Memory kernel theory

In this generalized oscillator equation, the frequency is related to the restoring force acting on a particle and Q is a friction constant. The key quantity of the theory is the memory kernel mq(l — t ), which involves higher order correlation functions and hence needs to be approximated. The memory kernel is expanded as a power series in terms of S(q, t)... 

The equations of motion in the extended hydrodynamic theory (Section IV) are obtained from the relaxation equation, where the correlation function is normalized. As mentioned before, in the extended hydrodynamic theory, the memory kernel matrix is considered to be independent of frequency thus the transport coefficients are replaced by their corresponding Enskog values. 

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. 

In this chapter a theory of encounter between pairs of particles has been develojjed and applied to a wide range of problems, which illustrates the utility and ease of application of the theory. The theoretical foundations of the model require further work—in particular an appropriate one-dimensional memory kernel should be develojjed that gives a kinematically weighted friction... 

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. 

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). 

Although it turned out that a number of essential features concerning dynamics of molecular liquids can be well captured by the theory of the previous subsection (see Secs. 3 and 5), an intense investigation through experimental, theoretical, and molecular-dynamics simulation studies for simple liquids has revealed that the microscopic processes underlying various time-dependent phenomena cannot be fully accounted for by a simplified memory-function approach [18, 19, 20]. In particular, the assumption that the decay of memory kernels is ruled by a simple exponential-type relaxation must be significantly revised in view of the results of the kinetic framework developed for dense liquids (see Sec. 5.1.4). This motivated us to further improve the theory for dynamics of polyatomic fluids. 

So far we have employed the exponential model for the memory kernel appearing in the GLE for the density correlation functions. In [60] the model based on the mode-coupling theory described in Sec. 5.2.4is applied to the calculation of the longitudinal current spectra of the same diatomic liquid as discussed here. It is found that the essential features of the results remained the same as far as the collective dynamics is concerned. It is also demonstrated that the results are in fair agreement with those determined from the molecular dynamics simulation. 

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 principle the memory kernel in the mode-coupling equation contains contributions from three particle correlations, however, for all systems studied so far in computer simulations, these only slightly modified the predictions of the theory and helped improve agreement between simulation and theory [31]. Also, there has been an extension of the theory taking chain connectivity into account [32] which improved agreement with the simulations of the bead-spring model, but it remains to be seen whether an application of this theory to the two models presented here can account for their strongly different dynamic behavior. 

Abstract. This article reviews from both theoretical and numerical aspects three non-equivalent complete second-order formulations of quantum dissipation theory, in which both the reduced dynamics and the initial canonical thermal equilibrium are properly treated in the weak system-bath coupling limit. Two of these formulations are rather familiar as the time-local and the memory-kernel prescriptions, while another which can be termed as correlated driving-dissipation equations of motion will be shown to have the combined merits of the two conventional formulations. By exploiting the exact solutions to the driven Brownian oscillator system, we demonstrate that the time-local and correlated driving-dissipation equations of motion formulations are usually better than their memory-kernel counterparts, in terms of their applicability to a broad range of system-bath coupling, non-Markovian, and temperature parameters. Numerical algorithms are detailed for an efficient evaluation of both the reduced canonical thermal equilibrium state and the non-Markovian evolution at any temperature, in the presence of arbitrary time-dependent external fields. 

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 ... 

The mode coupling approximation for m (0 yields a set of equations that needs to be solved self-consistently. Hereby the only input to the theory is the static equilibrium structure factor 5, that enters the memory kernel directly and via the direct correlation function that is given by the Ornstein-Zernicke expression = (1 - l/5,)/p, with p being the average density. In MCT, the dynamics of a fluid close to the glass transition is therefore completely determined by equilibrium quantities plus one time scale, here given by the short-time diffusion coefficient. The theory can thus make rather strong predictions as the only input, namely, the equilibrium structure factor, can often be calculated from the particle interactions, or even more directly can be taken from the simulations of the system whose dynamics is studied. 

Let us compare in detail the differential theory results with those obtained for rotational relaxation kinetics from the memory function formalism (integral theory). Using R(t) from Eq. (1.107) as a kernel of Eq. (1.71) we can see that in the low-density limit... 

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). 

The reaction rate in differential theories of bimolecular reactions is always the product of the reactant concentrations and the rate constant, does not matter whether the latter is truly the constant or the time-dependent quantity. In integral theories there are no such constants at all they give way to kernels (memory functions) of integral equations. However, there is a regular procedure that allows reduction the integral equations to differential equations under specificconditions [34,127]. This reduction can be carried out in full measure or partially, but the price for it should be well recognized. 

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. 

It is impossible to derive a general model that applies to all application domains and all computer systems." In theory, one can predict performance from first principles, but this would require a detailed understanding of every part of the computation and how its memory access (data communication) patterns interact with the computer s hardware and operating system. Except for small computational kernels, it is not feasible to acquire such understanding. In practice, a more satisfactory approach is to construct a fairly high level model using approximate functional forms for the amount of computation, load balance, and overheads. 

For polymeric fiuids, early Idnetic-theory workers (40) attempted to calculate the zero-shear-rate viscosity of dilute solutions by modeling the polymer molecules as elastic dumbbells. Later the constants in the Rivlin-Ericksen (17) expansion were obtained for dumbbells (41, 42) and other more complex models and only recently have the kernel functions in the memory integral expansions been obtained (43), This rapidly expanding field has been summarized recently in a monograph (44) here, too, molecular dynamics simulation may prove fhiitful (45),... 

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