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On Interactions for Large Scale Interacting Systems

Kenichi Bannai, Jun Koriki, Makiko Sasada, Hidetada Wachi, Shuji Yamamoto

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.

On Interactions for Large Scale Interacting Systems

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.

Paper Structure

This paper contains 6 sections, 26 theorems, 43 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Conserved Quantities and Interactions
  3. Wedge Sums and Box Products of Interactions
  4. Classification of Interactions
  5. Irreducibly Quantified Interactions
  6. A New Interaction

Key Result

Lemma 2.3

Let $(S,\phi)$ be an interaction. For any function $\xi\colon S\rightarrow\mathbb{R}$, we let $\xi_S\colon S\times S\rightarrow\mathbb{R}$ be the function defined by Then $\xi$ is a conserved quantity for the interaction $(S,\phi)$ if and only if the function $\xi_S$ is constant on the connected components of the associated graph $(S\times S,\phi)$. In other words, $\xi$ is a conserved quantity i

Figures (17)

  • Figure 1: The Exclusion Process on $\mathbb{Z}$.
  • Figure 2: Exclusion Interaction and the Associated Graph
  • Figure 3: $\kappa$-exclusion interaction for $\kappa=2$, underlying the process commonly referred to as the generalized exclusion process with maximal occupancy $\kappa=2$.
  • Figure 4: multi-species exclusion interaction for $\kappa=2$
  • Figure 5: The Glauber interaction $\phi_{\operatorname{G}}$ gives an example of an interaction such that ${c_\phi}=0$.
  • ...and 12 more figures

Theorems & Definitions (71)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Definition 2.8
  • Example 2.9
  • ...and 61 more