Memory kernel effects

In this equation g(t) represents the retarded effect of the frictional force, and /(f) is an external force including the random force from the solvent molecules. We see, in contrast to the simple Langevin equation with a constant friction coefficient, that the friction force at a given time t depends on all previous velocities along the trajectory. The friction force is no longer local in time and does not depend on the current velocity alone. The time-dependent friction coefficient is therefore also referred to as a memory kernel . A short-time expansion of the velocity correlation function based on the GLE gives (fcfiT/M)( 1 — (g/M)t2/(2r) + ), where r is the decay time of g(t), and it therefore does not have a discontinuous first derivative at t = 0. The discussion of the properties of the GLE is most easily accomplished by using so-called linear response theory, which forms the theoretical basis for the equation and is a powerful method that allows us to determine non-equilibrium transport coefficients from equilibrium properties of the systems. A discussion of this is, however, beyond the scope of this book. 

When dealing with a nonlinear case, even when a large time scale separation between the system of interest and thermal bath is available, a new preparation effect takes place related to the fact that the memory kernel depends on excitation. 

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. 

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... 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

The procedure above has not in any sense derived the macroscopic relaxation equations only some formal conditions have been stated under which the structures of the microscopic and macroscopic equations become the same. One crucial point, which certainly deserves further comment, is the physical basis of the Markov approximation. This approximation removes the memory effects from (5.5) so that the structures of the microscopic and macroscopic equations become similar. For this approximation to be useful, the memory kernel must decay much more rapidly than the density fields. The projected time evolution will guarantee that this is the case, provided these fields decay much more slowly than other variables in the system. 

M. Desouter-Lecomte, J. Lievin, Memory kernels and effective Flamiltonians from time dependent methods. 1. Predissociation with a curve crossing, J. Chem. Phys. 107... 

As described in Sec. 5.1.4, memory functions generally consist of the fast and slow portions the fast portion is due to the rapidly decaying binary collision contributions, whereas the slow portion stems from correlated collisional effects. Hence, a starting point of our argument here is that the memory kernel K(fc, t) for molecular systems also consists of its fast and slow portions ... 

The resonances determined by u l) are shifted and damped by the memory kernel K k, t). Here we analyze the collective excitations in our model liquid by solving the full GLE which incorporates the damping effect. As a model for the memory kernel we employ a simple exponential model described in Sec. 5.2.3. 

Longitudinal current spectra. Here we also employ a simple exponential model for the memory kernel described in Sec. 5.2.3 to account for the damping effect on the dynamics. Figures 5.16 and 5.17 exhibit the results based on this theory along with the MD simulation data performed on the same system. The results are reported in the... 

In Fig. 5.1 we plot the Ifont velocity vs rx for different values of y. The saturation effect at y/D(y - l)/ro is observed. Note that when y increases, the front velocity also increases due to the faster decay of the memory kernel tail. 

Recently we have constructed a complete second-order QDT (CS-QDT), in which all excessive approximations, except that of weak system-bath interaction, are removed [38]. Besides two forms of CS-QDT corresponding to the memory-kernel COP [Eq. (1.2)] and the time-local POP [Eq. (1.3)] formulations, respectively, we have also constructed a novel CS-QDT that is particularly suitable for studying the effects of correlated non-Markovian dissipation and external time-dependent field driving. This paper constitutes a review of the three nonequivalent CS-QDT formulations [38] from both theoretical and numerical aspects. Concrete comparisons will be carried out in connection with the exact results for driven Brownian oscillator systems, so that sensible comments on various forms of CS-QDT can be reached. Note that QDT shall describe not only the evolution of p(t), but also the reduced thermal equilibrium system as p t oo) = peq(7 )-... 

Here, can be easily evaluated via Eq. (3.7), but with TZs there being replaced by TZu [cf. Eq. (B.12b)] that may be considered as the Markovian dissipation superoperator. This statement may be supported by the arguments that the Markovian approximation amounts to the following two conditions (i) The bath correlation time is short compared with the reduced system dynamics (ii) The correlated effects of driving and dissipation can be neglected so that the Green s function G(t r) in the memory kernel can be replaced by its field-free counterpart Gs t t) = In this case, Eq. (B.9) reduces to... 

We suppose that the effective Hamiltonian is known. Let us first recall how it is directly related to the spectroscopical and dynamical observables [24,25]. Since the role of effective Hamiltonians in both line profiles and dynamics is already well documented the reader is referred to some review on this wide subject (see, e.g.. Refs. [16-18,26]). The reports [18] and [26] contain numerous references inside and outside chemical physics. Reference [17] is a review of time-dependent effective Hamiltonians. Earlier application can be found in references [27] and [28]. Memory kernels are discussed in references [14,15,18] with references to irreversible statistical mechanics. Here we briefly review the subject for introducing the basic concepts and the notations. In the second part of this section we will present corrections to the dynamics for taking into account the dependence on energy of the effective Hamiltonian. 

Note that the term open system refers here to exchange of energy and phase with the environment, as the number of particles is conserved throughout. The reduced density matrix p t) evolves coherently under the influence of the nuclear Hamiltonian, Hnuc, and the non-adiahatic effects enter the equation via the dissipative Liouvillian superoperator jSfn- The latter is also termed memory kernel , as it contains information about the entire history of the environmental evolution and its interaction with the system. The definition of the memory kernel is by no means unique nor straightforward. One possible solution is to start from the microscopic Hamiltonian of the total system, eqn (1). Using the projector formalism, it is possible to separate the evolution of the system, i.e., the... 

The first term in the sum on the right side of Eq. (548) describes the instantaneous response of the system to the displacements from thermal equilibrium. This response is characterized by the frequencies f2yi . The second term in the strm on the right side of Eq. (548) describes memory effects. More specifically, it relates the displacements from eqrtihbritrm at time t to earher values of these displacements through the memory kernels IKyi (f - t ), which are commonly called memory functiorrs. The last term on the right side of Eq. (548), denoted by Iy(f), is a sottrce term describing effects due to the initial preparation of the system. 

The vertices couple density fluctuations of different wavelengths and thereby capture the cage effect in dense fluids [2], It thus enters the theory as a nonlinear feedback mechanism where density fluctuations slow down because of increased friction, and where the friction (precisely the time integral over the memory kernel that dominates the long-time collective friction coefficient) increases because of slow density fluctuations. MCT is a first-principles approach as the vertices are calculated from the microscopic interactions ... 

Since the complications due to solvent structure have already been discussed, the remainder of this chapter is mainly devoted to a discussion of the complications introduced into the theory of reaction rates when the collision of solvent molecules does not lead to a complete loss of memory of the molecules about their former velocity. Nevertheless, while such effects are undoubtedly important over some time scale, the differences noted by Kapral and co-workers [37, 285, 286] between the rate kernel for reaction estimated from the diffusion and reaction Green s function and their extended analysis were rather small over times of 10 ps or more (see Chap. 8, Sect. 3.3 and Fig. 40). At this stage, it is a moot point whether the correlation of solvent velocity before collision with that after collision has a significant and experimentally measurable effect on the rate of reaction. The time scale of the loss of velocity correlation is typically less than 1 ps, while even rapid recombination of radicals formed in close proximity to each other occurs over times of 10 ps or more (see Chap. 6, Sect. 3.3). 

They may be obtained by means of the matrix IET but only together with the kernel E(f) = F(t) specified by its Laplace transformation (3.244), which is concentration-independent. However, from the more general point of view, Eqs. (3.707) are an implementation of the memory function formalism in chemical kinetics. The form of these equations shows the essentially non-Markovian character of the reversible reactions in solution the kernel holds the memory effect, and the convolution integrals entail the prehistoric evolution of the process. In the framework of ordinary chemical kinetics S(/j = d(t), so that the system (3.707) acquires the purely differential form. In fact, this is possible only in the limit when the reaction is entirely under kinetic control. 

Out of the detailed mathematical aspects, some of them summarized in this section, there is a more general physical concept at the heart of the theory of error bounds. It is a fact that the memory function formalism provides in a natural manner a framework by which Ae short-time behavior, via the kernel of the integral equations, makes its effects felt in the long-time tail. The mathematical apparatus of continued fractions can adequately describe memory effects, and this explains the central role of this tool in the theory of relaxation. 

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