Observing golden-mean universality class in the scaling of thermal transport

Abstract

We address the issue of whether the golden mean ψ = ( √ 5 + 1)/2 ≃ 1.618 universality class, as predicted by several theoretical models, can be observed in the dynamical scaling of thermal transport. Remarkably, we show estimate with unprecedented precision, that ψ appears to be the scaling exponent of heat mode correlation in a purely quartic anharmonic chain. This observation seems somewhat deviation from the previous expectation and we explain it by the unusual slow decay of the cross-correlation between heat and sound modes. Whenever the cubic anharmonicity is included, this cross-correlation is gradually died out and another universality class with scaling exponent γ = 5/3, as commonly predicted by theories, seems recovered. However, this recovery is accompanied by two interesting phase transition processes characterized by a change of symmetry of the potential and a clear variation of the dynamic structure factor, respectively. Due to these transitions, an additional exponent close to γ ≃ 1.580 emerges. All these evidences suggest that, to gain a full prediction of the scaling of thermal transport, more ingredients should be taken into account.Introduction.-Despite decades of intensive studies , our understanding of thermal transport in onedimensional (1D) systems is still scarce. Macroscopically, such transport is described by an empirical law, i.e., the Fourier's law: J = −κ∇T with J the heat current, ∇T the spatial temperature gradient, and κ a material constant named as thermal conductivity. Nevertheless, now it has been generally realized that Fourier's law is not always valid; instead, an anomalous transport will be shown in most cases [1][2][3]. In particular, in the momentum-conserving systems which are of particular interest, this anomaly is mainly characterized by [27]: α describing the divergence of κ with increasing space size L as κ ∼ L α with 0 < α ≤ 1 (normal transport, α = 0) [6, 8-11, 13-15, 17-19, 24, 26], and γ giving the space(m)-time(t) scaling of energy/heat correlation ρ E/Q (m, t) by t

Anomalous energy diffusion in two-dimensional nonlinear lattices

“…For the 1D nonlinear lattices i.e. N y = 1, our calculated ρ E (i, t) is qualitatively consistent with the previous results 15,27,[36][37][38][39][40]42 . In particular, on the 1D quartic and FPU-β lattices, there are two observable right/left-moving side peaks along both sides of the central heat mode.…”

“…Most majority of the studies on energy diffusion in nonlinear lattices 15,27,[36][37][38][39][40] has been devoted to 1D systems. So far, no investigation of energy diffusion or of momentum diffusion has been made in 2D nonlinear lattices.…”

“…Furthermore, if the profile of energy fluctuation correlation function ρ E belongs to the Lévy-stable distribution, the MSD will spread faster than the normal diffusion and exhibits the superdiffusion with β = 3 − γ given by the Lévy walk theory 45,46 . Anomalous heat conduction has been verified in the 1D momentum-conserving nonlinear lattices 15,27,[36][37][38][39][40] such as the FPU-β and purely quartic lattices. This relation 34 of energy diffusion to heat transport has been quantitatively verified in 1D nonlinear with symmetrical potential.…”

One-dimensional superdiffusive heat propagation induced by optical phonon-phonon interactions

“…In the ϕ 4 lattice, a final diffusive spread (γ = 2) will be reached. In the FPU-β chain, eventually γ will be between γ = 1.5 and γ ≃ 1.618 [49][50][51]. In comparison, in the AHsH model γ tends to stay at a value around 1.782.…”

Equilibration of sinusoidal modulation of temperature in linear and nonlinear chains

“…The Fourier law of thermal conductivity states that for macroscopic bodies, the heat flux is proportional to the temperature gradient with a proportionality coefficient κ, referred to as thermal conductivity. In a number of theoretical works [19][20][21][22][23][24][25][26][27][28][29][30][31], it was shown that in one-dimensional structures, Fourier's law does not work, in the sense that κ depends not only on the material, but also on the dimensions of the conductor, in particular, on its length L according to the power law κ ∼ L α , where the exponent is in the range 0 ≤ α ≤ 1. The case α = 1 corresponds to ballistic thermal conductivity, and for α = 0 we have normal, diffusive thermal conductivity obeying the Fourier law.…”