Dissipation model for extended environment
A unified model for "Diffusion Localization and Dissipation" (DLD), optionally termed "Diffusion with Local Dissipation", has been introduced for the study of "Quantal Brownian Motion" (QBM) in dynamical disorder. It can be regarded as a generalization of the familiar Caldeira-Leggett model. formula_1 formula_2 formula_3 where formula_4 denotes the dynamical coordinate of the formula_5 scatterer or bath mode. formula_6 is the interaction potential, and formula_7 are coupling constants. The spectral characterization of the bath is analogous to that of the Caldeira-Leggett model: formula_8 i.e. the oscillators that appear in the Hamiltonian are distributed uniformly  over space, and in each location have the same spectral distribution formula_9. Optionally the environment is characterized by the power spectrum of the fluctuations formula_10, which is determined by formula_9 and by the assumed interaction formula_12. See examples. The model can be used to describes the dynamics of a Brownian particle in an Ohmic environment whose fluctuations are uncorrelated in space. This should be contrasted with the Zwanzig-Caldeira-Leggett model, where the induced fluctuating force is assumed to be uniform in space (see figure). At high temperatures the propagator possesses a Markovian property and one can write down an  equivalent Master equation. Unlike the case of the Zwanzig-Caldeira-Leggett model, genuine quantum mechanical effects manifest themselves due to the disordered nature of the environment. Using the Wigner picture of the dynamics one can distinguish between two different mechanisms for destruction of coherence: scattering and smearing. The analysis of dephasing can be extended to the low temperature regime by using a semiclassical strategy. In this context the dephasing rate SP formula can be derived. Various results can be derived for ballistic, chaotic, diffusive, and both ergodic and non-ergodic motion.