Adaptive Numerical Solution of Kadanoff-Baym Equations

TL;DR

This paper introduces an adaptive time-stepping algorithm for solving Kadanoff-Baym equations, significantly reducing computational effort and enabling long-time simulations of complex quantum and stochastic systems.

Contribution

It presents a novel adaptive numerical scheme with variable step size and order for Kadanoff-Baym equations, improving efficiency over fixed-step methods.

Findings

Significant reduction in the number of time-steps needed

Enhanced ability to simulate long-time dynamics

Open-source Julia implementation available

Abstract

A time-stepping scheme with adaptivity in both the step size and the integration order is presented in the context of non-equilibrium dynamics described via Kadanoff-Baym equations. The accuracy and effectiveness of the algorithm are analysed by obtaining numerical solutions of exactly solvable models. We find a significant reduction in the number of time-steps compared to fixed-step methods. Due to the at least quadratic scaling of Kadanoff-Baym equations, reducing the amount of steps can dramatically increase the accessible integration time, opening the door for the study of long-time dynamics in interacting systems. A selection of illustrative examples is provided, among them interacting and open quantum systems as well as classical stochastic processes. An open-source implementation of our algorithm in the scientific-computing language Julia is made available.