Auxiliary density operators

The dynamics quantities in the HEOM formalism are a set of well-defined auxiliary density operators (ADOs), (p (r) = 0,1,...,, in which Po(t) = p t) is just the reduced system density operator. The hierarchy construction resolves not just system-bath coupling strengths but, more importantly, also memory time (l/yo) scales. ... 

Here, Spt denotes the field-dressed dissipation contribution and will be treated later in terms of a set of auxiliary operators [cf. Eq. (B.4)] that couple to the primary reduced density operator. C t) is the deterministic Liouvillian in the presence of external field [cf. Eq. (2.2)]. The field-free dissipation TZq assumes the second-order cumulant result and is given by [38]... 

The CODDE formulation of CS-QDT [Eqs. (2.21)] couples between p t) and a set of auxiliary operators K t) m > 0 that describe the effects of correlated driving and dissipation. The field-free dissipation action, TZs [Eq. (2.18)], can be evaluated relatively easily in terms of the causality spectral function Cab ( ) without going through the parameterization procedure of Eq. (2.24). The latter is required only for the correlated driving-dissipation effects described by the auxiliary operators. Methods of evaluating both the reduced dynamics p(t) and the reduced canonical density operator Peq(T) will be discussed in Sec. 3. 

Note that for the DBO system the correlated effects of driving and dissipation in POP-CS-QDT can be evaluated in terms of the effective driving field correction Se t) (cf. Sec. 4.2). For a general system one should evaluate the auxiliary operators Km dynamics, which are however uncoupled with the reduced density operator in POP-CS-QDT. This feature allows the POP-CS-QDT equations of motion to be integrated individually. However, the results obtained may be numerically reliable only for bounded or finite-level systems. For a general unbounded system the uncoupled evolution of the dynamical variables Km may introduce some uncontrollable... 

To construct the CODDE [Eq. (2.21)] in which all auxiliary operators couple with the reduced density operator p t), let us consider the following second-order approximation,... 

One point which has not been addressed in the example of the time-independent harmonic oscillator is the non-perturbative treatment of the time dependence in the system Hamiltonians. Both the TL and the TNL non-Markovian theories employ auxiliary operators or density matrices, respectively, and can be applied in strongly driven systems [29,32]. This point will be shown to be very important in the examples for the molecular wires under the influence of strong laser fields. 

Calculated Power and Efficiency. The simplified analytical models of thermionic characteristics have been used to project the converter efficiency and power density with the barrier index as a parameter. These projections are shown in Figures 8 and 9 as functions of the emitter temperature. The dashed lines in these two figures are for a constant current density of 10 A/cm. If the current density is adjusted to maximize the efficiency at each temperature, the calculated performance is represented by the solid lines. Typical present generation themionlc converters operate with Vg near 2.0. Ignited mode converters in laboratory experiments have demonstrated practical operation with 1.85 < Vg < 1.90. Other laboratory devices with auxiliary sources of ions and/or special electrode surfaces have achieved Vj < 1.5, but usually not under practical operating conditions. 

Cell design is also an important factor to improve the performance of SOFC stack/module [15-18]. Use of small diameter SOFC may also give opportunity to reduce operating temperature by increasing the volumetric power density [19]. Thus, they are expected to accelerate the progress of SOFC systems which can be applied to portable devices and auxiliary power units for automobile. 

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