Memory kernels

Zhu S-B, Lee J, Robinson G Wand Lin S H 1989 Theoretical study of memory kernel and velocity correlation function for condensed phase isomerization. I. Memory kernel J. Chem. Phys. 90 6335-9... 

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)... 

Exercise. The Ornstein-Uhlenbeck process (IV.3.10), (IV.3.11) satisfies the generalized Langevin equation with memory kernel ... 

There exists another prescription to extend the hydrodynamical modes to intermediate wavenumbers which provides similar results for dense fluids. This was done by Kirkpatrick [10], who replaced the transport coefficients appearing in the generalized hydrodynamics by their wavenumber and frequency-dependent analogs. He used the standard projection operator technique to derive generalized hydrodynamic equations for the equilibrium time correlation functions in a hard-sphere fluid. In the short-time approximation the frequency dependence of the memory kernel vanishes. The final result is a... 

Finally, note that the method used by Kadanoff and Swift is a very general scheme. For example, the expression of ris[1H is similar to the expression of viscosity derived later by Geszti [39]. In addition, the projection operator technique used in their study is the same used to derive the relaxation equation [20], and the expression of Ly and Uy are equivalent to the elements of the frequency and memory kernel matrices, respectively. 

The frequency matrix Qy and the memory function matrix Ty, in the relaxation equation are equivalent to the Liouville operator matrix Ly and the Uy matrix, respectively. The later two matrices were introduced by Kadanoff and Swift [37] (see Section V). Thus the frequency matrix can be identified with the static variables (the wavenumber-dependent thermodynamic quantities) associated with the nondissipative part, and the memory kernel matrix can be identified with the transport coefficients associated with the dissipative part. 

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. 

Finally, note that the relaxation equation [Eq. (76)] is usually written in terms of the hydrodynamic modes. In many problems of chemical interest, nonhydrodynamic modes such as intramolecular vibration, play an important role [50]. Presence of such coupling creates an extra channel for dissipation. Thus, the memory kernel, T, gets renormalized and acquires an additional frequency-dependent term [16, 43]. 

The above expression has been used by Leutheusser [34] and Kirkpatrick [30] in the study of liquid-glass transition. Leutheusser [34] has derived the expression of the dynamic structure factor from the nonlinear equation of motion for a damped oscillator. In their expression they refer to the memory kernel as the dynamic longitudinal viscosity. 

Equation (210) when compared with the viscoelastic model of the dynamic structure factor we can identify the memory kernel in the viscoelastic model, which is written as... 

Thus we note that the memory kernel has a short-time and a long-time part. It is the long-time part which is not present in the viscoelastic model, becomes important in the supercooled-liquid-near-glass transition, and gives rise to the long-time tail of the dynamic structure factor. 

The final expression for the long-time part of the memory kernel is thus given by... 

If the full memory kernel is replaced in Eq. (210), then F (q, t = 0) should become identical to Aq. However, in the expression of T (q, t = 0) the term (q2/f m) comes with a prefactor 1, whereas in Aq it comes with a prefactor 3. Thus, a factor of (2q2/f m) is found to be missing in the former. As has been discussed by Balucani and Zoppi [16], this extra term arises from the cou-... 

In the supercooled liquid, the important part of the memory kernel is its long-time part, r (q, t). The recollision term contains the contribution from the hydrodynamic modes. As discussed by many authors [3, 30, 34], among all the hydrodynamic modes the density fluctuation is found to yield the main contribution to the memory kernel in the supercooled fluid regime. 

The above equation is an integrodifferential equation that has an unusual structure. Here (co2)1 2 is the frequency of the free oscillator and y is the damping constant. The fourth term on the left-hand side of Eq. (226) has the form of the memory kernel, and its strength is controlled by the dimensionless coupling constant X which contains the contribution from the vertex function. 

The memory kernel for the p-region arising from its coupling to the electronic motions in the s-region is given in the IP by... 

As an illustration of the numerical treatment, the instantaneous dissipation due to fast electronic motions was constructed from the Lindblad expression in the treatment of CO/Cu(001), from decay and transition rates. The delayed dissipation, present for slow atomic vibrations of the medium, was given in terms of a memory kernel in the integrodifferential equation, calculated to second order in the coupling of the adsorbate and its environment. 

By considering the action of the operators given in Eq. (42) on functions to its right, the memory kernel operator may be reduced to a memory function. 

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. 

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