Brownian motion scaling dynamics

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

To begin, suppose that there exists a particle, P, whose nontrivial dynamics is on a time seale Xp. (In deseribing the dynamics as nontrivial, we mean that there is an appreeiable ehange in the given particle s non-averaged phase-spaee points.) Suppose further that the solute particle is moving within an environment of solvent partieles whose nontrivial dynamics is on a time scale Xe Xp, where the inequality is a result of mass separation, size separation, or some other meehanism. The motion of P can then be described as Brownian motion in whieh P is in some effeetive (averaged) uniform environment. 

Kramers idea was to give a more realistic description of the dynamics in the reaction coordinate by including dynamical effects of the solvent. Instead of giving a deterministic description, which is only possible in a large-scale molecular dynamics simulation, he proposed to give a stochastic description of the motion similar to that of the Brownian motion of a heavy particle in a solvent. From the normal coordinate analysis of the activated complex, a reduced mass pi has been associated with the motion in the reaction coordinate, so the proposal is to describe the motion in that coordinate as that of a Brownian particle of mass g in the solvent. 

Brownian Dynamics (BD) methods treat the short-term behavior of particles influenced by Brownian motion stochastically. The requirement must be met that time scales in these simulations are sufficiently long so that the random walk approximation is valid. Simultaneously, time steps must be sufficiently small such that external force fields can be considered constant (e.g., hydrodynamic forces and interfacial forces). Due to the inclusion of random elements, BD methods are not reversible as are the MD methods (i.e., a reverse trajectory will not, in general, be the same as the forward using BD methods). BD methods typically proceed by discretization and integration of the equation for motion in the Langevin form... 

The general principle of BD is based on Brownian motion, which is the random movement of solute molecules in dilute solution that result from repeated collisions of the solute with solvent molecules. In BD, solute molecules diffuse under the influence of systematic intermolecular and intramolecular forces, which are subject to frictional damping by the solvent, and the stochastic effects of the solvent, which is modeled as a continuum. The BD technique allows the generation of trajectories on much longer temporal and spatial scales than is feasible with molecular dynamics simulations, which are currently limited to a time of about 10 ns for medium-sized proteins. 

The best physical model is the simplest one that can explain all the available experimental time series, with the fewest number of assumptions. Alternative models are those that make predictions and which can assist in formulating new experiments that can discriminate between different hypotheses. We start our discussion of models with a simple random walk, which in its simplest form provides a physical picture of diffusion—that is, a dynamic variable with Gaussian statistics in time. Diffusive phenomena are shown to scale linearly in time and generalized random walks including long-term memory also scale, but they do so nonlinearly in time, as in the case of anomalous diffusion. Fractional diffusion operators are used to incorporate memory into the dynamics of a diffusive process and leads to fractional Brownian motion, among other things. The continuum form of these fractional operators is discussed in Section IV. 

Classic Brownian motion has been widely applied in the past to the interpretation of experiments sensitive to rotational dynamics. ESR and NMR measurements of T and Tj for small paramagnetic probes have been interpreted on the basis of a simple Debye model, in which the rotating solute is considered a rigid Brownian rotator, sueh that the time scale of the rotational motion is much slower than that of the angular momentum relaxation and of any other degree of freedom in the liquid system. It is usually accepted that a fairly accurate description of the molecular dynamics is given by a Smoluchowski equation (or the equivalent Langevin equation), that can be solved analytically in the absence of external mean potentials. 

When multiple scattering is discarded from the measured signal, DLS can be used to study the dynamics of concentrated suspensions, in which the Brownian motion of individual particles (self-diffusion) differs from the diffusive mass transport (gradient or collective diffusion), which causes local density fluctuations, and where the diffusion on very short time-scales (r < c lD) deviates from those on large time scales (r c D lones and Pusey 1991 Banchio et al. 2000). These different diffusion coefficients depend on the microstructure of the suspension, i.e. on the particle concentration and on the interparticle forces. For an unknown suspension it is not possible to state a priori which of them is probed by a DLS experiment. For this reason, a further concentration limit must be obeyed when DLS is used for basic characterisation tasks such as particle sizing. As a rule of thumb, such concentration effects vanish below concentrations of 0.01-0.1 vol%, but certainty can only be gained by experiment. 

Relaxation rates themselves are affected by dynamics and their measurement provide quantitative information on the rates at which relevant chemical processes are taking place. These include, but are not restricted to, the overall tumbling of the complexes in solution caused by Brownian motion and the internal motions taking place at comparable time scales. 

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