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



Tobias Brandes 2004-02-18