Mean displacement of a particle

Independently of Einstein, Smoluchowski arrived at a similar solution to the problem of the mean displacement of a particle. He obtained 2 32 R -a WSn-vr ... 

A common characteristic of small scale colloidal systems is the constant thermal movement of the components. Kinetic energy is transferred to suspended particles by collisions of solvent molecules, causing a random motion usually called Brownian motion. This motion ofpartides causes diffusion, which is a net transport of randomly moving partides along density gradients. The mean displacement of a particle by Brownian motion over time leads to the diffusion coefHdent, D (Equation 9.1) [22] ... 

Einstein s predictions by simply verifying that the mean displacement of a particle is doubled when the time is increased fourfold. 

Describe an experiment to determine Avogadro s number from the average root mean square displacement of a particle due to random walk. 

Ultrasonic waves are a mechanical disturbance which passes thru the medium by the progressive displacement of particles. The particles do not travel in the direction of the source but vibrate about their mean fixed position. The amplitude of the wave is the distance from peak to peak and therefore is the maximum displacement of a particle in the medium. The period (T) is the time required to complete one cycle and the frequency (f) refers to the number of cycles per unit time. The unit of frequency is the Hertz (Hz, one cycle per second) and it is the reciprocal of the period. The rate at which sound travels thru the medium is the velocity (c, meters per second). The wavelength (X, meters), is the distance between adjacent cycles. Therefore, the relation,between wavelength, velocity and frequency is given by... 

Equation 7.35 is a fundamental relationship between the diffusivity and the mean-square displacement of a particle diffusing for a time r. Because diffusion processes in condensed matter are comprised of a sequence of jumps, the mean-square displacement in Eq. 7.31 should be equivalent to Eq. 7.35. This equivalence, as demonstrated below, results in relations between macroscopic and microscopic diffusion parameters. 

We use the formula to write down the general expression for the mean square displacement of a particle in an arbitrary viscoelastic liquid... 

The asymptotic (xdj l) evaluations for the mean square displacement of a particle were found by Kokorin and Pokrovskii (1990, 1993). For a short time of observation, t more than the mobility of the macromolecule... 

For a binary mixture, if experimental diffusivities do not exist over the whole range of concentration, an interpolation of the diffusivities at infinite dilution D k] J is used. In calculating the diffusivities at infinite dilution by the Stokes-Einstein relation, we consider small isolated hard spheres, submerged in a liquid, that are subjected to Brownian motion The friction of the spheres in the liquid is given by the Stokes law Einstein used the Stokes law to calculate the mean-square displacement of a particle. The displacement increases linearly with time, and the proportionality constant is the Stokes-Einstein diffusivity... 

Technically, self-diffusion describes the displacement of a labeled molecule in a fluid of unlabeled but otherwise identical molecules. If this motion is chaotic, the mean square displacement will eventually obey the prediction of equation 13 and one can calculate the diffusion constant Dq for motion in direction g. This particular motion is difficult to observe in real adsorption systems so that simulation becomes of particular interest here. Before reviewing the literature, it is useful to consider the mean square displacement of a particle at short time rather than in the long time diffiisional limit. In the short time limit, one can carry out a Taylor series expansion to show that, after averaging, the mean square displacement in the q th direction q = x, y, z) is [60] ... 

The lack of long-range translational order in the 2D harmonic lattice is reflected in the mean-square displacement of a particle from its lattice site, which diverges logarithmically with increasing system size [47],... 

Diffusion Times. Brownian motion of molecules and particles is discussed in Section 5.2. The root-mean-square displacement of a particle is inversely proportional to the square root of its diameter. Examples are given in Table 9.4. The diffusion time for heat or matter into or out of a particle of diameter d is of the order of d2/ ()D where D is the diffusion coefficient. All this means that the length scale of a structural element, and the time scale needed for events to occur with or in such a structural element, generally are correlated. Such correlations are positive, but mostly not linear. 

In diffusion theory the mean-square displacement of a particle along a given axis is equal to 2Dt. Setting this equal to L% gives the time scale in Eq. (5.4.1a). The factor of 2 is omitted since we are interested only in the order of magnitude of the time scale. Likewise by setting the mean square displacement Dt equal to (t -1)2 we obtain Eq. (5.4.1b). 

Comparing the Eq. (67) with the known Einstein formula for the mean-square displacement of a particle making Brownian motion, the authors [83] have drawn the conclusion that the value of h (0 can be considered as )... 

The upper spatial boundary may be defined in a number of ways. Ideally we would define it as being infinitely far away from the electrode, i.e., a max = +00, such that changes in concentration at the electrode cannot have any effect on the concentration at the upper boundary on the time scale of the experiment. In practice, it transpires that it is not necessary to place the boundary infinitely far away from the electrode in order to meet this condition. Einstein s work on Brownian motion in 1905 [6] demonstrated that in one dimension, the root mean squared displacement of a particle from its starting position. 

In 1923, Wiener [24] proposed a rigorous mathematical model that exhibits a behavior similar to that observed in random motion. In this model, the displacement I is governed by a Gaussian (bell-shaped) probability distribution, with zero mean and unit variance. Therefore, if one plots the successive displacements of a particle (in one dimension for simplicity) over a certain period of time, the result (Figure 2.10a) corresponds to Gaussian, or white , noise. The sum of successive steps i, 2. , of the particle during n collision times is given by... 

Particles, chains, aggregates and floes were allowed to move in space according to their respective diffusion coefficients recomputed at each step from their masses and conformations. Brownian motion was represented by random walks. The number of random walks during a unit of (relative) time was proportional to the diffusion coefficients of the moving entities. The link between the relative and physical time was made by correlating the mean displacement of a reference particle (not participating in the aggregation process) and its diffusion coefficients. 

Brownian motion is defined as the irregular motion of microscopic particles suspended in a fluid due to collisions with the surrounding fluid molecules. The mean square displacement of a particle executing Brownian motion is directly proportional to the temperature of the fluid and inversely proportional to the viscosity of the fluid and the diameter of the particle. 

In the second part of his argument, Einstein relates the diffusion coefficient to a measurable property of Brownian motion such as the mean displacement of the particles. Let c(x, t) denote the probability for a Brownian particle to be at position X at time t. By following a probabilistic argument, the spatial and temporal changes in the concentration of the particles were shown to be related to the diffusion coefficient by... 

The mean-squared displacement of a particle in a solution subject to no external forces can be expressed as ... 

Einstein derived the following expression of the mean square displacement of a particle which executes random walk in a potential field, the profile of which is shown in Figure 2 A ... 

Equilibrium is a state of matter that results from spatial uniformity. In contrast, when there are concentration differences or gradients, particles will flow. In these cases, the rate of flow is proportional to the gradient. The proportionality constant between the flow rate and the gradient is a transport property for particle flow, this property is the diffusion constant. Diffusion can be modelled at the microscopic level as a random flight of the particle. The diffusion constant describes the mean square displacement of a particle per unit time. The fluctuation-dissipation theorem describes how transport properties are related to the ensemble-averaged fluctuations of the system in equilibrium. 

According to generally accepted theories (42, 43) the self-diffusion coefficient D of polymer melts is inversely proportional to the square of the chain length. For long chains, this leads to extremely low values of D. Since th mean square displacement of a particle in a specified direction, measure small D values is limited by the spatial resolution of the experimental technique. For instance, the first data on self-diffusion in a polymer melt were obtained following the diffusion of deuterated polyethylene into normal polyethylene by IR spectroscopy (44) and this technique could resolve only distances of 0.1 mm thus, experiments extending over a day could... 

A second possibility to calibrate trap stiffness is to use the equipartition theorem. The mean square displacement of a particle in a harmonic potential with stiffness Kt is given by the equipartition theorem as... 

Einstein determined, using osmotic pressure arguments, that the mean-squared displacement of a particle from an original position is proportional to time,... 

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