Brownian particle displacement

The relative errors e and as a result of visualizing the Brownian particle displacement in two dimensions are given by... 

Particulate diffusion does not play a significant role in the deposition of pharmaceutical aerosols. However, it is worth noting the mechanism by which diffusion of particles occurs in the lungs. The principle of Brownian motion is responsible for particle deposition under the influence of impaction with gas molecules in the airways. The amplitude of particle displacement is given by the following equation ... 

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]... 

Yet it turned out that this picture did not lead to agreement with the measurements of Brownian motion. The breakthrough came when Einstein and Smoluchowski realized that it is not this motion which is observed experimentally. Rather, between two successive observations of the position of the Brownian particle the velocity has grown and decayed many times the interval between two observations is much larger than the autocorrelation time of the velocity. What is observed is the net displacement resulting after many variations of the velocity. 

Suppose a series of observations of the same Brownian particle gives a sequence of positions Xl9X2,-..- Each displacement Xk + 1 — Xk is subject to chance, but its probability distribution does not depend on the previous history, i.e., it is independent of Xk-l9 Xk 2.Hence, not only the velocity... 

Debye62 showed that for a Brownian particle whose molecular orientation changes through small erratic angular displacements, < i(0) i(0) and 2(ll(0) p(0)) are als° exponentials. In particular under these conditions these functions are given by... 

Before calculating the probabilistic properties of the displacement X(t) of the Brownian particle, the long-time behavior of the statistical characteristics of the velocity V(t) should be discussed.156 159... 

The expression for the mean square end-to-end distance can be written as the mean square displacement of a Brownian particle after z steps of equal length / (Appendix A)... 

The time dependence of the displacement of a macromolecule, shown in Fig. 9 as a function of the ratio t/r, is typical for diffusion of Brownian particle in viscoelastic fluid (Zanten and Rufener 2000 Zanten et al. 2004). The function (5.5) for big values of B can be approximated as... 

Up to now, aging effects have mostly been discussed in a classical framework (see, however Refs. 46-48). In order to investigate quantum aging effects, it is interesting, to begin with, to study the displacement of a free Brownian particle, since, as quoted above, this variable displays aging in the classical case. Free... 

We compute below the velocity and displacement correlation functions, first, of a classical, then, of a quantal, Brownian particle. In contrast to its velocity, which thermalizes, the displacement x(t) — x(l0) of the particle with respect to its position at a given time never attains equilibrium (whatever the temperature, and even at T = 0). The model allows for a discussion of the corresponding modifications of the fluctuation-dissipation theorem. 

The displacement correlation function of the overdamped free Brownian particle is proportional to the waiting time [40,41],... 

Note, however, that the parameter 7// = 1 / (T t, if it allows one to write a modified FDT for the displacement of the free Brownian particle [Eq. (70)], cannot be given the full significance of a physical temperature. In particular, it does not control the thermalization in the absence of potential, the thermalization, which is solely linked with the behavior of the velocity, is effective once the limit f, —> —oo has been taken. 

A modified form of the quantum FDT in an out-of-equilibrium situation, allowing for the description of aging effects, has been proposed in [46,47] for mean-field spin-glass models. Here, we propose a modified form of the quantum FDT which can conveniently be applied to the displacement of the free quantum Brownian particle—that is, to the problem of diffusion.9... 

Let us now comment upon the results obtained (Figs. 5 to 8). For all values of x, and at any nonzero temperature, the ratio peff(x, tw)/ is equal to 1 when tw 0 and it differs from 1 when tw / 0. Thus, as in the classical case, aging is always present as far as the displacement of the quantum Brownian particle is concerned. For a given value of tw, the ratio perr(x, f ,)/p increases monotonously with x toward a finite limit X (tw) (Figs. 5 to 7). At T = 0, 7 cit(x, tw) decreases monotonously with x toward zero (Fig. 8). 

For a particle evolving in a thermal bath, we focused our interest on the particle displacement, a dynamic variable which does not equilibrate with the bath, even at large times. As far as this variable is concerned, the equilibrium FDT does not hold. We showed how one can instead write a modified FDT relating the displacement response and correlation functions, provided that one introduces an effective temperature, associated with this dynamical variable. Except in the classical limit, the effective temperature is not simply proportional to the bath temperature, so that the FDT violation cannot be reduced to a simple rescaling of the latter. In the classical limit and at large times, the fluctuation-dissipation ratio T/Teff, which is equal to 1 /2 for standard Brownian motion, is a self-similar function of the ratio of the observation time to the waiting time when the diffusion is anomalous. 

Brownian motion of a single noninteracting particle can be described in terms of self-diffusion characterized by Do, the particle self-diffusion coefficient in the infinite dilution limit. The probability / (Ar. r) of a particle displacement Ar in time r satisfies the diffusion equation... 

Thus the root mean square displacement in 1 s for a 1 pm particle settling in water, viscosity 0.001 Pa s, at an absolute temperature 300 K is 0.938 pm this is almost the same as the distance settled under gravity by a quartz particle (density 2650 kg m" ) in 1 s (0.90 pm). A comparison of Brownian movement displacement and gravitational settling displacement is given by Fuchs [8]. For a size determination to be meaningful the displacement of the particles due to Brownian diffusion must be much smaller than their displacement due to gravity, hence the condition ... 

Brownian motion between reflecting walk. This problem has been discussed by Oppenheim and Mazur. It is harder to visualize than the undamped motion since the Brownian particle may reverse its travel so as to return several times for reflection at one boundary before staggering oflF to the other. Nevertheless, if we consider a particular random walk executed by a particle starting at r = 0 at position c(0) with velocity k(0), we see that the sections of this walk between successive boundary reflections may be regarded as selected from one random walk executed from x(0), u(0) in the absence of boundaries, the same walk executed from displaced starting points, and the reverse walk, with initial velocity — k(0), from another set of displaced starting points. If x(0), k(0)] is the... 

We consider the pdf of the displacements of a Brownian particle in a process characterized by an equation like (532). The normal Fokker - Planck equation would now be (here the diffusion coefficient is denoted by B)... 

The displacement y(t) of a Brownian particle in the former (Wiener s) process is defined as... 

We now explicitly consider the waiting time distribution. First we reiterate that the Einstein theory of the Brownian motion relies on the central limit theorem that a sum of independent identically distributed random variables (the sum of the elementary displacements of the Brownian particle)... 

The Mean-Square Displacement of a Brownian Particle Langevin s Method Applied to Rotational Relaxation Application of Langevin s Method to Rotational Brownian Motion The Fokker-Planck Equation Method (Intuitive Treatment) Brown s Intuitive Derivation of the Fokker-Planck Equation... 

The advection velocity variability generates additional advective particle displacements , relative to the mean particle displacement v. Analogous to Brownian motion, the effect of a large number of independent... 

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