Brownian motion velocity correlation function

This is the velocity correlation function as obtained by the Wiener-Khinchin theorem. The above arguments may be extended to multidimensional systems, as discussed in some detail by Wang and Uhlenbeck [10] and by McConnell [57] in the context of rotational Brownian motion. 

To proceed we must determine the velocity-correlation function of a macromolecule. Because macromolecules are much more massive than solvent molecules, they have a much lower velocity on the average26 than the solvent molecules. The motion of massive molecules in solvents consisting of small molecules has received much attention. The theory that describes this situation is Brownian motion theory. In Brownian motion... 

Assuming the particles undergo Brownian motion, the velocity is by definition an isotropic random variable described by a real Gaussian process of zero mean. Hence all the odd-ordered velocity correlation functions are zero, and the factorization property for a real Gaussian process can be used to determine the even ordered velocity correlations in terms of the second order velocity correlation function. Noting in particular that... 

A qualitative discussion could be pursued in terms of any of the viscoelastic functions surveyed in Chapter 2. It is convenient for the moment to choose the components of the dynamic modulus, G and G". For a periodic strain, the energy storage per cycle depends on G, and is contributed by the polymeric solute molecules alone the energy dissipation depends on G", and has contributions from both solute and solvent. The relative contributions of the solute to G and G" depend on the extent to which the Brownian motions are correlated with the external forces. Force in phase with displacement corresponds to energy storage, but force in phase with velocity corresponds to energy dissipation. 

The Zimm model rests upon the Langevin equation for over-damped motion of the monomers, i.e., it applies for times larger than the Brownian time scale Tb 2> OTm/where is Stokes friction coefficient [12]. On such time scales, velocity correlation functions have decayed to zero and the monomer momenta are in equilibrium with the solvent Moreover, hydrodynamic interactions between the various parts of the polymer are assumed to propagate instantaneously. This is not the case in our simulations. First of all, the monomer inertia term is taken into account, which implies non-zero velocity autocorrelation functions. Secondly, the hydrodynamic interactions build up gradually. The center-of-mass velocity autocorrelation function displayed in Fig. 9 reflects these aspects. The correlation function exhibits a long-time tail, which decays as (vcm(t)vcm(O)) on larger time scales. The... 

In the previous section, the phenomenological description of Brownian motion was presented. The Langevin analysis leads to a velocity autocorrelation function which decays exponentially with time. This is characteristic of a Markovian process, as Doobs has shown (see ref. 490). Since it is known heyond question that the velocity autocorrelation function is far from such an exponential function, the effect that the solvent structure has on the progress of a chemical reaction cannot be assessed very reliably by means of phenomenological Langevin description. Since the velocity of a solute is correlated with its velocity a while before, a description which fails to consider solute and solvent velocities can hardly be satisfactory. Necessarily, the analysis requires a modification of the Langevin or Fokker—Plank description. In this section, some comments are made on this new and exciting area of research. 

As an application consider the velocity time correlation function for the simple Brownian motion. Using Eqs (7.76), (8.30), and (8.35) we get... 

Fig. 10.37 Measured correlation function (r) (a) and velocity distribution (b) computed from G (r) for living solid curve) and dead dashed curve) E. coli bacteria in solution. The dashed curve corresponds to the Poisson distribution of a random walk (Brownian motion) [1523]...

In the presence of a nonuniform velocity field, if there is a velocity gradient present across the interrogation window, the correlation function is deformed accordingly. In the case of a Poiseuille flow, the velocity gradient is nonumform across the channel. As a result, the probability distribution function / has a Brownian-motion component and a velocity distribution component. For simple flows, if the velocity profile is known, the probability distribution of velocities and the resulting broadening of the cross-correlation function can be computed. This can be subtracted from the total measured value to obtain the broadening due to... 

When a particle is subject to Brownian motion and irradiated, two frequencies of equal intensity are generated in addition to the frequency that would normally be scattered, inducing a positive and a negative Doppler shift proportional to the particle velocity. The interference between the nonshifted wave (photon reemission) and the two waves due to Brownian motion yields infinitesimal variations in intensity. Detection of these is the basic principle of DLS, which is therefore particularly suited to the study of properties of solutions. The scattered intensity is acquired as a function of time and is then self-correlated. This yields the relaxation time due to the Brownian motion and leads to the characterization of the particle size through hydrodynamic models of the diffusion coefficients. 

The velocity autocorrelation function tells you how fast a particle forgets its initial velocity, owing to Brownian randomization. When the time t is short relative to the correlation time of the physical process, the particle velocity will be nearly unchanged from time 0 to t, and v(t) will nearly equal v(0) so (u(O)v(t)) greater than the system s correlation time. Brownian motion will have had time to randomize the particle velocity relative to its initial velocity, so v it) will be uncorrelated with u (0). This means that v(0)v(t) will be negative just as often as it is positive, so the ensemble average will be zero, (u(O)u(t)) = 0. 

In PIV analysis, the cross-correlation function between two ideal images would have a single peak. The location of the peak gives an estimate of the mean velocity of the fluid, and the shape of the cross-correlation function contains information about the particle size, and about any fluctuations in the velocity. Brownian motion of the tracer particles introduces an error into the location of the correlation peak. Since Brownian motion is an unbiased noise source, ensemble averaging over multiple correlation peaks should help to minimize this error. However, as a result of the averaging process, the peak width of the correlation func-... 

The displacement and width of the correlation function depend on the probability function/ x , t2, x,t u (x),T), where / denotes the probability that a particle initially at (pc, q) will move into the volume (x ,x + dr) at 2 for a known velocity field u(x) and temperature T. In the absence of Brownian motion, Ax= u(x,t )At is the displacement undergone by a particle at (x,t ). Since for a given velocity field there can only be one final location for the particle, the probability function / becomes the delta function... 

The feasibility of this method will be illustrated here with the help of a numerical experiment, as described below. Figure 2a shows the velocity profile across the channel the red lines show the velocity gradients present in the interrogation window at different locations. A uniform Brownian motion (temperature profile) is assumed across the channel. Figure 2b shows the broadening in the cross-correlation function for different amounts of Brownian motion. The... 

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