Nonintegrability and the Fourier heat conduction law

Abstract

We study in momentum-conserving systems, how nonintegrable dynamics may affect thermal transport properties. As illustrating examples, two one-dimensional (1D) diatomic chains, representing 1D fluids and lattices, respectively, are numerically investigated. In both models, the two species of atoms are assigned two different masses and are arranged alternatively. The systems are nonintegrable unless the mass ratio is one. We find that when the mass ratio is slightly different from one, the heat conductivity may keep significantly unchanged over a certain range of the system size and as the mass ratio tends to one, this range may expand rapidly. These results establish a new connection between the macroscopic thermal transport properties and the underlying dynamics.

Anomalous Heat Transport in One Dimensional Systems: A Description Using Non-local Fractional-Type Diffusion Equation

“…(11), while for systems with AHT this has the scaling form in Eq. (12) and one again has super-diffusive growth of the MSD [21], now defined as…”

“…Several theoretical as well as large scale numerical studies have been performed on different mathematical model systems to understand the necessary and sufficient conditions needed in the microscopic description to validate Fourier's law at the macroscopic level [2][3][4]. Surprisingly, these studies suggest that Fourier's law is probably not valid in many one-dimensional systems and one finds that the thermal conductivity κ diverges with system size N as κ ∼ N α where 0 < α < 1 [2][3][4][5][6][7][8][9][10][11][12]. This is referred to as anomalous heat transport (AHT).…”

From Thermal Rectifiers to Thermoelectric Devices

“…As a consequence, ZT exhibits a rapid, liner growth with the system size. While the Fourier-like regime might be an intermediate (in the system size) regime, followed by an asymptotic regime of anomalous thermal conductivity κ ∼ Λ 1/3 [83,52], the range of validity of such regime may expand rapidly as an integrable limit is approached [42]. We point out that it is a priori not excluded that there exist models where the long-time limit t → ∞ and the thermodynamical limit Λ → ∞ do not commute when computing the Drude weights.…”

“…This conclusion, however, might be not correct as shown in [42], where the heat conductivity of the one-dimensional diatomic hard-point gas model was studied. As shown in Fig.…”

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

“…Similar features were observed in the trapped hard rod system [29], where it was observed that the system become chaotic after a characteristic time scale but fails to thermalize even at extremely large times. For momentum-conserving systems surprising features (e.g apparent diffusive transport) has been reported when a system is taken slightly out of integrability, for example arXiv:1812.11770v1 [cond-mat.stat-mech] 31 Dec 2018 in the Fermi-Pasta-Ulam chain at low temperatures [15 and 16] and the alternate mass hard particle gas, at a mass ratio close to one [12].…”