering hydrodynamic limits of large-scale interacting systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Such a classification might extend to larger numbers of states if computational methods are developed.
  • Interactions with the same conserved quantities may exhibit identical large-scale behaviors in simulations or experiments.
  • This framework could connect to other combinatorial objects in statistical mechanics beyond the listed processes.

Load-bearing premise

The equivalence relation defined by spaces of conserved quantities accurately reflects the corresponding macroscopic properties of the large-scale interacting systems.

What would settle it

Finding two separable interactions with the same conserved quantities but different hydrodynamic limits, or vice versa, would challenge the equivalence classification.

Figures

Figures reproduced from arXiv: 2410.06778 by Hidetada Wachi, Jun Koriki, Kenichi Bannai, Makiko Sasada, Shuji Yamamoto.

Figure 1
Figure 1. Figure 1: The Exclusion Process on Z. Our hopping rule is homogenous, in that it does not depend on the particular site. The hopping takes place on edges 𝑒 = (𝑥, 𝑥 + 1) between sites, for 𝑥 ∈ Z. Hence we can express the hopping rule as the set of permitted movement between local states on adjacent sites. More concretely, if we fix an edge 𝑒 = (𝑥, 𝑥 + 1), then the possible configurations on sites 𝑥 and 𝑥 + 1 can be e… view at source ↗
Figure 2
Figure 2. Figure 2: Exclusion Interaction and the Associated Graph We focus on the movement of particles on adjacent sites. The permitted movements can be expressed as edges of a graph with vertices 𝑆 × 𝑆. The hopping rule for the exclusion process is given by 𝜙ex = {( (1, 0), (0, 1)), ( (0, 1), (1, 0))} ⊂ (𝑆 × 𝑆) × (𝑆 × 𝑆), which we view as the set of edges of a graph with vertices 𝑆 × 𝑆. This rule is independent of our choi… view at source ↗
Figure 3
Figure 3. Figure 3: 𝜅-exclusion interaction for 𝜅 = 2, underlying the process commonly referred to as the generalized exclusion process with maximal occupancy 𝜅 = 2 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: multi-species exclusion interaction for 𝜅 = 2 Example 2.6. For an integer 𝜅 ≥ 1 and 𝑆𝜅 = {0, 1, . . . , 𝜅}, we let 𝜙 𝜅 ms B {( ( 𝑗, 𝑘), (𝑘, 𝑗)) ∈ 𝑆𝜅 × 𝑆𝜅 | 𝑗, 𝑘 ∈ 𝑆𝜅, 𝑗 ≠ 𝑘}. Then the pair (𝑆𝜅, 𝜙𝜅 ms) is an interaction, which we call the multi-species exclusion interaction. We have 𝑐𝜙 𝜅 ms = 𝜅, and we have a basis 𝜉 1 , . . . , 𝜉𝜅 of Consv𝜙 𝜅 ms (𝑆𝜅) called the standard basis defined as 𝜉 𝑖 (0) = 0 and 𝜉 𝑖… view at source ↗
Figure 5
Figure 5. Figure 5: The Glauber interaction 𝜙G gives an example of an interaction such that 𝑐𝜙 = 0. Next, we define the notion of equivalence and isomorphisms of interactions. For any set 𝑆, we denote by Map(𝑆, R) be the space of real valued functions on 𝑆. Definition 2.8. Let (𝑆, 𝜙) and (𝑆 ′ , 𝜙′ ) be interactions. (1) We say that (𝑆, 𝜙) and (𝑆 ′ , 𝜙′ ) are equivalent, if there exists a bijection 𝑆  𝑆 ′ such that the induce… view at source ↗
Figure 6
Figure 6. Figure 6: Interaction of Example 2.9 underlying the two-species exclusion pro￾cess with annihilation and creation studied in [23]. Note that the associated graph coincides with that of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The wedge sum 𝜙ex ∨ 𝜙ex of two exclusion interactions along ∗ gives the multi-species interaction 𝜙 2 ms. The blue arrow indicates the edge coming from the first exclusion, the red arrow indicates the edge coming from the second exclusion and the black arrow indicates the new edge added via definition of the wedge sum. We next consider the lattice gas with energy interaction, which is obtained as the wedge… view at source ↗
Figure 8
Figure 8. Figure 8: The wedge sum of the exclusion interaction 𝜙ex and the Glauber interaction 𝜙G is easily checked to be equivalent to the MIPS interaction described in the figure underlying the MIPS process studied in [1, 17]. We next define the notion of box products of interactions to construct new interactions. Definition 3.7. Let (𝑆1, 𝜙1) and (𝑆2, 𝜙2) be interactions. We let 𝑆1 × 𝑆2 be the product of sets, and we define… view at source ↗
Figure 10
Figure 10. Figure 10: The figure describes the two-lane 1-exclusion interaction 𝜙 □2 ex = 𝜙ex□𝜙ex. The blue particles are in the first lane and the red particles are in the second lane. The particles move in their respective lanes. The movements in each lane coincides with the exclusion interaction [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: For the multi-species exclusion interaction 𝜙 3 ms, Proposition 4.5 and Corollary 4.6 shows that any interaction connecting one or more of the red edges on the graph on the right are all equivalent. The numbers next to each connected component of the graphs are the values of the conserved quantities. Proof. By reordering the elements of 𝑆 = 𝑆𝜅 and taking equivalent interactions if necessary, we can assume… view at source ↗
Figure 12
Figure 12. Figure 12: The Classifications of Interactions for 𝑆 = {0, 1, 2}. The numbers next to each connected components of the graphs are the values of the conserved quantities. The Generalized Glauber Interaction 𝜙 2 G given via the graph in the above figure can be viewed as the generalization of the Glauber Interaction with spin {−1, 0, 1}. Proof. Suppose the interaction (𝑆, 𝜙) is not separable. By definition, there exist… view at source ↗
Figure 13
Figure 13. Figure 13: Classification of Equivalence Classes of Separable Interactions for 𝑆 = {0, 1, 2, 3} with 𝑐𝜙 ≥ 1. The numbers next to the connected components are the values of the conserved quantities of that component. The red arrows indicate addition of an extra edge connecting two distinct connected components of the associated graph of the interaction. (5) A new interaction, obtained from (2) by connecting the compo… view at source ↗
Figure 14
Figure 14. Figure 14: The above interaction is separable and exchangeable, but not irre￾ducibly quantified. The numbers next to the vertices expresses the value of the conserved quantity 𝜉 B 𝜉 1 + 2𝜉 2 + 3𝜉 3 . This is equivalent to the 3-exclusion process 𝜙3-ex, which is irreducibly quantified. Suppose now that (𝑆, 𝜙) is an interaction which is exchangeable. Then by definition, for any (𝑠1, 𝑠2) ∈ 𝑆 × 𝑆, we have (𝑠1, 𝑠2) ↭𝜙 (𝑠… view at source ↗
Figure 15
Figure 15. Figure 15: The figure represents the Lattice Gas with Energy on 𝑆 = {0, 1, 2, 3}. One can interpret the states as either empty (= 0) or occupied by a single particle with no energy (= 1), a particle with one energy (= 2), and a particle with two energy (= 3). The conserved quantity 𝜉 1 + 𝜉 2 + 𝜉 3 returns the total number of particles and 𝜉 2 + 2𝜉 3 returns the total energy. Fig16 (1,0) (2,0) (2,1) (0,1) (0,2) (1,2)… view at source ↗
Figure 16
Figure 16. Figure 16: The figure represents the New Interaction. It can be interpreted as the Lattice Gas with Energy with the additional transformation that two extra energy on a single particle adjacent to a vacant site can create a single particle with no extra energy on the vacant site. For any finite symmetric graph (𝑋, 𝐸), let (𝑆 𝑋 , Φ𝐸) and (𝑆 𝑋 , Φ′ 𝐸 ) be the configuration space with transition structure associated to… view at source ↗
Figure 17
Figure 17. Figure 17: The figure represents an interaction which is isomorphic to the New Interaction of Corollary 4.12 (5). The second interpretation of the New Interaction of Corollary 4.12 (5), obtained by exchanging 1 and 2 in 𝑆 = {0, 1, 2, 3}, is as follows. One can interpret the states as either empty (= 0), or occupied by a single green (=1), blue (=2) and red (=3) particles. Two green particles can combine to make a si… view at source ↗
read the original abstract

Statistical mechanics explains the properties of macroscopic phenomena based on the movements of microscopic particles such as atoms and molecules. Movements of microscopic particles can be represented by large-scale interacting systems. In this article, we study a combinatorial object which we call interactions, given as a symmetric directed graph representing the possible transition of states on adjacent sites of large-scale interacting systems. Such interactions underlie various standard processes such as the exclusion processes, generalized exclusion processes, multi-species exclusion processes, lattice-gas processes with energy, and the multi-lane particle processes. We introduce the notion of equivalences of interactions using their space of conserved quantities. This allows for the classification of interactions reflecting corresponding macroscopic properties. In particular, we prove that when the set of local states consists of two, three or four elements, then the number of equivalence classes of separable interactions are respectively one, two and five. We also define the wedge sums and box products of interactions, which give systematic methods for constructing new interactions from existing ones. Furthermore, we prove that the irreducibly quantified condition for interactions, which has implicitly played an important role in the theory of hydrodynamic limits, is preserved by wedge sums and box products. Our results provide a systematic method to construct and classify interactions, offering abundant examples suitable for considering hydrodynamic limits.

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.

Referee Report

2 major / 2 minor

Summary. The paper defines 'interactions' as symmetric directed graphs encoding allowed state transitions on adjacent sites of large-scale interacting particle systems. It introduces an equivalence relation on interactions via equality of their spaces of conserved quantities, proves that the number of equivalence classes of separable interactions is 1, 2, and 5 when the local state set S has cardinality 2, 3, and 4 respectively, and shows that the wedge sum and box product operations preserve the irreducibly quantified condition. The results are positioned as providing systematic construction and classification tools for examples suitable for hydrodynamic limit analysis.

Significance. If the combinatorial classification is correct, the explicit counts for small |S| together with the closure properties under wedge sums and box products supply a concrete supply of inequivalent interactions with controlled conserved quantities. This is a useful contribution to the literature on interacting particle systems, as it organizes families (exclusion, multi-species, energy-conserving, multi-lane) under a single framework and identifies which constructions preserve a condition known to be relevant for hydrodynamic limits.

major comments (2)
  1. [Introduction] Introduction, paragraph 3 and abstract: the claim that the equivalence 'reflects corresponding macroscopic properties' is not supported by any explicit derivation or comparison of hydrodynamic PDEs (or even fluxes) for representatives of distinct classes. The combinatorial definition via conserved quantities is load-bearing for the classification, but the reflection step to distinct scaling limits remains an unverified bridge; a concrete test would be to compute the hydrodynamic limit for at least two inequivalent interactions with |S|=3 and exhibit different conservation laws or fluxes.
  2. [§4] §4, Theorem 4.3 (classification for |S|=4): the enumeration of the five equivalence classes is asserted via exhaustive checking of conserved-quantity spaces, but the manuscript provides no explicit list of graph representatives or the dimension of the conserved-quantity spaces for each class. Without these data, independent verification of the count of five (as opposed to four or six) is not possible from the text alone.
minor comments (2)
  1. [§5] Notation for the wedge sum and box product is introduced in §5 but the associativity or commutativity properties are not stated explicitly; adding a short remark or table summarizing the algebraic structure would improve readability.
  2. [§3] The definition of 'separable interaction' (used throughout the classification theorems) appears only in §3; moving a concise restatement to the introduction would help readers who focus on the main results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Introduction] Introduction, paragraph 3 and abstract: the claim that the equivalence 'reflects corresponding macroscopic properties' is not supported by any explicit derivation or comparison of hydrodynamic PDEs (or even fluxes) for representatives of distinct classes. The combinatorial definition via conserved quantities is load-bearing for the classification, but the reflection step to distinct scaling limits remains an unverified bridge; a concrete test would be to compute the hydrodynamic limit for at least two inequivalent interactions with |S|=3 and exhibit different conservation laws or fluxes.

    Authors: The equivalence relation is defined precisely by equality of the spaces of conserved quantities. Any hydrodynamic limit, if it exists, must conserve exactly those quantities that are conserved microscopically; hence distinct spaces of conserved quantities necessarily produce hydrodynamic equations with distinct conservation laws. We will add a short clarifying paragraph in the introduction (and a corresponding sentence in the abstract) that makes this direct implication explicit. Performing a full hydrodynamic-limit computation for two |S|=3 representatives lies outside the combinatorial scope of the present work and would require separate analytic effort; we therefore view the added explanatory paragraph as the appropriate strengthening of the manuscript. revision: partial

  2. Referee: [§4] §4, Theorem 4.3 (classification for |S|=4): the enumeration of the five equivalence classes is asserted via exhaustive checking of conserved-quantity spaces, but the manuscript provides no explicit list of graph representatives or the dimension of the conserved-quantity spaces for each class. Without these data, independent verification of the count of five (as opposed to four or six) is not possible from the text alone.

    Authors: We agree that the current text does not supply the explicit representatives or dimensions needed for independent verification. In the revised manuscript we will insert a table (or enumerated list) in §4 that, for each of the five classes, gives (i) one concrete graph representative on four vertices and (ii) the dimension of its space of conserved quantities. This addition will make the classification fully verifiable from the text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; combinatorial classification is self-contained

full rationale

The paper defines interactions as symmetric directed graphs and an equivalence relation via the space of conserved quantities, then enumerates the resulting classes for |S|=2,3,4 and proves preservation of the irreducibly quantified condition under wedge sums and box products. These counts and preservation statements follow directly from the definitions and combinatorial arguments internal to the paper. No load-bearing self-citations, no fitted parameters renamed as predictions, no ansatzes imported via citation, and no self-definitional loops are present. The results are independent mathematical statements about the defined objects.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work is a combinatorial classification resting on definitional choices for interactions and equivalences rather than external data or fitted constants. No free parameters appear. Axioms are standard background assumptions in graph theory and interacting particle systems.

axioms (2)
  • domain assumption Interactions are modeled as symmetric directed graphs on the set of local states.
    This is the foundational representation introduced for transition rules in the systems studied.
  • domain assumption Equivalence classes are determined by the space of conserved quantities.
    This definition is used to classify interactions and link them to macroscopic properties.
invented entities (1)
  • separable interactions no independent evidence
    purpose: A subclass of interactions for which the equivalence classification and counts are established.
    Introduced as the objects to which the main classification theorems apply.

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