Liouville equation correlation function

The main lines of the Prigogine theory14-16-17 are presented in this section. A perturbation calculation is employed to study the IV-body problem. We are interested in the asymptotic solution of the Liouville equation in the limit of a large system. The resolvent method is used (the resolvent is the Laplace transform of the evolution operator of the N particles). We recall the equation of evolution for the distribution function of the velocities. It contains, first, a part which describes the destruction of the initial correlations this process is achieved after a finite time if the correlations have a finite range. The other part is a collision term which expresses the variation of the distribution function at time t in terms of the value of this function at time t, where t > t t—Tc. This expresses the fact that the system has a memory because of the finite duration of the collisions which renders the equations non-instantaneous. 

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

In view of the fact that the correlation function for the random force, as given by Eq. [16], is a Dirac 8 function, the strict Langevin equation (Eq. [15]) is not amenable to computer simulation. In order to circumvent the above difficulty, it is convenient to describe the motion of the fictitious particle by the generalized Langevin equation. The generalized Langevin equation, which can be derived from the Liouville equation (11), along with the supplementary conditions are... 

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. 

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). 

Let us make a final comment, concerning the violation of the Green-Kubo relation. There is a close connection between the breakdown of this fundamental prescription of nonequilibrium statistical physics and the breakdown of the agreement between the density and trajectory approach. We have seen that the CTRW theory, which rests on trajectories undergoing abrupt and unpredictable jumps, establishes the pdf time evolution on the basis of v /(f), whereas the density approach to GME, resting on the Liouville equation, either classical or quantum, and on the convenient contraction over the irrelevant degrees of freedom, eventually establishes the pdf time evolution on the basis of a correlation function, the correlation function in the dynamical case... 

The two previous secfions were devoted to modeling quantum resonances by means of effective Hamiltonians. From the mathematical point of view we have used two principal tools projection operators that permit to focus on a few states of interest and analytic continuation that allows to uncover the complex energies. Because the time-dependent Schrodinger equation is formally equivalent to the Liouville equation, it is attractive to try to solve the Liouville equation using the same tools and thus establishing a link between the dynamics and the nonequilibrium thermodynamics. For that purpose we will briefly recall the definition of the correlation functions which are similar to the survival and transition amplitudes of quantum mechanics. Then two models of regression of a fluctuation and of a chemical kinetic equation including a transition state will be presented. 

The physical description of the functional derivative Vee (r) requires knowledge of the wavefunction 4 for the determination of the electron-interaction component W e(r) = Wnlr) -i- W (r), and knowledge of both the wavefunction P and the Kohn-Sham orbitals < i(x) for the correlation-kinetic-energy component W, (r). The corresponding Kohn-Sham wavefunction is then a single Slater determinant. It has, however, also been proposed [42,52,53] that the wavefunction V be determined by solution of the Sturm-Liouville equation... 

There are a number of different formulations of the time correlation function method, all of which lead to the same results for the linearized hydrodynamic equations. One way is to generalize the Chapman-Enskog normal solution method so as to apply it to the Liouville equations, and obtain the N-particle distribution function for a system near a local equilibrium state. " Expressions for the heat current and pressure tensor for a general fluid system can be obtained, which have the form of the macroscopic linear laws, with explicit expressions for the various transport coefficients. These expressions for the transport coefficients have the form of time integrals of equilibrium correlation functions of microscopic currents, viz., a transport coefficient t is given by... 

The time-dependent classical statistical mechanics of systems of simple molecules is reviewed. The Liouville equation is derived the relationship between the generalized susceptibility and time-correlation function of molecular variables is obtained and a derivation of the generalized Langevin equation from the Liouville equation is given. The G.L.E. is then simplified and/or approximated by introducing physical assumptions that are appropriate to the problem of rotational motion in a dense fluid. Finally, the well-known expressions for spectral intensity of infrared and Raman vibration-rotation bands are reformulated in terms of time correlation functions. As an illustration, a brief discussion of the application of these results to the analysis of spectral data for liquid benzene is presented. 

Several remarks are in order here. First, the fact that the correlation functions need to be evaluated at the transition frequency reminds us of the earlier classical results for barrier transitions. A friction coefficient is also related to solvent fluctuations in fact, it can be written as a velocity-velocity correlation function, and as such the Redfleld theory is not much different from classical theories. This is to be expected because these can be derived from a very similar formalism based on the classical Liouville equation, albeit without the problems encountered here. In the Kramers case, we needed to evaluate the friction at zero frequency, which sometimes severely overestimates its role. It would be better to evaluate it at the barrier frequency. In the theory developed by Grote and Hynes, the friction needs to be evaluated not at the original barrier frequency but at the frequency the barrier transition actually takes place as Eq. (9.13) shows. The reason is simple in the latter case the back reaction of the system on the solvent is taken into account, something that was not allowed in the Redfield theory. 

Thus, we see that in order to obtain the mean field equations of motion, the density matrix of the entire system is assumed to factor into a product of subsystem and environmental contributions with neglect of correlations. The quantum dynamics then evolves as a pure state wave function depending on the coordinates evolving in the mean field generated by the quantum density. As we have seen in the previous sections, these approximations are not valid and no simple representation of the quantum-classical dynamics is possible in terms of single effective trajectories. Consequently, in contrast to claims made in the literature [54], quantum-classical Liouville dynamics is not equivalent to mean field dynamics. 

---