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Topological Structures of Large Scale Interacting Systems via Uniform Functions and Forms

Kenichi Bannai, Yukio Kametani, Makiko Sasada

TL;DR

The paper develops a new, purely algebraic framework—uniform cohomology—for analyzing macroscopic observables in large-scale interacting systems on infinite graphs. By introducing uniform functions and forms on the configuration space S^X, it proves that the zeroth uniform cohomology H^0_unif(S^X) naturally corresponds to conserved quantities and that higher uniform cohomology vanishes, facilitating a clean separation of local and global effects. The central result is a Varadhan-type decomposition in a broad, group-action setting: for irreducibly quantified interactions and locales with a free G-action, the space of shift-invariant closed uniform forms modulo exact forms is canonically isomorphic to Hom(G, Consv^φ(S)), with a decompositional split C ≅ E ⊕ Hom(G, Consv^φ(S)). This framework, which avoids spectral-gap arguments, provides a universal, cohomological lens to understand hydrodynamic limits and macroscopic evolution across a wide class of nongradient models, tying microscopic conserved quantities to macroscopic flows through a group-cohomology perspective.

Abstract

In this article, we investigate the topological structure of large scale interacting systems on infinite graphs, by constructing a suitable cohomology which we call the uniform cohomology. The central idea for the construction is the introduction of a class of functions called uniform functions. Uniform cohomology provides a new perspective for the identification of macroscopic observables from the microscopic system. As a straightforward application of our theory when the underlying graph has a free action of a group, we prove a certain decomposition theorem for shift-invariant closed uniform forms. This result is a uniform version in a very general setting of the decomposition result for shift-invariant closed $L^2$-forms originally proposed by Varadhan, which has repeatedly played a key role in the proof of the hydrodynamic limits of nongradient large scale interacting systems. In a subsequent article, we use this result as a key to prove Varadhan's decomposition theorem for a general class of large scale interacting systems.

Topological Structures of Large Scale Interacting Systems via Uniform Functions and Forms

TL;DR

The paper develops a new, purely algebraic framework—uniform cohomology—for analyzing macroscopic observables in large-scale interacting systems on infinite graphs. By introducing uniform functions and forms on the configuration space S^X, it proves that the zeroth uniform cohomology H^0_unif(S^X) naturally corresponds to conserved quantities and that higher uniform cohomology vanishes, facilitating a clean separation of local and global effects. The central result is a Varadhan-type decomposition in a broad, group-action setting: for irreducibly quantified interactions and locales with a free G-action, the space of shift-invariant closed uniform forms modulo exact forms is canonically isomorphic to Hom(G, Consv^φ(S)), with a decompositional split C ≅ E ⊕ Hom(G, Consv^φ(S)). This framework, which avoids spectral-gap arguments, provides a universal, cohomological lens to understand hydrodynamic limits and macroscopic evolution across a wide class of nongradient models, tying microscopic conserved quantities to macroscopic flows through a group-cohomology perspective.

Abstract

In this article, we investigate the topological structure of large scale interacting systems on infinite graphs, by constructing a suitable cohomology which we call the uniform cohomology. The central idea for the construction is the introduction of a class of functions called uniform functions. Uniform cohomology provides a new perspective for the identification of macroscopic observables from the microscopic system. As a straightforward application of our theory when the underlying graph has a free action of a group, we prove a certain decomposition theorem for shift-invariant closed uniform forms. This result is a uniform version in a very general setting of the decomposition result for shift-invariant closed -forms originally proposed by Varadhan, which has repeatedly played a key role in the proof of the hydrodynamic limits of nongradient large scale interacting systems. In a subsequent article, we use this result as a key to prove Varadhan's decomposition theorem for a general class of large scale interacting systems.

Paper Structure

This paper contains 27 sections, 55 theorems, 211 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Introduction
  3. The Large Scale Interacting System
  4. Main Theorem and Relation to Varadhan's Decomposition
  5. Overview
  6. Configuration Space and Conserved Quantities
  7. Configuration Space and Transition Structure
  8. Conserved Quantities and Irreducibly Quantified Interactions
  9. The Cohomology of the Configuration Space
  10. Uniform Functions and Uniform Forms
  11. Uniform Functions on the Configuration Space
  12. Horizontal Uniform Functions
  13. Pairings for Functions with Uniform Differentials
  14. Criterion for Uniformity
  15. Pairing and Criterion in the Weakly Transferable Case
  16. ...and 12 more sections

Key Result

Theorem 1

For the system $(X,S,\phi)$, assume that the interaction $\phi$ is irreducibly quantified, and that $X$ has a free action of a group $G$. If $X$ is transferable, or if the interaction $\phi$ is simple and $X$ is weakly transferable, then we have a canonical isomorphism Moreover, a choice of a fundamental domain for the action of $G$ on $X$ gives a natural decomposition of $\mathbb{R}$-linear spa

Figures (2)

  • Figure 1: The Triangular and Hexagonal Lattices, and the Cayley graph for the free group $G$ generated by $\sigma_1$ and $\sigma_2$ (see Example \ref{['example: locale']} (4))
  • Figure 7: Example of Calculation of Homology

Theorems & Definitions (159)

  • Definition 1.1
  • Remark 1.2
  • Example 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1: =Theorem \ref{['thm: main']}
  • Corollary 2: =Corollary \ref{['cor: main']}
  • Theorem 3: FUY96*Theorem 4.1
  • Theorem 4
  • Theorem 5
  • ...and 149 more