Universal Markovian reduction of Brownian particle dynamics

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 J₀, 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 J₀ and defines a sequence \{J_n\}_{ninmathbb{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 J₀, convergence is fast enough to be useful in practice for an effective Markovian reduction of quantum dissipative dynamics.