Parabolic spectral density

As an illustrative example of the procedures described, we consider the parabolic spectral density, normalized to 1, given by... 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

Room temperature simulations at the classical level yielded values of all AX° and quantities. Quantal results for G were also obtained from path integral simulations (in conjunction with Eq. 41) or semiclassical results based on spectral densities from the classical simulations [36]. The calculated classical G and 7. values and the AG° value were found to conform well to Eq. 112, and Gi t]) and Gf t]) functions were close to parabolic in form. Due to systematic deficiencies in the aqueous solvent model (a nonpolarizable force field and finite Coulombic cut-off, ca. 10 A), corrections to / were made on the basis of reference dielectric continuum models, and then G was corrected accordingly on the basis of Eq. 112. 

Another way is to calculate the power spectral density (PSD) associated with the anodic current fluctuations. Let us consider a Poissonian series of birth and death events (birth frequency A, death frequency x), with a parabolic growth law between birth and death. This situation can be modeled as follows ... 

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