Fermi-Pasta-Ulamβlattice: Peierls equation and anomalous heat conductivity

Abstract

The Peierls equation is considered for the Fermi-Pasta-Ulam β lattice. Explicit form of the linearized collision operator is obtained. Using this form the decay rate of the normal mode energy as a function of wave vector k is estimated to be proportional to k 5/3 . This leads to the t −3/5 long time behavior of the current correlation function, and, therefore, to the divergent coefficient of heat conductivity. These results are in good agreement with the results of recent computer simulations. Compared to the results obtained through the mode coupling theory our estimations give the same k dependence of the decay rate but a different temperature dependence. Using our estimations we argue that adding a harmonic on-site potential to the Fermi-Pasta-Ulam β lattice may lead to finite heat conductivity in this model.

Energy Transport in Weakly Anharmonic Chains

“…Thus one expects that the following limit exists, 15) and that the limit Wigner function W (k, t) evolves according to the spatially homogeneous Boltzmann equation…”

“…As we learned later on, Pereverzev [15] has already applied kinetic theory to the FPU β-lattice, for which the potential energy depends only on the relative displacements. For this model he obtains the non-perturbative solution and argues, based on the linearized transport equation, that the energy current correlation has a power law decay as t −3/5 .…”

Anomalous kinetics and transport from 1D self-consistent mode-coupling theory

“…This estimate was later criticized as inconsistent in [12], where renormalization group arguments were instead shown to yield α = 1/3. Nevertheless, the 2/5 value has been later derived both from a kinetic-theory calculation for the quartic Fermi-PastaUlam (FPU-β) model [16] and from a solution of the MCT by means of an ad hoc Ansatz [17]. It was thereby conjectured [17] that 2/5 is found for a purely longitudinal dynamics, while a crossover towards 1/3 is to be observed only in the presence of a coupling to transversal motion.…”

Understanding Divergent Thermal Conductivity in Single Polythiophene Chains Using Green–Kubo Modal Analysis and Sonification

“…This was done largely to examine how small the higher order terms (e.g., cubic, quartic) must be in order to preserve the phenomenon. In addition, the many questions it spawned have been the subject of experts in non-linear dynamics for decades and tremendous progress has been made [15][16][17] . Excellent reviews of that progress are available elsewhere 14,[18][19][20] and one of the prevailing theories for understanding the origin of the phenomenon is that of mode-coupling theory, which was pioneered by Lepri, Livi, and Politi 19 , and the notion that the reduced dimensionality is the origin of the anomalous thermal transport 19 .…”