Energy Transport in Weakly Anharmonic Chains

Abstract

Abstract.We investigate the energy transport in a one-dimensional lattice of oscillators with a harmonic nearest neighbor coupling and a harmonic plus quartic on-site potential. As numerically observed for particular coupling parameters before, and confirmed by our study, such chains satisfy Fourier's law: a chain of length N coupled to thermal reservoirs at both ends has an average steady state energy current proportional to 1/N. On the theoretical level we employ the Peierls transport equation for phonons and note that beyond a mere exchange of labels it admits nondegenerate phonon collisions. These collisions are responsible for a finite heat conductivity. The predictions of kinetic theory are compared with molecular dynamics simulations. In the range of weak anharmonicity, respectively low temperatures, reasonable agreement is observed.

Non-Equilibrium Steady States for Chains of Four Rotors

“…6 the time-dependence of p 2 , p (1) 2 and p (2) 2 (slightly shifted for better readability). 2 Clearly, the oscillations of p (1) 2 are much smaller than those of p 2 , and we barely perceive the oscillations of p (2) 2 , since they are smaller than the random fluctuations.…”

“…A similar argument applies to f = −p 4 W L w C . Therefore, applying Proposition 3.16, we find that on the set A 2 ∪ A 3 , the functions U 1 (x) and U 4 (x) are O c (1), and that…”

“…In order to see a net effect, we need to average again. We consider now the observable f = p (1) 2 , and apply the same procedure. Instead of averaging the righthand side of (2.6) in one step, we first deal only with the terms of order −1 in p 2 , by introducing…”

“…In this region, p 2 and p 3 are close to each other, and are both large in absolute value. It is then easy to find a counterterm for p 1 …”

“…See for example [2,3,7,13,15,19,21] for chains of rotors and [1][2][3][4]16,20,21] for chains of oscillators. From a rigorous point of view however, the mere existence of an invariant measure is not evident, and has been proved only in special cases.…”