Brownian approximation

It is well-known that the electrophoretic effect involves the hydrodynamical properties of the solvent in a very crucial way for this reason, the theory of this effect is rather difficult. However, using a Brownian approximation for the ions, we have been able to obtain recently a microscopic description of this effect. This problem, together with the more general question of long-range hydrodynamical correlations, is discussed in Section VI. 

From now on, when we talk about a Brownian approximation the limit (378) will always be understood. 

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. 

Statistically, in a high-pressure region, an ion will be struck by neutral molecules randomly from all angles. The ion receives as many collisions from behind as in front and as many collisions from one side as from the other. Therefore, it can be expected that the overall forward motion of the ion will be maintained but that the trajectory will be chaotic and similar to Brownian motion (Figure 49.4b). Overall, the ion trajectory can be expected to be approximately along the line of its initial velocity direction, since it is still influenced by the applied potential difference V. 

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] ... 

If the Brownian particles were macroscopic in size, the solvent could be treated as a viscous continuum, and the particles would couple to the continuum solvent through appropriate boundary conditions. Then the two-particle friction may be calculated by solving the Navier-Stokes equations in the presence of the two fixed particles. The simplest approximation for hydrodynamic interactions is through the Oseen tensor [54],... 

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. 

After the jump, the particle is taken to have reacted with a given probability if its distance from another particle is within the reaction radius. For fully diffusion-controlled reactions, this probability is unity for partially diffusion-controlled reactions, this reaction probability has to be consistent with the specific rate by a defined procedure. The probability that the particle may have reacted while executing the jump is approximated for binary encounters by a Brownian bridge—that is, it is assumed to be given by exp[—(x — a)(y — a)/D St], where a is the reaction radius, x andy are the interparticle separations before and after the jump, and D is the mutual diffusion coefficient of the reactants. After all... 

The first paper that was devoted to the escape problem in the context of the kinetics of chemical reactions and that presented approximate, but complete, analytic results was the paper by Kramers [11]. Kramers considered the mechanism of the transition process as noise-assisted reaction and used the Fokker-Planck equation for the probability density of Brownian particles to obtain several approximate expressions for the desired transition rates. The main approach of the Kramers method is the assumption that the probability current over a potential barrier is small and thus constant. This condition is valid only if a potential barrier is sufficiently high in comparison with the noise intensity. For obtaining exact timescales and probability densities, it is necessary to solve the Fokker-Planck equation, which is the main difficulty of the problem of investigating diffusion transition processes. 

One aspect of MD simulations is that all molecules, including the solvent, are specified in full detail. As detailed above, much of the CPU time in such a simulation is used up by following all the solvent (water) molecules. An alternative to the MD simulations is Brownian dynamics (BD) simulation. In this method, the solvent molecules are removed from the simulations. The effects of the solvent molecules are then reintroduced into the problem in an approximate way. Firstly, of course, the interaction parameters are adjusted, because the interactions should now include the effect of the solvent molecules. Furthermore, it is necessary to include a fluctuating force acting on the beads (atoms). These fluctuations represent the stochastic forces that result from the collisions of solvent molecules with the atoms. We know of no results using this technique on lipid bilayers. 

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. 

Let us consider a Brownian particle (B-particle) characterized by a mass Ma and coordinates Ra and ua. If this B-particle were so large that the fluid could be described by a hydrodynamic approximation (without fluctuations), its motion would be described by ... 

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.). 

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. 

Inserting Eq. (385) into Eqs. (365) and (312), it is then easy to obtain the following expression for the current in the Brownian-static approximation ... 

We want to analyze here the effect of these long-range Coulomb forces in the large friction limit (396) we shall thus consider the Brownian-dynamic approximation, which, as we shall see presently, gives exactly the same result as the classical semi-phenomenological theory developed in Section V-A. 

For the weak coupling case with Eq. (32), our master equation reduces to the well-known quantum master equation, obtained through the approximation, widely used in quantum optics. This equation describes, among other things, quantum decoherence due to Brownian motion. Hence, we have derived an exact quantum master equation for the transformed density operator p that describes exact decoherence. Furthermore, our master equation cannot keep the purity of the transformed density matrix. Indeed, one can show that if p(t) is factorized into a product of transformed wave functions at t = 0, it will not be factorized into their product for t > 0. This is consistent the nondistributivity of the nonunitary transformation (18). 

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 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. 

Deposition efficiencies for particles in the respiratory tract are generally presented as a function of their aerodynamic diameter (e.g. [8,12]). Large particles (> 10 pm) are removed from the airstream with nearly 100% efficiency by inertial impaction, mainly in the oropharynx. But as sedimentation becomes more dominant, the deposition efficiency decreases to a minimum of approximately 20% for particles with an aerodynamic diameter of 0.5 pm. When particles are smaller than 0.1 pm, the deposition efficiency increases again as a result of dif-fusional displacement. It is believed that 100% deposition due to Brownian motion might be possible for particles in the nanometer range. 

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