Correlation functions memory function equation

The memory function equation for the time-correlation function of a dynamical operator Ut can be cast into the form of a continued fraction as was first pointed out by Mori.43 We prove this in a different way than Mori. In order to proceed it is necessary to define the set of memory functions K0 t),. .., Kn t). .., such that... 

Before discussing other results it is informative to first consider some correlation and memory functions obtained from a few simple models of rotational and translational motion in liquids. One might expect a fluid molecule to behave in some respects like a Brownian particle. That is, its actual motion is very erratic due to the rapidly varying forces and torques that other molecules exert on it. To a first approximation its motion might then be governed by the Langevin equations for a Brownian particle 61... 

The mean square torque is taken from computer experiments. Nevertheless, it could have been found from the infrared bandshapes. Likewise the integral in this expression can be found from the experimental spin rotation relaxation time, or it can be found directly from the computer experiment as it is here. The memory function equation can be solved for this memory. The corresponding angular momentum correlation function has the same form as v /(0 in Eq. (302) with... 

The parameter F mimics tire short time, microscopic dynamics, and depends on structural and hydrodynamic correlations. The memory function describes stress fiuctuations which become more sluggish together with density fiuctuations, because slow structural rearrangements dominate all quantities. A self consistent approximation closing the equations of motion is made mimicking (14a). In the... 

It is the presence of the non-zero elements mi2 which brings about a coupling of the memory function equations for the correlation functions with different m subscripts (but a fixed j superscript). It can be seen that the matrix of "memory constant" is diagonal whenf (y these symmetric diffusers, the equations for each m decouple and one recovers the well-known result ... 

As mentioned above, a KS-LCAO calculation adds one additional step to each iteration of a standard HF-LCAO calculation a quadrature to calculate the exchange and correlation functionals. The accuracy of such calculations therefore depends on the number of grid points used, and this has a memory resource implication. The Kohn-Sham equations are very similar to the HF-LCAO ones and most cases converge readily. 

In the impact approximation (tc = 0) this equation is identical to Eq. (1.21), angular momentum relaxation is exponential at any times and t = tj. In the non-Markovian approach there is always a difference between asymptotic decay time t and angular momentum correlation time tj defined in Eq. (1.74). In integral (memory function) theory Rotc is equal to 1/t j whereas in differential theory it is 1/t. We shall see that the difference between non-Markovian theories is not only in times but also in long-time relaxation kinetics, especially in dense media. 

For small wavevectors the test particle density is a nearly conserved variable and will vary slowly in time. The correlation function in the memory term in the above equation involves evolution, where this slow mode is projected... 

The set of equations (50) can be formally considered as generalized Langevin equations if the operator Fj(t) can be interpreted as a stochastic quantity in the statistical mechanical sense. If the memory function does not correlate different solute modes, namely, if Kjj =Sjj Kj, then a Langevin-type equation follows for each mode ... 

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)... 

Physically this description corresponds to putting an atom (mass M) in an external time-dependent harmonic potential (frequency co0). The potential relaxes exponentially in time (time constant l/x0) so that eventually the atom experiences only a frictional force. Compared with other models2 which have been proposed for neutron scattering calculation, the present model treats oscillatory and diffusive motions of an atom in terms of a single equation. Both types of motion are governed by the shape of the potential and the manner in which it decays. The model yields the same velocity auto-correlation function v /(r) as that obtained by Berne, Boon, and Rice2 using the memory function approach. 

There have been a number of attempts to calculate time-correlation functions on the basis of simple models. Notable among these is the non-Markovian kinetic equation, the memory function equation for time-correlation functions first derived by Zwanzig33 and studied in great detail by Berne et al.34 This approach is reviewed in this article. Its relation to other methods is pointed out and its applicability is extended to other areas. The results of this theory are compared with the results of molecular dynamics. 

Linear response theory is reviewed in Section II in order to establish contact between experiment and time-correlation functions. In Section III the memory function equation is derived and applied in Section IV to the calculation of time-correlation functions. Section V shows how time-correlation functions can be used to guess time-dependent distribution functions and similar methods are then applied in Section VI to the determination of time-correlation functions. In Section VII a succinct review is given of other exact and experimental calculations of time-correlation functions. 

Ku(t) is called the memory function/ and the equation for the time-correlation function that we derived is called the memory function equation.33,34,42 Note that the propagator in this equation contains the projection operator Pt. Further note that the memory function is an even function of the time,... 

Time-correlation functions Cu(t) obey the memory function equation... 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

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... 

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. 

Equation (228) is the normalized density correlation function in the Fourier frequency plane and has the same structure as Eq. (210), which is the density correlation function in the Laplace frequency plane. i/ (z) in Eq. (228) is the memory function in the Fourier frequency plane and can be identified as the dynamical longitudinal frequency. Equation (228) provides the expression of the density correlation function in terms of the longitudinal viscosity. On the other hand, t]l itself is dependent on the density correlation function [Eq. (229)]. Thus the density correlation function should be calculated self-consistently. To make the analysis simpler, the frequency and the time are scaled by (cu2)1 2 and (cu2)-1 2, respectively. As the initial guess for rjh the coupling constant X is considered to be weak. Thus rjt in zeroth order is... 

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 order to explain the non-Debye response (134) it is possible to use the memory function approach [22,23,31,266-268]. Thus, the normalized dipole correlation function k(f) (22) corresponding to a nonexponential dielectric relaxation process obeys the equation... 

One considers a particle interacting linearly with an environment constituted by an infinite number of independent harmonic oscillators in thermal equilibrium. The particle equation of motion, which can be derived exactly, takes the form of a generalized Langevin equation, in which the memory kernel and the correlation function of the random force are assigned well-defined microscopic expressions in terms of the bath operators. 

The Ohmic model memory kernel admits an infinitely short memory limit y(t) = 2y5(t), which is obtained by taking the limit a>c —> oo in the memory kernel y(t) = yG)ce c [this amounts to the use of the dissipation model as defined by Eq. (23) for any value of ]. Note that the corresponding limit must also be taken in the Langevin force correlation function (29). In this limit, Eq. (22) reduces to the nonretarded Langevin equation ... 

A second problem with the GME derived from the contraction over a Liouville equation, either classical or quantum, has to do with the correct evaluation of the memory kernel. Within the density perspective this memory kernel can be expressed in terms of correlation functions. If the linear response assumption is made, the two-time correlation function affords an exhaustive representation of the statistical process under study. In Section III.B we shall see with a simple quantum mechanical example, based on the Anderson localization, that the second-order approximation might lead to results conflicting with quantum mechanical coherence. 

In the case where the correlation function <3> (f) has the form of Eq. (148), with p fitting the condition 2 < p < 3, the generalized diffusion equation is irreducibly non-Markovian, thereby precluding any procedure to establish a Markov condition, which would be foreign to its nature. The source of this fundamental difficulty is that the density method converts the infinite memory of a non-Poisson renewal process into a different type of memory. The former type of memory is compatible with the occurrence of critical events resetting to zero the systems memory. The second type of memory, on the contrary, implies that the single trajectories, if they exist, are determined by their initial conditions. 

It is now well understood that this is not an approximation rather it is a way to force an equation with infinite memory to become compatible with Levy diffusion. The assumption (152) makes it possible for us to get rid of the time convolution nature of the generalized diffusion equation (133). At the same time, this key relation replaces the correlation function (t) with its second-order derivative and, as a consequence of Eq. (147), with the waiting time distribution /( ). This fact is very important. In fact, any Liouville-like approach makes the correlation function 3F(f) enter into play. The CTRW is a perspective resting on trajectories and consequently on /(f). Establishing a connection between the two pictures implies the conversion of 4> (f) into j/(r), or vice versa. Here, this conversion has been realized paying the price of altering the physics of the generalized diffusion equation (133). 

We can easily prove that these three equations yield the same time evolution for the second moment (x2(t)) if the first and the second refer to the same correlation function and if the memory kernel (t) of the third equation... 

We apply to this equation the same remarks as those adopted for the comparison among Eqs. (316)—(318). We note, first of all, that the structure of Eq. (325) is very attractive, because it implies a time convolution with a Lindblad form, thereby yielding the condition of positivity that many quantum GME violate. However, if we identify the memory kernel with the correlation function of the 1/2-spin operator ux, assumed to be identical to the dichotomous fluctuation E, studied in Section XIV, we get a reliable result only if this correlation function is exponential. In the non-Poisson case, this equation has the same weakness as the generalized diffusion equation (133). This structure is... 

It is interesting to notice that Eq. (325) can also be derived from the Lindblad master equation using the same subordination approach as that adopted to derive Eq. (318). Here, however, the memory kernel of this master equation does not have the meaning of a correlation function. 

It is convenient to express the memory kernel of the generalized master equation in terms of the correlation functions of the variable of interest. To do that, rather than using Eq. (2.1) we prefer to have recourse to the corresponding interaction picture ... 

The corresponding memory functions were then computed, inverting the integral equation of the previous section, and verified by recomputing the correlation functions from the integral equation. The correlation... 

The linearized transport equations (7), the equations for the equilibrium time correlation functions (13), and the equation for collective mode spectrum (14) form a general basis for the study of the dynamic behavior of a multicomponent fluid in the memory function formalism. 

The pair kinetic equation in Section VII.D follows directly from these results if the dynamic memory function " xbs.abs neglected, and the static structural correlations in (D.3) to (D.6) are approximated so that all binary collisions are calculated in the Enskog approximation. [This is the singly independent disconnected (SID) approximation, which is discussed in detail in Ref. 53.] We have also used the static hierarchy to obtain the final form involving the mean force, given in (7.32). This latter reduction involving the static hierarchy is carried out below in the context of a comparison of the singlet and doublet formulations. 

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