Brownian motion theory

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

In liquid solution. Brownian motion theory provides the relation between diffiision and friction coefficient... 

Brownian motion theory may be generalized to treat systems with many interacting B particles. Such many-particle Langevin equations have been investigated at a molecular level by Deutch and Oppenheim [58], A simple system in which to study hydrodynamic interactions is two particles fixed in solution at a distance Rn- The Langevin equations for the momenta P, (i = 1,2)... 

P. Espanol and I. Zuniga, Force autocorrelation function in Brownian motion theory, J. Chem. Phys. 98, 574 (1993). 

IV. Brownian Motion Theory a Model for the Zeroth-Order Con-... 

IV. BROWNIAN MOTION THEORY A MODEL FOR THE ZEROTH-ORDER CONDUCTANCE... 

The field TCP can be calculated on the basis of Brownian motion theory. Its initial time dependence is in every case described by a single exponential decay... 

In recent articles1,2 on the dynamics of stiff polymer chains, the Langevin version of Brownian motion theory was used instead of the more common Fokker-Planck approach. These investigations were made, however, only in the free-draining limit. 

In that year, he published three important papers the theory of Brownian motion an explanation of the photoelectric effect and the special theory of relativity. According to his Brownian motion theory, the fluid molecules move at random by thermal movement, and therefore a small particle would receive a random number of impacts of random strength and from random directions in any short period. If the particle is sufficiently small, this random collision by the fluid molecules would cause... 

Upon inserting these last equations into Eq. 20, the continuity equation becomes formally identical in form with Eq. 19 obtained by Chandrasekhar. However, it differs in that an expression for C has been obtained which is explicit in the intermolecular forces in contrast to the older Brownian motion theory in which C is a phenomenological constant only. 

Eisenschitz12 has calculated a cell probability density perturbation for viscous flow and thermal conduction using Brownian motion theory. The viscosity and thermal conductivity coefficients are then rewritten in terms of the displacement of the single molecule within the cell in place of intermolecular distances. The use of Brownian motion theory, however, leads to transport coefficients in terms of the frictional coefficient which again is not easy to evaluate. 

Brownian motion theory was verified by many scientists (T. Svedberg, A. Westgren, J.Perrin, L.de Broglie and others), who both observed individual particles and followed the diffusion in disperse systems [5]. The influence of various factors, such as the temperature, dispersion medium viscosity, and particle size on the value of the Brownian displacement, was evaluated. It was shown that the Einstein-Smoluchowski theory describes the experimental data adequately and with high precision. 

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... 

In this section it isshown that arbitrary dynamical properties in complicated systems can be described by equations which are analogous to the Langevin equation of Brownian motion theory [cf. Section (5.9)]. For example, the arbitrary property A is described by the equation... 

The first pubUshed criticism of the binary collision model was due to Fixman he retained the approximation that the relaxation rate is the product of a collision rate and a transition probabihty, but argued that the transition probability should be density dependent due to the interactions of the colliding pair with surrounding molecules. He took the force on the relaxing molecule to be the sum of the force from the neighbor with which it is undergoing a hard binary collision, and a random force mA t). This latter force was taken to be the random force of Brownian motion theory, with a delta-function time correlation ... 

Espanol, Pep. Zuniga, Ignacio. Force autocorrelation function in brownian motion theory. The Journal of Chemical Chem. Physics Phys., 1993, 98(1), 574-580. Yuan-Hui, Li. Gregory, Sandra. Diffusion of ions in sea water and in deep-sea sediments. Geochimica et Cosmochimica Acta, 1974, 38(5), 703-714. 

We can now derive an exact equation of motion for an arbitrary function A(t) which has the form of the Langevin equation of Brownian motion theory. To this end, the rate of change of A is written as... 

In classical Brownian motion theory, one usually assumes that the time evolution of the probability density p(F, t) is governed by the Chapman-Kolmogorov... 

According to the second law of thermodynamics, ary spontaneous process in an isolated system out of equilibrium will lead to an increase in the errtropy inside that system. In spite of the general validity of the second law, we have yet to fully rmderstand the problem of irreversible time evolution. Of course. Brownian motion theory does provide some insight into the direction that might be pursued in seeking answers to this problerrr. The most impor-tarrt idea to emerge from Brownian motion theory is the notion that dissipative or irreversible behavior arises from spontaneous equilibrium fluctuations. 

As in classical Brownian motion theory, the phenomenological time derivative is a coarse-grained time derivative obtained by time averaging the instantaneous time derivative (d/dt ) Sdj(t )) over the time scale At of macroscopic measurements (time resolution of the macroscopic observer). 

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