Non-equilibrium BBGKY Hierarchy from the Redfield Equation

TL;DR

This paper derives a BBGKY-like hierarchy from the non-equilibrium Redfield equation and introduces two approximation methods to truncate and solve it efficiently, demonstrated on small fermionic systems.

Contribution

The paper presents a novel BBGKY-like hierarchy from the Redfield equation and develops two approximation techniques for its solution, improving computational efficiency for non-equilibrium quantum systems.

Findings

Methods produce consistent results for small systems (N=4).

Eigenvalue and linear system dimensions are significantly reduced compared to existing methods.

Potential for extension to larger systems and higher levels of approximation.

Abstract

A BBGKY-like hierarchy is derived from the non-equilibrium Redfield equation. Two further approximations are introduced and each can be used to truncate and solve the hierarchy. In the first approximation such a truncation is performed by replacing two-particle Green's functions (GFs) in the hierarchy by their values at equilibrium. The second method is developed based on the cluster expansion, which constructs two-particle GFs from one-particle GFs and neglects the correlation part. A non-equilibrium Wick's Theorem is proved to provide a basis for this non-equilibrium cluster expansion. Using those two approximations, our method of solving the Redfield equation, for instance, of an N-site chain of interacting spinless fermions, involves an eigenvalue problem with dimension $2^{N}$ and a linear system with dimension $N^2$ in the first case, and a nonlinear equation with dimension $N^2$ in the second case, which can be solved iteratively via a sequence of $N^2$ linear systems. Other currently available direct methods correspond to a linear system or an eigenvalue system with dimension $4^N$ plus an eigenvalue system with dimension $2^N$. As a test of the methods, for small systems with size N=4, results are found to be consistent with results made available by other direct methods. Although not discussed here, extending both methods to their next levels is straightforward. This indicates a promising potential for this BBGKY-like approach of non-equilibrium kinetic equations.