Average perturbation theorem

In the second period, which was ended by review GT after the average perturbation theorem was proved, it became possible to get the Kubo-like expression for the spectral function L(z) (GT, p. 150). This expression is applicable to any axially symmetric potential well. Several collision models were also considered, and the susceptibility was expressed through the same spectral function L(z) (GT, p. 188). The law of motion of the particles should now be determined only by the steady state. So, calculations became much simpler than in the period (1). The best achievements of the period (2) concern the cone-confined rotator model (GT, p. 231), in which the dipoles were assumed to librate in space in an infinitely deep rectangular well, and applications of the theory to nonassociated liquids (GT, p. 329). 

The formulas for the susceptibility of a harmonic oscillator, presented above, were first derived in Ref. 18 with neglect of correlation between the particles orientations and velocities. This derivation was based on an early version of the ACF method, in which the average perturbation theorem was not employed, so that the expression equivalent to Eq. (14c) was used. (The integrand of the latter involves the quantities perturbed by an a.c. field.) For a specific case of the parabolic potential, the above-mentioned theory is simple however, it becomes extremely cumbersome for more realistic forms of the potential well. 

To find the component Xdyn( ) we aPPly the average perturbation (AP) theorem, formulated and proved in GT, pp. 373-376 (see also VIG, pp. 82-87). This theorem allows us to express the ensemble average value for 8p , induced by a.c. electric field, through an integral, including unperturbed time dependence p (f). The formulation of the theorem is8... 

Our final value for the first-order correction is the average value theorem (Eq. 2.10) applied to the perturbation Hamiltonian H and integrated over the zero-order wavefunction if/Q. 

One particular strength of perturbation theory is its intuitive simplicity. That may not be apparent when you ve just emerged from two pages of calculus, but the first- and second-order corrections to the energy are very useful conceptually. As we ve seen, the first-order correction is essentially the quantum average value theorem applied to the perturbation Hamiltonian H. In many cases. 

To get the first-order correction to this energy, we apply the average value theorem to get the average perturbation energy using the zero-order wavefunction ... 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

The other correlations which are neglected in the Hartree-Fock model are the Coulomb correlations, due to the approximate treatment implied by using an averaged central field. Often, they are small. The Hartree-Fock model is fairly robust, because the next higher order contribution to the many-body perturbation series is zero (Brillouin s theorem). 

The nonautonomous perturbed component in (3.33d) will not survive the KAM averaging theorem, and some of the nonautonomous terms may be suspended by applying the following Hamilton-Jacobian nonautonomous canonical transformation. 

The manifold M- a has locally stable and unstable manifolds that are close to the unperturbed locally stable and unstable manifolds and if these manifolds intersect transversely, then the Smale-Birkhoff theorem predicts the existence of horseshoes and their chaotic dynamics in the perturbed dissipative system. A 2D hyperbolic invariant torus Tja(Pi, P2) may be located on by averaging the perturbed dissipative vector field 7 > 0 and a > 0 restricted to M q, over the angular variables Qi and Q2- The averaged equations have a unique stable hyperbolic fixed point (Pi,P2) = (0,0) with two negative eigenvalues provided that the... 

The first paper in which the separability of the total infrared correlation function into its vibrational and rotational components was questioned is the paper by Van Woerkom, de Bleyser et al, (27). Their theory is based on use of the generalized cumulant expansion theorem for non-commuting quantum mechanical operators and involves the following assumptions, (i) The Hamiltonian of the liquid sample is written in the form H = E + F + G where E is the Hamiltonian for vibrational degrees of freedom, F is the bath Hamiltonian for rotational and vibrational degrees of freedom whereas G is the interaction Hamiltonian, (ii) G is small with respect to E + F. An interaction representation is used in which G is considered as a perturbation, (iii) The averaged ordered exponential is developped into a truncated cumulant expansion series. 

Within the variational time-dependent approach of Sect. 3.1.1, the molecular response functions (3.13) are determined by expanding the time-dependent wave-function >/ (t) > and the time-averaged free-energy functional (3.10) in orders of the perturbation, and by imposing that the variational condition (3.9) is satisfied at the various order. The response functions are then identified by means of the Hellmann-Feynman theorem (3.11), as terms of the expansion of the quasi-free-energy. 

---