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Asymptotic Scattering Relation for the Toda Lattice

Amol Aggarwal

TL;DR

This work formulates and justifies a dense-quasiparticle framework for the Toda lattice at thermal equilibrium by tying quasiparticle spectral data to the Lax-matrix eigenstructure. It defines localization centers from exponentially localized eigenvectors to assign quasiparticle positions and proves that local observables approximate sums over these quasiparticles. The core result is a rigorous asymptotic scattering relation that governs the long-time evolution of quasiparticle locations, incorporating explicit logarithmic scattering shifts derived from spectral data. The approach relies on inverse scattering, resolvent perturbations, and transfer-matrix analysis for random tridiagonal Lax matrices, along with finite- to infinite-volume comparison arguments. The findings provide a mathematically solid foundation for the soliton-gas picture in a classical integrable system under rough, random initial data and offer a precise link between microscopic Lax-data and macroscopic transport properties, with potential extensions to other integrable models and invariant measures.

Abstract

In this paper we consider the Toda lattice $(\boldsymbol{p}(t); \boldsymbol{q}(t))$ at thermal equilibrium, meaning that its variables $(p_i)$ and $(e^{q_i-q_{i+1}})$ are independent Gaussian and Gamma random variables, respectively. We justify the notion from the physics literature that this model can be thought of as a dense collection of ``quasiparticles'' that act as solitons by, (i) precisely defining the locations of these quasiparticles; (ii) showing that local charges and currents for the Toda lattice are well-approximated by simple functions of the quasiparticle data; and (iii) proving an asymptotic scattering relation that governs the dynamics of the quasiparticle locations. Our arguments are based on analyzing properties about eigenvector entries of the Toda lattice's (random) Lax matrix, particularly, their rates of exponential decay and their evolutions under inverse scattering.

Asymptotic Scattering Relation for the Toda Lattice

TL;DR

This work formulates and justifies a dense-quasiparticle framework for the Toda lattice at thermal equilibrium by tying quasiparticle spectral data to the Lax-matrix eigenstructure. It defines localization centers from exponentially localized eigenvectors to assign quasiparticle positions and proves that local observables approximate sums over these quasiparticles. The core result is a rigorous asymptotic scattering relation that governs the long-time evolution of quasiparticle locations, incorporating explicit logarithmic scattering shifts derived from spectral data. The approach relies on inverse scattering, resolvent perturbations, and transfer-matrix analysis for random tridiagonal Lax matrices, along with finite- to infinite-volume comparison arguments. The findings provide a mathematically solid foundation for the soliton-gas picture in a classical integrable system under rough, random initial data and offer a precise link between microscopic Lax-data and macroscopic transport properties, with potential extensions to other integrable models and invariant measures.

Abstract

In this paper we consider the Toda lattice at thermal equilibrium, meaning that its variables and are independent Gaussian and Gamma random variables, respectively. We justify the notion from the physics literature that this model can be thought of as a dense collection of ``quasiparticles'' that act as solitons by, (i) precisely defining the locations of these quasiparticles; (ii) showing that local charges and currents for the Toda lattice are well-approximated by simple functions of the quasiparticle data; and (iii) proving an asymptotic scattering relation that governs the dynamics of the quasiparticle locations. Our arguments are based on analyzing properties about eigenvector entries of the Toda lattice's (random) Lax matrix, particularly, their rates of exponential decay and their evolutions under inverse scattering.

Paper Structure

This paper contains 41 sections, 46 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Toda Lattice
  4. Open Toda Lattice
  5. Lax Matrices, Locally Conserved Quantities, and Currents
  6. Thermal Equilibrium
  7. Localization Centers
  8. Asymptotic Scattering Relation
  9. Notation
  10. Outline
  11. Miscellaneous Preliminaries
  12. Periodic Toda Lattice
  13. Lax Matrix Eigenvector Evolution
  14. Resolvents
  15. Eigenvectors of Tridiagonal Matrices
  16. ...and 26 more sections

Key Result

Lemma 2.2

For any real numbers $t, t' \in \mathbb{R}_{\ge 0}$, we have $\mathop{\mathrm{eig}}\nolimits \bm{L} (t) = \mathop{\mathrm{eig}}\nolimits \bm{L}(t')$.

Theorems & Definitions (93)

  • Definition 2.1: Lax matrix
  • Lemma 2.2: LEIFMPL
  • Definition 2.3: Local charges and currents
  • Definition 2.4: Open thermal equilibrium
  • Proposition 2.5
  • Definition 2.6: Localization centers
  • Lemma 2.7
  • Proposition 2.9
  • Proposition 2.10
  • Theorem 2.11: Asymptotic scattering relation
  • ...and 83 more