On the Propagation of a Perturbation in an Anharmonic System

Abstract

We give a not trivial upper bound on the velocity of disturbances in an infinitely extended anharmonic system at thermal equilibrium. The proof is achieved by combining a control on the non equilibrium dynamics with an explicit use of the state invariance with respect to the time evolution.

Classical Lieb-Robinson Bound for Estimating Equilibration Timescales of Isolated Quantum Systems

“…The above strategy of proof could also be applied directly to the l-nested commutator (16), without resorting to (17), by iteratively counting the number of nonvanishing terms in . Surprisingly, because of excessive multiple-counting of terms, the resulting bound is less tight than (21) and entails divergences in the subsequent calculations.…”

“…At this point we substitute the upper bound (21), derived in Sec. II of this Supplemental Material, into (5) to obtain {li|j} a {li|j} K l1 K l2 · · · K lj H n−j ≤ p j (n, l)(2 O ) l (kgr) j H n−j .…”

“…which quantifies the effect of a perturbation of the amplitude x k at time 0 on the amplitude x j at a later time t. Upper bounds B jk Λ jk are known for fairly general Hamiltonian systems as classical analogs of Lieb-Robinson bounds [20][21][22][23][24]. For network nodes j and k separated by a large graph distance, Λ jk is small at early times, but will usually become non-negligible at later times.…”

Dynamics of Infinite Classical Anharmonic Crystals

“…However, the results in these papers do not exclude that the initial data could change very much their local energy as time goes by. A better control of this growth, which has been proved in [8] in the case σ 1 ≥ 2σ 2 − 1, is the principal task of the present investigation. As we will see in the next section, this generalization needs a different approach to the problem.…”

“…In this respect, in [8] a not trivial bound on the velocity of propagation of a perturbation has been given, provided that the system be in an equilibrium state. In the present case, in which we require only a positive σ 1 , noticing that the existence of thermodynamic equilibria (i.e., the infinite volume Gibbs states) has been proved in [2], we are able to extend the bound of [8] to this more general situation. We refer to the next section for a precise formulation.…”