Universal Markovian reduction of Brownian particle dynamics

TL;DR

This paper presents a method to convert non-Markovian Brownian particle dynamics into Markovian form by transforming the bath variables into a chain of effective modes, enabling practical Markovian approximations.

Contribution

The authors introduce a unique orthogonal transformation based on the bath spectral density that systematically reduces non-Markovian dynamics to Markovian form with a simple recurrence relation.

Findings

Convergence of the residual spectral densities is rapid for various bath types.

The method reproduces the Rubin model's quasi-Ohmic dissipation limit.

Numerical results demonstrate practical effectiveness for quantum dissipative systems.

Abstract

Non-Markovian processes can often be turned Markovian by enlarging the set of variables. Here we show, by an explicit construction, how this can be done for the dynamics of a Brownian particle obeying the generalized Langevin equation. Given an arbitrary bath spectral density $J_{0}$, we introduce an orthogonal transformation of the bath variables into effective modes, leading stepwise to a semi-infinite chain with nearest-neighbor interactions. The transformation is uniquely determined by $J_{0}$ and defines a sequence $\{J_{n}\}_{n\in\mathbb{N}}$ of residual spectral densities describing the interaction of the terminal chain mode, at each step, with the remaining bath. We derive a simple, one-term recurrence relation for this sequence, and show that its limit is the quasi-Ohmic expression provided by the Rubin model of dissipation. Numerical calculations show that, irrespective of the details of $J_{0}$, convergence is fast enough to be useful in practice for an effective Markovian reduction of quantum dissipative dynamics.