Brownian approximation model

This problem is very difficult to solve in general however, we have to keep conditions (303) in mind, which we used in order to obtain the Fokker-Planck collision term (304). With this approximation, it is expected that the ions will exhibit random Brownian motion instead of free particle motion between two successive coulombic interactions. We shall thus refer to this model as the Brownian-static model (B.s.). 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

The dumbbell relaxation time (t) in the preceding model is coil deformation dependent. Neglecting Brownian forces, the dumbbell relaxation time is given by t ssf H/fs. Equation (45) is then tantamount to saying that t increases approximately in proportion to the root mean square end-to-end separation distance R [52] ... 

To work out the time-dependence requires a specific model for the movement of the paramagnet, for example, Brownian motion, or lateral diffusion in a membrane, or axial rotation on a protein, or jumping between two conformers, etc. That theory is beyond the scope of this book the math can become quite hairy and can easily fill another book or two. We limit the treatment here to a few simple approximations that are frequently used in practice. 

It is possible however to analyze mathematically well defined models which we hope will give a correct approximation to real physical systems. In this section, we shall be concerned with the simplest case the zeroth-order conductance of electrolytes in an infinitely dilute solution. We shall describe this situation by assuming that the ions—which are so far from each other that their mutual interaction may be completely neglected—have a very large mass with respect to the solvent molecules we are then confronted with a typical Brownian motion problem. 

This result gives in fact the mathematical limitation for the validity of the plasma approximation developed in the two preceding sections even with solvent molecules interacting with the ions, the plasma model will he valid in the limit of Brownian ions, provided that conditions (376) holds. 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

A further development is possible by noting that the high frequency shear modulus Goo is related to the mean square particle displacement (m ) of caged fluid particles (monomers) that are transiently localized on time scales ranging between an average molecular collision time and the structural relaxation time r. Specifically, if the viscoelasticity of a supercooled liquid is approximated below Ti by a simple Maxwell model in conjunction with a Langevin model for Brownian motion, then (m ) is given by [188]... 

For the calculation of the Maxwell-constant an assembly of frozen random conformations is considered. Brownian motion is taken into account only so far as rotary diffusion of the rigid conformations is concerned. In this way a first order approximation of the distribution function with respect to shear rate is obtained. This distribution function is used for the calculation of the Maxwell-constant, [cf. the calculation of the Maxwell-constant of an assembly of frozen dumb-bell models, as sketched in Section 5.I.3., eq. (5.22)]. Intrinsic viscosity is calculated for the same free-draining model, using average dimensions [cf. also Peter-lin (101)]. As for the initial deviation of the extinction angle curve from 45° a second order approximation of the distribution function is required, no extinction angles are given. 

Before discussing other results it is informative to first consider some correlation and memory functions obtained from a few simple models of rotational and translational motion in liquids. One might expect a fluid molecule to behave in some respects like a Brownian particle. That is, its actual motion is very erratic due to the rapidly varying forces and torques that other molecules exert on it. To a first approximation its motion might then be governed by the Langevin equations for a Brownian particle 61... 

The molecular theory of Doi [63,166] has been successfully applied to the description of many nonlinear rheological phenomena in PLCs. This theory assumes an un-textured monodomain and describes the molecular scale orientation of rigid rod molecules subject to the combined influence of hydrodynamic and Brownian torques, along with a potential of interaction (a Maier-Saupe potential is used) to account for the tendency for nematic alignment of the molecules. This theory is able to predict shear thinning viscosity, as well as predictions of the Leslie viscosity coefficients used in the LE theory. The original calculations by Doi for this model employed a preaveraging approximation that was later... 

Electro- and magnetooptical phenomena in colloids and suspensions are widely used for structure and kinetics analysis of those media as well as practical applications in optoelectronics [143,144]. The basic theoretical model used to study optical anisotropy of the disperse systems is the noninteracting Brownian particle ensemble. In the frame of this general approximation, several special cases according to the actual type of particle polarization response to the applied field may be distinguished (1) particles with permanent dipole moments, (2) linearly polarizable particles, (3) nonlinearly polarizable particles, and (4) particles with hysteretic dipole moment reorientation. 

Monte Carlo techniques were first applied to colloidal dispersions by van Megen and Snook (1975). Included in their analysis was Brownian motion as well as van der Waals and double-layer forces, although hydrodynamic interactions were not incorporated in this first study. Order-disorder transitions, arising from the existence of these forces, were calculated. Approximate methods, such as first-order perturbation theory for the disordered state and the so-called cell model for the ordered state, were used to calculate the latter transition, exhibiting relatively good agreement with the exact Monte Carlo computations. Other quantities of interest, such as the radial distribution function and the excess pressure, were also calculated. This type of approach appears attractive for future studies of suspension properties. 

Wc note that P E,t) must have the property that i Efi) = 0 and P(Eyt) —> 1 as > oo. H. A. Kramers, Physics 7,284 (1940), has given an approximate solution to this problem based on a simplified analogy to a model for brownian motion in one dimension. 

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