Excitation spectrum of the Toda lattice for finite temperatures

Kinetic theory of quantum and classical Toda lattices

“…Thus for the purposes of considering local equilibrium dynamics in the Toda lattice, it suffices to restrict attention to phononic degrees of freedom; this will be justified more carefully below. The phonon dispersion relation is given by [26] ∆E ph p = ωe −β cl p , ∆P ph p = 2π ∞ p dp ρ p ,…”

Solitons and Nondissipative Diffusion

“…of the phonon mode l. On the average, this process leads to a frequency "drift"v l ; in the case of zero-wavelength phonons, we have computed this average drift and found it to be identical with the BA frequency [13,14]. In addition, if the densities fluctuate according to ͗dn a ͑t͒dn a 0 ͑t 0 ͒͘ ͗͑dn a ͒ 2 ͘d͑a 2 a 0 ͒d͑t 2 t 0 ͒ ,…”

“…The statistical distribution of the two types of objects has been determined both numerically [7] and theoretically [8]. The fundamental (zero temperature) dynamics is governed by the inverse scattering theory [9]; the thermodynamics is described by the classical limit of the Bethe ansatz (BA) [10][11][12]; there is as yet no consensus regarding the dynamical significance of quasienergies derived within the BA formalism [13,14]. An analysis of dynamical correlation spectra in terms of soliton phenomenology [15] (i.e., by exploiting the availability of exact equilibrium statistical mechanical properties in phonon and soliton "sectors" of phase space, in order to formulate an approximate theory of dynamical correlations) yields two essential results: (i) it shows that nondissipative (soliton and phonon) diffusion exists and is observable in a nonlinear lattice, and (ii) it allows us to identify unambiguously a soliton peak in the kinetic energy autocorrelation spectra; this assignment allows us in turn to give a precise dynamical meaning to the equilibrium BA solution 0031-9007͞99͞83(12)͞2293(4)$15.00 (in particular, the soliton dispersion relation derived within the BA).…”

“…(7) the static structure factor S rr ͑q͒ is given by the product v qnq , where v q andn q are, respectively, the harmonic frequency and phonon occupation number of the mode q;ṽ q is the thermally renormalized frequency obtained in the BA solution [13,14]; the phonon occupation number has been identified as the number of holes in the BA solution. The total phonon phase diffusion constant G qp͞a 0.120 is a sum of both soliton (0.094) and phonon (0.026) contributions.…”

“…[8], where the low-temperature T 1͞3 dependence of the total soliton density is derived). The soliton velocity follows from a literal interpretation of the BA quasimomenta h͑k͒ and quasienergies e͑k͒ above the Fermi level as soliton momenta [8,21] and energies, respectively; the resulting slope ≠e͞≠h defines the soliton velocity at finite temperatures [13]. This thermally renormalized velocity is systematically higher than the bare soliton velocity.…”