Lieb-Robinson bounds in classical oscillating lattice systems
Ian Koot, C. J. F. van de Ven
TL;DR
The paper addresses the propagation of information in infinite classical oscillator lattices with inhomogeneous parameters by proving a robust classical Lieb-Robinson bound that bounds the Poisson bracket of time-evolved and local observables in terms of a space–time decay D(X,Y) and an exponential in time. The authors develop a comprehensive mathematical framework for inhomogeneous phase spaces and finite-volume Hamiltonians, derive a second-order variational system whose Dyson expansions yield explicit decay bounds, and use these to establish a global infinite-volume dynamics on the commutative resolvent algebra, a classical analog of resolvent algebras. The main contributions are (i) a rigorous LR-type locality bound for classical oscillators with general interaction structures, (ii) explicit Dyson-series-based estimates for the Jacobians of the flow, and (iii) the construction of a well-behaved global C*-dynamical system in the infinite-volume limit, enabling rigorous study of locality and dynamics in broad classical settings with potential applications to transport and thermalization in solids.
Abstract
The aim of this paper is two-fold. First, we prove the existence of Lieb-Robinson bounds for classical particle systems describing harmonic oscillators interacting with arbitrarily many neighbors, both on lattices and on more general structures. Second, we prove the existence of a global dynamical system on the commutative resolvent algebra, a C*-algebra of bounded continuous functions on an infinite dimensional vector space, which serves as the classical analog of the Buchholz--Grundling resolvent algebra.