Grabert

Poliak E, Grabert H and Hanggi P 1989 Theory of activated rate processes for arbitrary frequency dependent friction solution of the turnover problem J. Chem. Phys. 91 4073... 

H. Grabert and M. H. Devoret, eds.. Single Charge Tunneling. Coulomb Blockade Phenomena in Nanostructures, Plenum Press, New York, 1992. 

The influence functional theory, as it was formulated by Feyman and Vernon, relies on the additional assumption concerning factorization of the total (system and bath) density matrix in the past. Without this assumption the theory requires a triple path integral, with one thermal integration over the imaginary time axis [Grabert et al. 1988]. 

The last term in (5.44) accounts for quantum corrections to the classical escape rate (5.31) [Wolynes 1981 Melnikov and Meshkov 1983 Grabert and Weiss 1984 Dakhnovskii and Ovchinnikov 1985]. 

The instanton action behaves in accord with the scaling predictions and is independent of coq. Loosely speaking, the frequency ojq is replaced by the friction coefficient rj. Grabert et al. [1984b] have studied the energy loss A tunneling process and found that is... 

Moving downward to the molecular level, a number of lines of research flowed from Onsager s seminal work on the reciprocal relations. The symmetry rule was extended to cases of mixed parity by Casimir [24], and to nonlinear transport by Grabert et al. [25] Onsager, in his second paper [10], expressed the linear transport coefficient as an equilibrium average of the product of the present and future macrostates. Nowadays, this is called a time correlation function, and the expression is called Green-Kubo theory [26-30]. 

The asymmetric part of the transport matrix gives zero contribution to the scalar product and so does not contribute to the steady-state rate of first entropy production [7]. This was also observed by Casimir [24] and by Grabert et al. [25], Eq. (17). 

These relations are the same as the parity rules obeyed by the second derivative of the second entropy, Eqs. (94) and (95). This effectively is the nonlinear version of Casimir s [24] generalization to the case of mixed parity of Onsager s reciprocal relation [10] for the linear transport coefficients, Eq. (55). The nonlinear result was also asserted by Grabert et al., (Eq. (2.5) of Ref. 25), following the assertion of Onsager s regression hypothesis with a state-dependent transport matrix. 

E. Hershkovitz and E. Poliak, Multidimensional generalization of the Pollak-Grabert-Hanggi turnover theory for activated rate processes, J. Chem. Phys. 106, 7678 (1997). 

Beck, T. L. Marchioro, T. L., The quantum potential distribution theorem, in Path Integrals from meV to MeV Tutzing 1992, Grabert, H. Inomata, A. Schulman, L. Weiss, U., Eds., World Scientific Singapore, 1993, pp. 238-243... 

Grabert H, Devoret MH (eds) (1992) Single charge tunneling. Coulomb blockade phenomena in nanostructures. Plenum, New York... 

Vorontsov 11, Kovalevsky AY, Chen Y-S, GraberT, Gembicky M, Novozhilova IV, Omary MA, Coppens P (2005) Shedding light on the structure of a photoinduced transient excimer by time-resolved diffraction. Phys Rev Lett 94 193003-1-193003-4... 

Grabert E, Kloos R (1999) Der SensorBSB der zweiten Generation als schnelle und ein-fache Alternativmethode zm- herkommlichen BSEj-Bestimmimg. Kommunalwirtschaft 1999/4 176-180... 

Poliak and Eckhardt have shown that the QTST expression for the rate (Eq. 52) may be analyzed within a semiclassical context. The result is though not very good at very low temperatures, it does not reduce to the low temperature ImF result. The most recent and best resultthus far is the recent theory of Ankerhold and Grabert," who study in detail the semiclassical limit of the time evolution of the density matrix and extract from it the semiclassical rate. Application to the symmetric one dimensional Eckart barrier gives very good results. It remains to be seen how their theory works for asymmetric and dissipative systems. 

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