Solving the Peierls-Boltzmann transport equation with matrix product states

TL;DR

This paper introduces a matrix product state approach to efficiently solve the high-dimensional Peierls-Boltzmann transport equation, significantly reducing computational costs in phonon transport simulations.

Contribution

It demonstrates that MPS with optimized index ordering can effectively reduce dimensionality and computational cost in solving the PBE across different transport regimes.

Findings

MPS achieves high fidelity with reference solutions using a compression ratio of 10^{-3}

Sublinear scaling of computational cost with grid points in real and modal spaces

Order of magnitude reduction in computational time compared to traditional FVM

Abstract

The Peierls-Boltzmann transport equation (PBE), which governs non-equilibrium phonon transport, suffers from the curse of dimensionality due to its high-dimensional phase space including both real and modal spaces. We explore the use of matrix product states (MPS) for numerical simulation of the PBE. We show that an MPS configuration based on scattering events combined with a dimensionless form of the solution can drastically increase the locality of correlations between tensors in the MPS representation, enhancing its effectiveness in dimension reduction. We further examine the effects of index ordering in an MPS and find that the highest locality is achieved when tensor chains associated with both real and modal spaces are connected from the coarsest grid to each other in the center of the MPS. Using this optimal configuration and a solver inspired by the density matrix renormalization group, we solve the PBE discretized by a finite volume method (FVM). The solution is obtained for crystalline silicon across ballistic, quasi-ballistic, and diffusive transport regimes. An MPS truncated to the compression ratio of $10^{-3}$ suffices to reproduce reference solutions with high fidelity. The computational cost scales sublinearly with the number of grid points in both real and modal spaces, achieving roughly an order of magnitude reduction in computational time compared to the FVM with sparse matrix operation.