Transport in perturbed classical integrable systems: The pinned Toda chain

Abstract

Nonequilibrium and thermal transport properties of the Toda chain, a prototype of classically integrable system, subject to additional (nonintegrable) terms are considered. In particular, we study via equilibrium and nonequilibrium simulations, the Toda lattice with a power-law pinning potential, recently analyzed by Lebowitz and Scaramazza [arXiv:1801.07153]. We show that, according to general expectations, even the case with quadratic pinning is genuinely non-integrable, as demonstrated by computing the Lyapunov exponents, and displays normal (diffusive) conductivity for very long chains. However, the model has unexpected dynamical features and displays strong finite-size effects and slow decay of correlations to be traced back to the propagation of soliton-like excitations, weakly affected by the harmonic pinning potential. Some novel results on current correlations for the standard integrable Toda model are also reported.

Transport Properties of the Classical Toda Chain: Effect of a Pinning Potential

“…This was then attributed to the fact that solitons are the main energy carriers in this system and the quadratic pinning potential affects them rather weakly. In the present study, we confirm the finding in [31] but propose a somewhat different understanding for the slow crossover. We show that at low temperatures and small system sizes, the harmonically pinned Toda chain, in fact, behaves like a pinned harmonic chain, and more so at strong pinning.…”

“…It has been observed in a few recent studies that depending on the form of the integrability-breaking term and other parameters such as temperature, the crossover can occur at extremely large system sizes. Following some preliminary results by two of the present authors [30], this was further investigated in [31] which demonstrated that the crossover to diffusive transport in the quadratically pinned Toda chain occurs at very large length scales. This was then attributed to the fact that solitons are the main energy carriers in this system and the quadratic pinning potential affects them rather weakly.…”

Anomalous Heat Transport in Classical Many-Body Systems: Overview and Perspectives

“…The special role played by the quadratic pinning potential (3) has been revealed also by a recent study of heat transport in the Toda chain [55]. It is worth recalling that the unpinned Toda chain is an integrable Hamiltonian model of the form (1) with V(x) = e −x + x − 1; in this model heat transport is ballistic due to the finite-speed propagation of solitons (rather than phonons, as in the harmonic chain).…”

“…When the pinning term (3) is included, the Toda chain becomes chaotic, as one can easily see by measuring the spectrum of Lyapunov exponents [55]. Despite this, not only the energy but also the "center of mass"…”

Equilibrium time-correlation functions of the long-range interacting Fermi–Pasta–Ulam model

“…To conclude, let us also mention that the non monotonic behaviour of λ max with a control parameter α at fixed energy density e has been observed also for the shortrange model obtained by adding to the Toda Hamiltonian a term proportional to i |q i | α (see [55,56,57]). In this case however, contrary to what observed here for the long-range FPU chain, such non monotonicity (again with a relative minimum for α = 2) disappears for increasing values of e at fixed N (see Fig.…”