Correlation functions in finite memory-time reservoir theory

Abstract

Articles you may be interested inTime-dependent exchange-correlation current density functionals with memory Time correlation functions for finite systems using secondquantized molecular time scale generalized Langevin equation theory: Effect of rotating wave approximationInteraction of a small system S with a large reservoir R amounts to thermal relaxation of the reduced system density operator Ps (t). The presence of the reservoir enters the equation of motion for p s (t) through the reservoir correlation functions fkl ( 'T) (defined in the text), which decay to zero for 'T ..... 00 on a time scale 'Tc' Commonly, this 'Tc is much smaller than the inverse relaxation constants for the time evolution of Ps (t). Then a series of approximations can be made, which lead to a Markovian equation of motion for Ps (t). In this paper the assumption of a small reservoir correlation time is removed. The equation of motion for p s (t) is solved, and it appears that the memory effect, due to 'Tc <#0, can be incorporated in a frequency dependence of the relaxation operator f(w). Subsequently, (unequal-time) quantum correlation functions of two system operators are considered, where explicit expressions for (the Laplace transform of) the correlation functions are obtained. They involve again the relaxation operator f(w), which accounts for the time regression. Additionally it is found that an initial-correlation operator Y(w) arises, as a consequence of the fact that the equal-time correlation functions do not factorize as Ps (t) times the reservoir density operator.It is pointed out that the frequency dependence of f (w) and the occurrence of a nonzero Y (w ) both arise as a result of'Tc <#0, and should therefore be treated on an equal footing. Explicit evaluation of f (w) and Y (w) shows that their matrix elements can be expressed entirely in ikl (w ), just as in the Markov approximation. Hence no essential complications appear if one should go beyond the limits of a small reservoir correlation time 'T c .

Memory-induced extra resonances of adsorbates

“…2 give rise to this line, and its line shape is given by Eq. (6). In a zero memory-time approximation this line vanishes.…”

“…Its profile is given by Eq. (6). As a last remark we mention that the detailed shape of the line at COO+COD depends crucially on the adopted model for the phonon dispersion relation.…”

“…this line [(b) and (c)] can only occur for | co -coo I < ^D, since otherwise the the phonon frequency would become larger than COD-This can be shown explicitly from Eq. (6). If we work out the real part and use Reg(z) =0 for \z\ > 1, we find I(co) =0 for | co -cool > COD-…”

“…In order to investigate this problem quantitatively, we have developed a finite-memory-time relaxation theory, 6 with no restriction on the time scales. Subsequently, we have applied this theory to the evaluation of the absorption line shape, with the formal result 7…”

Phonon Relaxation and Line Shapes of Adsorbates

“…In order to resolve these problems, we have developed a generalized finite-memory-time reservoir theory which puts no restrictions on the decay time of f(r) and the order of magnitude of the decay constants and the relevant frequencies (Arnoldus and George 1987). It appeared that the relaxation in the equation of motion could again be expressed entirely in terms of f(w), but now f(w) appears for all w, rather than only at the adsorbate resonances.…”