Stochastic equation

Using equations of this type, the expressions for the average transition probability at an arbitrary value of the electron resonance integral V were obtained.77 For the symmetric transition, Wlf has the form 

(174) gives the well-known expression for the transition probability [see Eqs. (9) and (10)]. If the condition opposite to Eq. (175) holds, the transition probability for the adiabatic process takes the form 

in this case the pre-exponential factor (the frequency factor) in the transition probability depends on the relaxation time rq. Various refinements of this simple model were made in many papers by considering the change of the intramolecular structure of the reactants,79,80 cases of several relaxation times,88 etc. 

The influence of the fluctuational motion along the reaction coordinates on the probability of the electron transition has been considered recently in the framework of the Landau-Zener method.102 A Hamiltonian of the form 

In this case, P is equal to the Landau-Zener probability PLZ. 

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By substituting the stochastic equations into Eq, (26-58), taking an average, and then using Eq, (26-59), the following result is obtained ... 

Depending on the desired level of accuracy, the equation of motion to be numerically solved may be the classical equation of motion (Newton s), a stochastic equation of motion (Langevin s), a Brownian equation of motion, or even a combination of quantum and classical mechanics (QM/MM, see Chapter 11). 

Thompson, P.D., 1968. A transfomiation of the stochastic equation for droplet coalescence. In Proceedings of the international conference on cloud physics, Toronto, Canada, pp. 1115-1126. 

Unlike (2.19), in the present case the angular velocity vector random noise, but, in its turn, satisfies the stochastic equation... 

Stochastic equation (A8.7) is linear over SP and contains the operators La and V.co of differentiation over time-independent variables Q and co. Therefore, if we assume that the time fluctuations of the liquid cage axis orientation Z(t) are Markovian, then the method used in Chapter 7 yields a kinetic equation for the partially averaged distribution function P(Q, co, t, E). The latter allows us to calculate the searched averaged distribution function... 

If we denote by cf>(r, t) the magnetization density, the dynamics of the system is governed by the following stochastic equation ... 

We will later further analyze the members of (3.7) as they stand, but it is useful for our subsequent discussion to now simply add a generalized dissipative term to the solvent equation of motion to obtain the stochastic equation of motion set... 

Previously, stochastic Schrodinger equations for a quantum Brownian motion have been derived only for the particle component through approximated equations, such as the master equation obtained by the Markovian approximation [18]. In contrast, our stochastic Schrodinger equation is exact. Moreover, our stochastic equation includes both the particle and the field components, so it does not rely on integrating out the field bath modes. 

A large class of reduced-dimensional stochastic equations may be written in the form,... 

Numerical simulations of these stochastic equations under fast temperature ramping conditions indicate that the correlations in the random forces obtained by way of the adiabatic method do not satisfy the equipartition theorem whereas the proposed iGLE version does. Thus though this new version is phenomenological, it is consistent with the physical interpretation that 0(t) specifies the effective temperature of the nonstationary solvent. 

Other approaches to genetic networks include study of small circuits with either differential equations or stochastic differential equations. The use of stochastic equations emphasizes the point that noise is a central factor in the dynamics. This is of conceptual importance as well as practical importance. In all the families of models studied, the non-linear dynamical systems typically exhibit a number of dynamical attractors. These are subregions of the system s state space to which the system flows and in which it thereafter remains. A plausible interpretation is that these attractors correspond to the cell types of the organism. However, in the presence of noise, attractors can be destabilized. 

Zwanzig s diffusion equation [444], eqn. (211), can be reduced to the stochastic equation used by Clifford et al. [442, 443] [eqn. (183)] to describe the probability that N identical reactant particles exist at time t (see also McQuarrie [502]), Let us consider the case where U — 0, with a static solvent, for a constant homogeneous diffusion coefficient. This is a major simplification of eqn. (211). Now, rather than represent the reaction between two reactants k and j by a boundary condition which requires the... 

In this section, we discuss briefly how the Langevin equation, which is a stochastic equation, can be derived from the molecular equations of motion. The stochastic model described by the Langevin equation has been of great use in interpreting a large number of experiments and physical systems. The stochastic model is extremely simple but, as always, its ultimate justification rests on the molecular dynamical laws. 

In a stochastic approach, one replaces the difficult mechanical equations by stochastic equations, such as a diffusion equation, Langevin equation, master equation, or Fokker-Planck equations.5 These stochastic equations have fewer variables and are generally much easier to solve than the mechanical equations, One then hopes that the stochastic equations include the significant aspects of the physical equations of motion, so that their solutions will display the relevant features of the physical motion. 

This calculation illustrates a point made in the introduction. We have been able to calculate a well-defined theoretical spectrum to a specified range of reliability. We used the rigorous mechanical equations of motion for a large spin system without actually undertaking the hopeless task of solving them, and without resorting to a phenomenological or stochastic equation of motion. 

When the constant field is weak and the fluctuating field is comparable to or even larger than the constant field, the above decomposition becomes meaningless. There is no way of distinguishing between the adiabatic and nonadiabatic effects. In order to obtain an understanding of this rather complex situation, we have examined a stochastic model,14 extending the theory in Section II. The stochastic equation of motion of a spin in a random local field is written as... 

Equation (57) is the stochastic equation of motion for r(/), in which the matrix element 2(/) is a random process. This is similar to Eq. (2). This may be written as a stochastic Liouville equation in the form... 

The stochastic equation of motion of v(t), Eq. (77), can be transformed into a stochastic Liouville equation of the type Eq. (7) if a Markovian process can be properly defined to generate the process of H(t). Then we again obtain Eq. (63) for the conditional expectation V(t) defined by Eq. (60). The line shape function is then given by... 

Remark. The following difference with ordinary, non-stochastic differential equations needs to be emphasized. All solutions of a non-stochastic equation are obtained by imposing at an arbitrary t0 an initial condition u t0) = a, and then considering all possible values of a. For a stochastic differential equation, however, one gets in this way only a subclass of solutions, namely those that happen to have no dispersion at this particular t0. 

Having treated in the previous two sections linear stochastic differential equations we now return to the general case (1.1). Just as normal differential equations, it can be translated into a linear equation by going to the associated Liouville equation. To do this we temporarily take a single realization y(t) of Y(t) and consider the non-stochastic equation... 

Let us apply general stochastic equations (2.2.15) to the simple A+B —> C reaction with particle creation - the model problem discussed more than once ([84] to [93]). A relevant set of kinetic equations reads... 

The GLE is a stochastic equation of motion for the coordinate z (see Figure 24). The left-hand side of Eq. (41) is the inertial force along z in terms of the effective polarization mass m of the solvent and the acceleration z. The term... 

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