Motion of a Brownian Particle

We need, however, a more refined and realistic description, since this equation predicts an exponential decay of the initial velocity to zero, in contrast to the observed incessant motion of a Brownian particle. Therefore, we must add to the systematic friction force the action of all individual solvent molecules on the Brownian particle, which results in an additional term F(t) ... 

As a result of AEP, the initial system of the set of Eqs. (81) is reduced to the equation describing the diffusional motion of a Brownian particle which undergoes the action of an additive and a multiplicative noise (with intensities D and Q, respectively) in the presence of a renormalized bounding potential, Eq. (90). The Markovian l t corresponds to X 00. If we take such a limit at a ed value of y, = d, and the case studied by Htoggi is recovered. Of course, having neglected the condition X c y we have reduced the problem to a trivial diffusional (lowest-order) approximation. 

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... 

Linear Motion of a Brownian Particle. In the amplest case of Brownian motion, a massive particle is immersed in a mediiun of lighter partides whose rapid thermal motion produces a quickly fluctuating force on the massive Brownian particle. This force will be in part correlated with the motion of the Brownian particle itself. Langevin s simplifying hypothesis is that the correlated part of the force exerted by the medium is propor< tional and opposed to the velocity u of the particle. Langevin s equation of motion then has the form... 

Next, Fig. 5 shows a typical center of mass trajectory for the case of a full chaotic internal phase space. The eyecatching new feature is that the motion is no more restricted to some bounded volume of phase space. The trajectory of the CM motion of the hydrogen atom in the plane perpendicular to the magnetic field now closely resembles the random motion of a Brownian particle. In fact, the underlying equation of motion at Eq. (35) for the CM motion is a Langevin-type equation without friction. The corresponding stochastic Lan-gevin force is replaced by our intrinsic chaotic force — e B x r). A main characteristic of random Brownian motion is the diffusion law, i.e. the linear dependence of the travelled mean-square distance on time. We have plotted in Fig. 6 for our case of a chaotic force for 500 CM trajectories the mean-square distance as a function of time. Within statistical accuracy the plot shows a linear dependence. The mean square distance

of the CM after time t, therefore, obeys the diffusion equation... 

Here x is the position of the slider relative to the substrate, h /Zn is the substrate s period, and/o is the (zero-temperature) static friction force, whose scaling with the area of contact and normal load we just discussed. In order to incorporate the effects of thermal fluctuations on the motion of the slider, one can exploit the isomorphism to the motion of a Brownian particle moving on a substrate. A nice description of that problem is given by Risken in Chapter 11 of Ref. 64. Here we will discuss some of the aspects that we believe to be important for friction. 

The translational and rotational motion of a Brownian particle immersed in a fluid continuum is well described by the Stokes-Einstein and Debye equations, respectively. 

In his treatment of Brownian motion, Langevin began by writing down the equation of motion of a Brownian particle in a suspension. He assumed the forces acting on it could be divided into two parts ... 

As the third application, we consider the motion of a Brownian particle in a periodic potential. Schmid [69] was the first to study this problem, and many others have since then applied a variety of techniques to this model [70-72]. The importance of the model stems from its widespread... 

Equation (1.20) has many applications in this book, such as the diffusion motion of a Brownian particle (Appendix 3.D) and the probability distribution of the end-to-end vector of a long polymer chain. The latter case will be studied in this chapter. Essentially, (x ) / jjj gq (1.20) can be regarded as the mean projection of an independent segment (bond) vector in one of the three coordinate directions (i.e. x, y or z all three directions are equivalent). As long as the considered polymer chain is very long, we can always apply the central limit theorem, regardless of the local chemical structure. 

Lukic B et al (2005) Direct observation of nondiffusive motion of a brownian particle. Phys Rev Lett 95 160601... 

The motion of a Brownian particle of the chosen macromolecule agitates a volume with a size of through its adjacent chain particles. This volume... 

When we expressed Eq. (2.18) for the distribution function p R), we did this based on the assumption that the random walk carried out by a chain of freely jointed segments should be equivalent to the motion of a Brownian particle. We pointed out that the equivalence is lost in the presence of excluded volume forces, however, this is not the only possible deficiency in the treatment. Checking the properties for large values of R we find that the Gaussian function never vanishes and actually extends to infinity. For the model chain, on the other hand, an upper limit exists, and it is reached for... 

If = 0, Eq.24 describes the motion of a Brownian particle undergoing spontaneous decay and from Eqs. 19 and 24 it is easy to compute that... 

In Chapter 11, we discussed how Paul Langevin devised an equation that describes the motion of a Brownian particle (Eq. 11.33). Similar equations can be devised to describe the time evolution of any stochastic process X t). 

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