Validity of Markov Assumption

With explicit expressions like Eq. (7.59) and Eq. (7.61), one can now directly assess the validity of the Markov assumption (Assumption 2a above): ` the bath correlation function $C_{kl}(\tau)$ is strongly peaked around $\tau=0$ with a peak width $\delta \tau \ll \gamma^{-1}$, where $\gamma$ is a typical rate of change of $\tilde{\rho}(t')$.' For example, for $T=0$, $\gamma=2\pi \rho(\Omega)$, and within the model $\rho(\omega) = {2}\alpha \omega_c^{1-s} \omega^{s}e^{-{\omega}/{\omega_c}}$, Eq.(7.54), one has $\delta \tau \sim \omega_c^{-1}$, cf. Eq.(7.61). This would mean

$$\omega_c^{-1}4\pi \alpha \omega_c^{1-s} \Omega^{s}e^{-{\Omega}/{\omega_c}} \ll 1$$

$$4\pi \alpha \left(\Omega/\omega_c\right)^s e^{-{\Omega}/{\omega_c}} \ll 1,$$ (62)

which is fulfilled for large $\omega_c$ ( $\Omega/{\omega_c}\lesssim 1$), $s>0$, and small $\alpha$. The condition of small $\alpha$ is consistent with the Born approximation (perturbation theory in the coupling to the bath).