Green function equilibrium approach

We will now develop the transport equations in L-space from the above Green functions. Following the Keldysh approach in //-space, the transport equations for non-equilibrium plasmas and radiation have been given by DuBois [29]. A similar transport equation for a system of ions may be found in Kwok [30], which is based on the Green function associated with ion positions. In a separate paper [31], we will derive the appropriate transport equations for the coupled system of electrons, ions, and electromagnetic fields. 

As noted above, the elastic Green function yields the elastic fields induced by a point load. The ideas here resemble those from the setting of electrodynamics where point sources of charge are the basis of the Green function approach. Recall from earlier that the equilibrium equations that govern the continuum are given by... 

The most comprehensive description of the tunneling problem is based either on a self-consistent solution of the Lippman-Schwinger equation [3] or on the non-equilibrium Green s function approach [4-8]. Inelastic effects within e.g. a molecule-surface interface can be included by considering multiple electron paths from the vacuum into the surface substrate [9], The current between two leads with the chemical potentials /ja and hb is given by the energy integral ... 

For simplicity, the QD and SM indices in the e-ph constants have been omitted in Eq. (118) however, the frequencies and e-ph constants are obviously different in the both subsystems. In the proposed description, we assume that equilibrium Green s functions of the semiconductor and the quantum dot are known. However, to find QD equilibrium Green s function in a time-dependent field is not an easy task because it is not even clear whether Dyson equations for SM and QD Keldysh functions exist for different types of fermions interacting with each other. This problem is rather complicated even for molecular wires [54], Thus, we expect this problem to be even more complicated for solar cell systems where the interaction with light makes the problem essentially time dependent. In this section, we prove that Dyson equations for nonequilibrium Green s functions do exist. In our description, we adopt a graduate approach to the problem introducing different approximations step by step. As the first and the easiest step, we consider only uncorrelated electrons. 

Moisture Sorption Isotherms. Green wood loses moisture to the atmosphere and approaches a moisture content designated as the equilibrium moisture content (EMC) for the particular atmospheric conditions. The EMC is a function of relative humidity, temperature, previous exposure history (hysteresis), species, and other miscellaneous factors. 

A computational approach was not only very successful in fullerene research, but advances in these studies have created the demand for the development of new theoretical methods. The last part of this chapter describes the application of the non-equilibrium Green s function formalism to the investigation of the current-voltage dependence of the fullerene molecule. This method can be also q>plied to a wide range of nanomolecular devices. 

The concentration dependence of ionic conductivity needs to be understood from a molecular viewpoint [4]. In this approach, a time correlation function representation of viscosity derived long ago by Green and Kubo was used along with a molecular-level description of the equilibrium correlation between the positions of ions and water... 

Viscosity. The molecular modeling techniques used for determining the other transport coefficients, viscosity and thermal conductivity, in addition to dif-fusivity that was discussed above, fall into two categories. In the first category, the coefficients are calculated from Green-Kubo relations, with the required correlation fimctions being evaluated with equilibrium simulations at the desired state point. Indeed, this was the approach followed traditionally. However, the calculation of correlation functions is known to yield large statistical uncertainties. This... 

Another approach to calculate thermal conductivity is equilibrium molecular dynamics (EMD) [125] that uses the Green-Kubo relation derived from linear response theory to extract thermal conductivity from heat current correlation functions. The thermal conductivity X is calculated by integrating the time autocorrelation function of the heat flux vector and is given by... 

This equilibrium MD approach uses current fluctuations to calculate the thermal conductivity k via the fluctuation dissipation theorem. The MD approach is used to compute the autocorrelation function of the heat flux, which is related to the thermal conductivity by the Green-Kubo formula given by ... 

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