Brownian stochastic force

The driving force of the drop—drop collisions (F in Fig. 2) can be the Brownian stochastic force, the buoyancy force, or some attractive surface force (say, the van der Waals interaction) in stirred vessels an important role is played by the hydrodynamic (including turbulent) forces. The mutual approach of two emulsion drops (step A—in Fig. 2) is decelerated by the viscous friction due to the expulsion of the liquid from the gap between the drops. If a doublet of two drops (Fig. 2D) is sufficiently stable, it can grow by attachment of additional drops thus, aggregates of drops (floes) are produced. 

The Brownian stochastic force dominates the gravitational body force for droplets, which are smaller than 1 pm. Thus, the Brownian collision of two droplets becomes a prerequisite for flieir flocculation and/or coalescence, which is termedperikinetic coagulation. 

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. 

Let us further analyze the stochastic force F(t) with respect to the dynamics of the system in the pre-Brownian regime (t< /0). Before we introduce models, let us clarify the role of F(t) by averaging the particle velocities v. To this end, we rewrite Eqn. (5.37) in the following form... 

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... 

To recover the ideal case of Eq. (1.1) we would have to assume that (u ), vanishes. The analog simulation of Section III, however, will involve additive stochastic forces, which are an unavoidable characteristic of any electric circuit. It is therefore convenient to regard as a parameter the value of which will be determined so as to fit the experimental results. In the absence of the coupling with the variable Eq. (1.7) would describe the standard motion of a Brownian particle in an external potential field G(x). This potential is modulated by a fluctuating field The stochastic motion of in turn, is driven by the last equation of the set of Eq. (1.7), which is a standard Langevin equation with a white Gaussian noise defined by... 

The friction coefficient C of the Brownian particle was determined by means of the stochastic force autocorrelation function (FACT) [4] ... 

Different from the molecular dynamics (MD) simulation method (Sect. 4.5), the Brownian dynamics approach does not directly simulate the inter-particle collision. Instead, in the Brownian dynamics, the pseudorandom motion characteristic of the effect of particle-particle interactions is mimicked by a stochastic force generated from random numbers. This makes the Brownian dynamics more efficient than the... 

As the name suggests, this method simulates the Brownian motion of macro-molecular or colloidal particles due to random collisions with the surrounding molecules. The collisions are simulated by a random stochastic force, so... 

Rapidly fluctuating stochastic forces associated with Brownian diffusion. 

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. 

For small colloidal particles, which are subject to random Brownian motion, a stochastic approach is more appropriate. These methods are based on the formulation and solution of the diffusion equation in a force field, in the presence of convection... 

Having specified the interactions (i.e., the model of the system), the actual simulation then constructs a sequence of states (or the system trajectory) in some statistical mechanical ensemble. Simulations can be stochastic (Monte Carlo (MC)) or deterministic (MD), or they can combine elements of both, such as force-biased MC, Brownian dynamics, or generalized Lan-gevin dynamics. It is usually assumed that the laws of classical mechanics (i.e., Newton s second law) may adequately describe the atoms and molecules in the physical system. 

Brownian dynamics (BD), which is stochastic dynamics in the over-damped limit, can just as well be understood as force-biased (dynamic) MC employing collective moves only [100,101]. 

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