Brownian motion linearization

The relationships follow from considerations of Brownian motions and thermal fiuctuations which also infiuence the internal motions in flexible objects. is the translation coefficient of the particle s center of mass where the subscript indicates the z-average over the molar mass distribution. The first bracket in Eq. (12) describes the concentration dependence which often is well represented by a linear dependence... 

It is, perhaps, less known that the concepts of complementarity and indeterminacy also arise naturally in the theory of Brownian motion. In fact, position and apparent velocity of a Brownian particle are complementary in the sense of Bohr they are subject to an indeterminacy relation formally similar to that of quantum mechanics, but physically of a different origin. Position and apparent velocity are not conjugate variables in the sense of mechanics. The indeterminacy is due to the statistical character of the apparent velocity, which, incidentally, obeys a non-linear (Burgers ) equation. This is discussed in part I. 

The stochastic motion of particles in condensed matter is the fundamental concept that underlies diffusion. We will therefore discuss its basic ideas in some depth. The classical approach to Brownian motion aims at calculating the number of ways in which a particle arrives at a distinct point m steps from the origin while performing a sequence of z° random steps in total. Consider a linear motion in which the probability of forward and backward hopping is equal (= 1/2). The probability for any sequence is thus (1/2). Point m can be reached by z° + m)/2 forward plus (z° m)/2 backward steps. The number of distinct sequences to arrive at m is therefore... 

In another paper, R. Kuho (Kcio University, Japan) illustrates in a rather technical and mathematical fashion tire relationship between Brownian motion and non-equilibrium statistical mechanics, in this paper, the author describes the linear response theory, Einstein s theory of Brownian motion, course-graining and stochastization, and the Langevin equations and their generalizations. 

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... 

This result implies that the energy equipartition relationship of Eq. (2.S) applies as well as the general definitions of Chapter I. Note that for Af m the variable turns out to be coupled weakly to the thermal bath. This condition generates that time-scale separation which is indispensable for recovering an exponential time decay. To recover the standard Brownian motion we have therefore to assiune that the Brownian particle be given a macroscopic size. In the linear case, when M = w we have no chance of recovering the properties of the standard Brownian motion. In the next two sections we shall show that microscopic nonlinearity, on the contrary, may allow that the Markov characters of the standard Brownian motion be recovered with increasing temperature. 

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... 

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