Effective Velocities in the Toda Lattice

2025/03/14 by Aggarwal, Amol · 2 citations
#Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.11407

Abstract

In this paper we consider the Toda lattice (\boldsymbolp(t); \boldsymbolq(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. This model can be thought of a dense collection of many ``quasiparticles'' that act as solitons. We establish a law of large numbers for the trajectory of these quasiparticles, showing that they travel with approximately constant velocities, which are explicit. Our proof is based on a direct analysis of the asymptotic scattering relation, an equation (proven in previous work of the author) that approximately governs the dynamics of quasiparticles locations. This makes use of a regularization argument that essentially linearizes this relation, together with concentration estimates for the Toda lattice's (random) Lax matrix.

Cited by

• Stochastic dynamics of quasiparticles in the hard rod gas
• Large deviations at the edge for 1D gases and tridiagonal random matrices at high temperature

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