Markovian limit

In non-Markovian theory, the off-diagonal elements of G cannot be looked upon as transfer rates between two or more discrete eigenfrequen-cies of the system as they are functions of the continuous variable to. The transfer rates concept is only acceptable in the Markovian limit of the theory (t tc) when co-dependence is eliminated. To obtain this limit, we must first pass to differential formulation of non-Markovian theory and after that let t -> oo. In the literature there is complete unanimity on how the transition from integral to differential formalism can be carried out correctly. According to Eq. (4.28) the integrand in Eq. (4.26) may be written as... 

Much simpler is the situation in the Markovian limit. The spectrum is governed by the co-independent operator... 

If e t) corresponds to sufficiently slow rates of interruption/modulation v, the spectrum of Ffco) is much narrower than the interval of change of G( ) around , the resonance frequency of the system. Then Ffco) can be replaced by 8 co), so that the spectral width of G plays no role in determining R, and we may as well replace G(a> mf) by a spectrally finite, flat white-noise) reservoir, that is, take the Markovian limit. The result is that Eq. (4.49) coincides with the GR rate, Eq. (4.35) (Figure 4.4a) ... 

In the Markovian limit [ (f) = 0S(t), oj(t) = Kramers equation (4.160). It is of importance to note that the long-time behavior of the solution does not necessarily approach the Markovian result. 

Here we followed the notation of Ref. [52] for <9+(t, tofu,) and used Eqs. (5) and (6) for the correlation functions a(t) and b(t). In the case of non-Markovian treatments 6>+ (t, ojija,) has to be calculated at every moment in time instead of only once as in the Markovian limit. The Markovian limit of these expressions can easily be obtained by moving the upper integration boundary in Eq. (29) to infinity [16]. In both, the Markovian as well as the non-Markovian case, the evaluation of the matrix elements (30) does not contain any numerical integration anymore. 

Fig. 1 Population dynamics of a damped harmonic oscillator. The populations of the ground state (n = 0) up to the third excited state (n = 3) are shown while initially all population is in the third excited state. The parameters are o-v = tao/2, / = 0.544ivo, and (3 = l/ivo. The results for the TNL theory are shown by the solid curve, those for the TL approach by the dotted curve and those of the Markovian limit by the dashed curve. (Reproduced from Ref. [29]. Copyright 2004, American Institute of Physics.)...

As follows from Eq. (3.13) in the Markovian limit (r = oo), the Stern-Volmer constant of irreversible quenching Ko is identical to the stationary ionization constant k. This is not the case for the reversible quenching whose Stern-Volmer constant (3.85) can be essentially reduced, due to backward transfer to the excited state. As seen from Eq. (3.86), icq k only at fast exciplex recombination, when... 

In the case of irreversible energy transfer (kb = 0), this result reduces to the conventional one, given by Eq. (3.22) with /c0 = ka. However, when energy transfer is reversible (kh > 0), it comes to the stationary (Markovian) limit only at... 

In view of Eq. (3.245), the Markovian limit of this rate, k s (oo) = kdF — 1 /rexc), is the complex quantity as well as the rate constant (3.150) at x = oo. This means that the stationary dissociation of the exciplex does not exist. Unlike feeff (f), its DET analog, fcdis(f) from Eq. (3.253), diverges after changing the sign, thus indicating that the rate description of the delayed fluorescence is not appropriate at long times. 

The recombination term of the reduced integral equation can be transformed into its differential analog as in Eq. (3.278). In the Markovian limit this differential equation reduces to the conventional one with the rate constant kr = fes(oo) [48] ... 

After the differentiation expression (3.422) with respect to time, one can substitute the result into Eq. (3.418b). Then in the Markovian limit t — oo the whole set (3.418) is transformed to... 

Some numerical examples are shown in Fig. 7a. The nonexponential part of 4>v,b(t) for short times is readily seen and leads to a delayed onset of the exponential decay. The latter represents the Markovian limit (straight line in the semi-log plot). A time interval of approximately 2 x rc is required to establish the asymptotic dephasing rate 1/T2. The purely exponential case, rc = 0, is fictitious and shown in the figure only for comparison. 

Figure 9 describes the results obtained by applying the CFP. The most remarkable feature of these results, is the increase in the rate k as the parameter g increases. A further remarkable finding is that for g -> 0 (Markovian limit) the accurate value of Larson and Kostin is attained within a precision of a few percent. 

Equation (72) coincides with the analytical result of Grote and Hynes. In the Markovian limit, v(/) = 2yS(t),... 

The Markovian limit corresponds to Xi - oo. By solving Eq. (73) to the lowest order in Xf it is easy to see that in this case the non-Markovian dynamics leads to an enhancement of the decay rate k. In the notation of ref. 22b, an approximate expression for Eq. (69) may then be written as... 

In the Markovian limit, Zg t) = 2yjj (t) and jj(A) = y. co is then given by Eq. (2.24), and Eq. (6.16b) becomes the well-known Markovian multidimensional result.If moreover QS, = Qfl, that is, the nonreactive subsystem is not affected by the state of the reactive mode, this becomes the Kramers one-dimensional result. 

The simplest problem of this kind is the escape from a truncated harmonic well in the low-friction Markovian limit. This problem was treated by Ben Jacob et al. The model is defined by... 

Figure 1. Different regimes of the system-environment coupling. Here a represents the coupling strength between a quantum system and its environment and k is the environment relaxation rate. Note that, in the Markovian limit, the system decouples from the environment altogether for a/n — 0 unless a2/ remain finite.

In this form we may now consider the Markovian limit 7 relation... 

Figure 5. Probability of the excited state evolution of an initially excited state due to faked-continuum dissipation. In this figure, we consider the limit T = 0 only, (a) Detuning-dominated, (b) weak-damping, (c) strong-damping, and (d) Markovian limit. Due to the finite values of all parameters, even the case of the Markovian limit has a non exponential decay within a very short initial slip, although it is hardly visible here.

The Markovian limit. The correct memory-less Markovian limit is the following... 

Another problem is that memory kernels seem to be delicate entities. Erroneous kernels can destroy the physical sense of the time evolution of an initially acceptable density matrix. We do not have a general criterion to help us judge from the Master Equation with memory if the evolution is acceptable. In the Markovian limit, we know that the Lindblad form is certain to preserve the physical interpretation. It is a challenge for the theory of irreversibility in quantum systems to find such a criterion when memory effects are important. 

This equation can also be used for obtaining the master equation in the Markovian limit. In that case, the exponential decays faster than the state p evolves p(s) may therefore be replaced with pit) and taken out of the integral. Assuming still Re(f n) < 0, the integral may be performed explicitly and the master equation is... 

Equation (8.54) is a stochastic equation of motion similarto Eq. (8.13). However, we see an important difference Eq. (8.54) is an integro-differential equation in which the term yx of Eq. (8.13) is replaced by the integral /J drZ t — r)x(r). At the same time the relationship between the random force R t) and the damping, Eq. (8.20), is now replaced by (8.59). Equation (8.54) is in fact the non-Markovian generalization of Eq. (8.13), where the effect of the thermal environment on the system is not instantaneous but characterized by a memory—at time t it depends on the past interactions between them. These past interactions are important during a memory time, given by the lifetime of the memory kernel Z t). The Markovian limit is obtained when this kernel is instantaneous... 

Coming back to the non-Markovian equations (8.61) and (8.62), and their Markovian limiting form obtained when Z(Z) satisfies Eq. (8.60), we next seek to quantify the properties of the thermal environment that will determine its Markovian or non-Markovian nature. 

This result is akin to Eq. (13.22), which relates the vibrational relaxation rate of a single harmonic oscillator of frequency m to the Fourier transform of the force autocorrelation function at that frequency. In the Markovian limit, where (cf. Eq. (8.60)) Z(Z) = 2y<5(Z), we recover Eq. (14.78). 

The Marcus theory, as described above, is a transition state theory (TST, see Section 14.3) by which the rate of an electron transfer process (in both the adiabatic and nonadiabatic limits) is assumed to be determined by the probability to reach a subset of solvent configurations defined by a certain value of the reaction coordinate. The rate expressions (16.50) for adiabatic, and (16.59) or (16.51) for nonadiabatic electron transfer were obtained by making the TST assumptions that (1) the probability to reach transition state configuration(s) is thermal, and (2) once the reaction coordinate reaches its transition state value, the electron transfer reaction proceeds to completion. Both assumptions rely on the supposition that the overall reaction is slow relative to the thermal relaxation of the nuclear environment. We have seen in Sections 14.4.2 and 14.4.4 that the breakdown of this picture leads to dynamic solvent effects, that in the Markovian limit can be characterized by a friction coefficient y The rate is proportional to y in the low friction, y 0, limit where assumption (1) breaks down, and varies like y when y oo and assumption (2) does. What stands in common to these situations is that in these opposing limits the solvent affects dynamically the reaction rate. Solvent effects in TST appear only through its effect on the free energy surface of the reactant subspace. 

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