Anharmonic interaction as random field for thermal transport in FPU-$β$ lattice

TL;DR

This work introduces an open quantum framework for thermal transport in a one-dimensional FPU-beta lattice by treating localized bosons (LBs) as transport carriers and mapping anharmonic interactions to a random field via the Hubbard-Stratonovich transformation. Using path-integral and coherent-state formalisms, the authors derive stochastic differential equations for LB amplitudes and compute quantum averages through Monte Carlo sampling of Gaussian fields, linking LB dynamics to steady-state currents. They show that anharmonicity nontrivially modulates thermal transport: it can trap LBs and suppress current, yet also increase LB numbers and enhance transport, yielding a non-monotonic dependence of thermal conductivity on the anharmonic strength, with clear finite-size effects captured. The results simultaneously provide a framework to dissect hopping versus diffusion contributions to heat flow and point to future work extending the approach to larger systems and to uncover the origin of possible power-law divergences in κ for the FPU chain.

Abstract

We present an open quantum theory for the thermal transport in the Fermi-Pasta-Ulam-$β$(FPU-$β$) lattice. In the theory, local bosons(LBs) are introduced as carriers for the transport. The LBs are stimulated by individual atoms in the lattice, which are different from the phonons that are collective motions of the atoms. The LBs move in the FPU chain and are governed by a set of stochastic differential equations(SDEs). The anharmonic interaction between the atoms in the lattice is transformed to a random field by the Hubbard-Stratonovich transformation, and has been implemented in the set of SDEs. By solving the set of SDEs at the steady state, we study the influence of the anharmonic interaction on the thermal transport. Results show that the anharmonic interaction decreases the thermal current by trapping the LBs on the lattice sites, as well as increase the thermal current by enhancing the amount of the LBs for the transport. The competition between these two mechanisms makes the thermal conductivity of the lattice dependent on the anharmonic interaction non-monotonically. The finite size effect of the thermal conductivity has also been captured by the theory.

Figures (4)

• Figure 1: Linearized distribution of the LB numbers in the system of $L=10$. The atoms of the system are indexed from 1 to 10. The atoms of the reservoirs have also been indicated. The data for various C overlap.
• Figure 2: Influence of the anharmonic coefficient $C$ on the transport of the LBs. (a) The anharmonic interaction blocks the motion of the LBs in the system and tends to trap the LBs on the lattice sites. (b) The anharmonic interaction weakens the correlations between the atoms.
• Figure 3: Thermal conductivity $\kappa$ as functions of the anharmonic coefficient $C$ and the length $L$ of the system. (a)$\kappa$ is dependent on the anharmonic coefficient $C$ non-monotonically. (b)The finite size effect of $\kappa$ has been observed.
• Figure 4: Contributions of the hopping and the diffusing of the LBs in the system to the thermal conductivity. (a)The thermal conductivity is dominated by the hopping of the LBs in the system. (b) The thermal conductivity attributed to the diffusing of the LBs in the system increases with the anharmonic coefficient $C$.