Stochastic perturbation theory

The change in a molecule s orientation in space as a result of rotation is described by the dynamic equation of motion 

When perturbation is fast enough in comparison with the molecular response to it, the averaging procedure proposed in [91] is justified. After substitution of a formal solution of Eq. (2.20) 

This means that the theory may be applied only to dense media where rotational relaxation proceeds at a higher rate than does orientational relaxation. 

Multiplying (2.22) by (— l)q d q 0) and summing over q according to (2.11), we obtain an equation that connects the correlation functions of the perturbation to those of the response  

It is commonly believed that K (t ) may be carried outside the integral without lack of accuracy if inequality (2.23) is satisfied. This is the same way that was used in Chapter 1 to obtain the non-Markovian differential equation 

---

In order to describe the shape of the Q-branch after its collapse, it is sufficient to use the stochastic perturbation theory expounded in the... 

One possibility for this was demonstrated in Chapter 3. If impact theory is still valid in a moderately dense fluid where non-model stochastic perturbation theory has been already found applicable, then evidently the continuation of the theory to liquid densities is justified. This simplest opportunity of unified description of nitrogen isotropic Q-branch from rarefied gas to liquid is validated due to the small enough frequency scale of rotation-vibration interaction. The frequency scales corresponding to IR and anisotropic Raman spectra are much larger. So the common applicability region for perturbation and impact theories hardly exists. The analysis of numerous experimental data proves that in simple (non-associated) systems there are three different scenarios of linear rotator spectral transformation. The IR spectrum in rarefied gas is a P-R doublet with either resolved or unresolved rotational structure. In the process of condensation the following may happen. 

Hassan, A.E., J.H. Cushman, and J.W. Delleur. 1997. Monte Carlo studies of flow and transport in fractal conductivity fields comparison with stochastic perturbation theory. Water Resour. Res. 33 2519-2534. 

In recent years the diagrammatic technique of the perturbation theory found wide application in solving the stochastic differential equations, e.g., see a review article by Mikhailov and Uporov [68]. 

The code FORMOS A-P has been developed over a number of years at North Carolina State University [2-4] for the purpose of automating the process of determining the family of near optimum fuel and BP LPs, while taking into account, with a minimum of assumptions, the complexities of the reload design problem. FORMOSA-P couples the stochastic optimization technique of Simulated Annealing (SA) [5] with a computationally efficient neutronics solver based on second-order accurate, nodal generalized perturbation theory (GPT) [6-7] for evaluating core physics characteristics over the cycle. 

A most interesting recent development is the work of Augustin and Rabitz, who obtained a transition between statistical and perturbation theories for any type of collision, not only complex-forming ones. More general stochastic aspects of unimolecular reactions have been discussed by Sole and Widom. An application of a phase-space model to electronic transitions in atomic collisions has been reported, as well as a simple RRKM model for electronic to vibrational energy transfer in 0( Z)) -I- Nj collisions. ... 

EDMD and thermodynamic perturbation theory. Donev et developed a novd stochastic event-driven molecular dynamics (SEDMD) algorithm for simulating polymer chains in a solvent. This hybrid algorithm combines EDMD with the direa simulation Monte Carlo (DSMC) method. The chain beads are hard spheres tethered by square-wells and interact with the surrounding solvent with hard-core potentials. EDMD is used for the simulation of the polymer and solvent, but the solvent-solvent interaction is determined stochastically using DSMC. 

It should be emphasized that the Langevin equations in the form given by Eqs. 2 and 146 are not simple to solve because one needs to average over both the stochastic Brownian forces fi(t) and the random part of the locaUza-tion parameter 8qo(R) (the crosslink density SM R)). Such calculations were performed in [137,138] with the use of perturbation theory, taking Sqo(R) as being a small parameter. As an illustration, here we present the final expression for the mean square displacement of a network bead [ 137,138] ... 

In many cases composite materials cannot be characterized by deterministic material properties. Often stochastically distributed geometrical parameters (e.g. characteristic length of fibres or cracks) of the specimens or the construction elements have to be taken into account. Here the so-called stochastic F.E.M. is a well suited numerical tool to compute stochastic distributions of displacements, stresses and other related quantities instead of the deterministic ones. Actually a lihear elastic version including stochastic material parameters and geometries as well can be used. It is based on the perturbation theory (18). 

The form of the stochastic equations (105) and (106) is motivated by the following considerations. The first term in (105), dF/da, is the thermodynamic force due to bending energy and volume constraints it is calculated from the free energy F a). The second term of (105) is the deformation force due to the shear flow. Since the hydrodynamic forces elongate the vesicle for 0 < 0 < r/2 but push to reduce the elongation for - r/2 < 0 < 0, the flow forces should be proportional to sin(20) to leading order. The amplitude A is assumed to be independent of the asphericity a. C,a and A can be estimated [205] from the results of a perturbation theory [199] in the quasi-spherical limit. Equation (106) is adapted from KeUer-Skalak theory. While B is a constant in KeUer-Skalak theory, it is now a function of the (time-dependent) asphericity a in (106). 

I now consider statement 3 How should an extension of dynamics be understood In the MPC theory the problem does not exist For the intrinsically stochastic systems there is no need for modifying the laws of dynamics. As for the LPS theory, one notes the presence of two essentially new concepts. The introduction of non-Hilbert functional spaces only concerns the definition of the states of the dynamical system, and not at all the law governing their evolution. It is an important precision introduced in statistical mechanics. The extension of dynamics thus only appears in the operation of regularization of the resonances. This step is also the one that is most difficult to justify rigorously it is related to the (practical) necessity to use perturbation calculus (see Appendix). 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

---