Nonuniversal heat conduction of one-dimensional lattices

Abstract

For one-dimensional nonlinear lattices with momentum conserving interparticle interactions, intensive studies have suggested that the heat conductivity κ diverges with the system size L as κ~L(α) and the value of α is universal. But in the Fermi-Pasta-Ulam-β lattices with nearest-neighbor (NN) and next-nearest-neighbor (NNN) coupling, we find that α strongly depends on γ, the ratio of the NNN coupling to the NN coupling. The correlation between the γ-dependent heat conduction behavior and the in-band discrete breathers is also analyzed.

Crossover between different universality classes: Scaling for thermal transport in one dimension

“…It has been well proposed [12][13][14][15] Lévy walks description, in our opinion, the walks may not always be the Lévy type as the phonon dispersions could be more complicated. In fact, our quite recent simulations of a FPU-β model chain with both nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings having a phonon dispersion of a high order form [25] [the Hamiltonian is H = L k p 2 k /2+V (q k+1 −q k )+rV (q k+2 −q k ) with r the ratio of the NNN to NN couplings and the potential V (ξ) = ξ 2 /2 + βξ 4 /4; note that here the dispersion depends on the ratio r], indeed support a more complicated density shape which has not yet been covered by the existing progress of Lévy walks theory [12] (see Fig. 6, the central ranges imply some localizations, which may be due to the effects of the peculiar dispersions along with nonlinearity enabling us to excite the intraband DBs [25]; note that here the Lévy walks scaling might not be satisfied, so we are unable to show the exact scaling exponent γ).…”

“…The underlying picture is suggested to be understood by phonons performing various kinds of continuous-time random walks (in most cases, be the Lévy walks but not always), probably induced by the peculiar phonon dispersions along with nonlinearity. The results and suggested mechanisms may provide insights into controlling the transport of heat in some 1D materials.Transport in one dimension has, for a long time, been realized to be anomalous in most cases [1,2], with signatures of a universal power-law scaling of transport coefficients, among which the heat transport has been extensively investigated in the recent decades, both by various theoretical techniques, such as the renormalization group [3], mode coupling [4,5] or cascade [6][7][8], nonlinear fluctuating hydrodynamics [9][10][11], and Lévy walks [12][13][14][15]; and also by computer simulations [16][17][18][19][20][21][22][23][24][25][26]. For all studied cases two main scaling exponents have been given the most focus, i.e., α describing the divergence of heat conductivity with space size L as L α and γ characterizing the space(x)-time(t) scaling of heat spreading density ρ(x, t) as t −1/γ ρ(t −1/γ x, t).…”

“…Unfortunately, however, depending on the focused system's different parameter regimes, different theoretical models have been employed, and different predictions have been suggested. Thus, the universality classes of both scaling exponents and their relationship [27,28] remain controversial: (i) for α, two classes: α = 1/3 [3,9,[14][15][16]19] and α = 1/2 [4,5,9] have been reported; however, the universality has been doubted [25,26] and a Fibonacci sequence of α values converging on α * = (3 − √ 5)/2 (≃ 0.382) [6][7][8] has been suggested; (ii) for γ, (a) E-mail: phyxiongdx@fzu.edu.cn two universality classes, γ = 5/3 [9-11, 13-15, 20-23] and γ = 3/2 [9-11, 22, 23] have been predicted recently. The discussion of the latter scaling exponent γ is currently very hot [9-15, 20-23, 29, 30] because it involves more detailed space and time information [31], thus it can present a very detailed prediction for heat transport.…”

Solitons as candidates for energy carriers in Fermi-Pasta-Ulam lattices

“…Breathers can thus only influence the energy transport by scattering the energy carriers [48][49][50][51][52].…”

Discrete breathers: possible effects on heat transport

“…In this work we briefly introduce our recent results [14][15][16] on the possible roles of DBs in thermal transport. We will mainly employ two one-dimensional (1D) lattice models to describe our viewpoints.…”