Stochastic equation force

The different theoretical models for analyzing particle deposition kinetics from suspensions can be classified as either deterministic or stochastic. The deterministic methods are based on the formulation and solution of the equations arising from the application of Newton s second law to a particle whose trajectory is followed in time, until it makes contact with the collector or leaves the system. In the stochastic methods, forces are freed of their classic duty of determining directly the motion of particles and instead the probability of finding a particle in a certain place at a certain time is determined. A more detailed classification scheme can be found in an overview article [72]. 

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. 

Note here that the relation between mesoscopic and microscopic approaches is not trivial. In fact, the former is closer to the macroscopic treatment (Section 2.1.1) which neglects the structural characteristics of a system. Passing from the micro- to meso- and, finally, to macroscopic level we loose also the initial statement of a stochastic model of the Markov process. Indeed, the disadvantages of deterministic equations used for rather simplified treatment of bimolecular kinetics (Section 2.1) lead to the macro- and mesoscopic models (Section 2.2) where the stochasticity is kept either by adding the stochastic external forces (Section 2.2.1) or by postulating the master equation itself for the relevant Markov process (Section 2.2.2). In the former case the fluctuation source is assumed to be external, whereas in the latter kinetics of bimolecular reaction and fluctuations are coupled and mutually related. Section 2.3.1.2 is aimed to consider the relation between these three levels as well as to discuss problem of how determinicity and stochasticity can coexist. 

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... 

The fourth term on the right hand side of (3.4) represents the elastic forces on each Brownian particle due to its neighbours along the chain the forces ensure the integrity of the macromolecule. Note that this term in equation (3.4) can be taken to be identical to the similar term in equation for dynamic of a single macromolecule due to a remarkable phenomenon - screening of intramolecular interactions, which was already discussed in Section 1.6.2. The last term on the right hand side of (3.4) represents a stochastic thermal force. The correlation function of the stochastic forces is connected... 

The set of stochastic equations given by (3.37) is equivalent (in the linear case) to equations (3.11) with the memory functions defined in Section 3.3, but, in contrast to equations (3.11), set (3.37) is written as a set of Markov stochastic equations. This enables us to determine the variables that describe the collective motion of the set of macromolecules. In this particular approximation, the interaction between neighbouring macromolecules ensures that the phase variables of the elementary motion are co-ordinates, velocities, and some other vector variables - the extra forces. This set of phase variables describes the dynamics of the entire set of entangled macromolecules. Note that the Markovian representation of the equation of macromolecular dynamics cannot be made for any arbitrary case, but only for some simple approximations of the memory functions. We are considering the case with a single relaxation time, but generalisation for a case with a few relaxation times is possible. 

Both equations give good results for the description of mass and heat transport without forced flow. Here, it is important to notice that the Fokker-Plank-Kolmo-gorov equation corresponds to a Markov process for a stochastic connection. Consequently, it can be observed as a solution to the stochastic equations written below ... 

An(t)/n, the transiently induced birefringence. By this means, deterministic equations of motion, without stochastic terms, can be used via computer simulation to produce spectral features. As we have seen, a stochastic equation such as Eq. (1) is based on assumptions which are supported neither by spectral analysis nor by computer simulation of free molecular diffusion. The field-on simulation allows us the direct use of more realistic fimctions for the description of intermolecular interaction than any diffusional equation which uses stochastically generated intermolecular force fields. 

Equation (8.54) is a stochastic equation of motion similarto Eq. (8.13). However, we see an important difference Eq. (8.54) is an integro-differential equation in which the term yx of Eq. (8.13) is replaced by the integral /J drZ t — r)x(r). At the same time the relationship between the random force R t) and the damping, Eq. (8.20), is now replaced by (8.59). Equation (8.54) is in fact the non-Markovian generalization of Eq. (8.13), where the effect of the thermal environment on the system is not instantaneous but characterized by a memory—at time t it depends on the past interactions between them. These past interactions are important during a memory time, given by the lifetime of the memory kernel Z t). The Markovian limit is obtained when this kernel is instantaneous... 

To generate the trajectories that result from stochastic equations of motion (14.39) and (14.40) one needs to be able to properly address the stochastic input. For Eqs (14.39) and (14.40) we have to move the particle Linder the influence of the potential T(.v), the friction force—yvm and a time-dependent random force R(t). The latter is obtained by generating a Gaussian random variable at each time step. Algorithms for generating realizations of such variables are available in the applied mathematics or numerical methods hterature. The needed input for these algorithms are the two moments, (2J) and In our case (7 ) = 0, and (cf. Eq. (8.19)) = liiiyk/jT/At. where Ai is the time interval... 

Third is the stochastic Brownian force Fx<3> (t) which plays the same role here as it does in the Langevin equation (Section 5.10). 

The stochastic equation for ji [Eq. (8.A.1)] is used along with suitable statistical assumptions to evaluate the moments A and B. We assume that the random force in Eq. (8.A.1) is a Gaussian random process with the average values... 

Equations (9.35) and (9.47) provide a convenient framework for the analysis of forces acting on particles. These equations simply state that the acceleration experienced by the particle is proportional to the sum of forces acting on the particle. We have used this equation so far for deterministic forces, namely, the gravity, drag, and electrical forces. We now need to use the stochastic Brownian force, which is simply the product of the particle mass mp and the random acceleration a caused by the bombardment by the fluid molecules. Then the equation of motion is... 

Thus far, we have described the time-dependent nature of polymerizing environments both through stochastic [49-51] and lattice [52,53] models capable of addressing this kind of dynamics in a complex environment. The current article focuses on the former approach, but now rephrases the earlier justification of the use of the irreversible Langevin equation, iGLE, to the polymerization problem in the context of kinetic models, and specifically the chemical stochastic equation. The nonstationarity in the solvent response due to the collective polymerization of the dense solvent now appears naturally. This leads to a clear recipe for the construction of the requisite terms in the iGLE. Namely the potential of mean force and the friction kernel as described in Section 3. With these tools in hand, the iGLE is used... 

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] ... 

Let a limit cycle oscillator be exposed to some weak random forces which may depend on the state variable X. The governing equation is a nonlinear stochastic equation ... 

RUELLE [1] suggested more than a decade ago that since nonequilibrium chemical reactions are described by coupled nonlinear differential equations, for some conditions they might exhibit nonperiodic behavior. The nonperiodic behavior that arises from the nonlinear nature of a system rather than from stochastic driving forces is now called chaos, a term that we will define more carefully later. 

The relation between the Langevin equations (3.49) [assuming -correlated gaussian random forces obeying (3.59, 61)] and the stochastic equation (3.29) has now to be established. The constitutive quantities of the latter equation are the moments, see (3.23, 28),... 

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). 

---