Smoluchowski equation motion

Diflfiision-controlled reactions between ions in solution are strongly influenced by the Coulomb interaction accelerating or retarding ion diffiision. In this case, die dififiision equation for p concerning motion of one reactant about the other stationary reactant, the Debye-Smoluchowski equation. 

Equation (2.6) is called the Fokker-Planck equation (FPE) or forward Kolmogorov equation, because it contains time derivative of final moment of time t > to. This equation is also known as Smoluchowski equation. The second equation (2.7) is called the backward Kolmogorov equation, because it contains the time derivative of the initial moment of time to < t. These names are associated with the fact that the first equation used Fokker (1914) [44] and Planck (1917) [45] for the description of Brownian motion, but Kolmogorov [46] was the first to give rigorous mathematical argumentation for Eq. (2.6) and he was first to derive Eq. (2.7). The derivation of the FPE may be found, for example, in textbooks [2,15,17,18],... 

For example, in the case of PS and applying the Smoluchowski equation [333], it is possible to estimate the precipitation time, fpr, of globules of radius R and translation diffusion coefficient D in solutions of polymer concentration cp (the number of chains per unit volume) [334]. Assuming a standard diffusion-limited aggregation process, two globules merge every time they collide in the course of Brownian motion. Thus, one can write Eq. 2 ... 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

Using the SI units, the velocity of the EOF is expressed in meters/second (m s ) and the electric held in volts/meter (V m ). Consequently, the electroosmotic mobility has the dimension of m V s. Since electroosmotic and electrophoretic mobility are converse manifestations of the same underlying phenomena, the Helmholtz-von Smoluchowski equation applies to electroosmosis, as well as to electrophoresis (see below). In fact, it describes the motion of a solution in contact with a charged surface or the motion of ions relative to a solution, both under the action of an electric held, in the case of electroosmosis and electrophoresis, respectively. 

The central problem in the theory of geminate ion recombination is to describe the relative motion and reaction with each other of two oppositely charged particles initially separated by a distance ro- If we assume that the particles perform an ideal diffusive motion, the time evolution of the probability density, w(r,t), that the two species are separated by r at time t, may be described by the Smoluchowski equation [1,2]... 

When the motion of electrons and positive ions in a particular system may be described as ideal diffusion, the process of bulk recombination of these particles is described by the diffusion equation. The mathematical formalism of the bulk recombination theory is very similar to that used in the theory of geminate electron-ion recombination, which was described in Sec. 10.1.2 ( Diffusion-Controlled Geminate Ion Recombination ). Geminate recombination is described by the Smoluchowski equation for the probability density w(r,i) [cf. Eq. (2)], while the bulk recombination is described by the diffusion equation for the space and time-dependent concentration of electrons around a cation (or vice versa), c(r,i). 

The breakdown of the diffusion theory of bulk ion recombination in high-mobility systems has been clearly demonstrated by the results of the computer simulations by Tachiya [39]. In his method, it was assumed that the electron motion may be described by the Smoluchowski equation only at distances from the cation, which are much larger than the electron mean free path. At shorter distances, individual trajectories of electrons were simulated, and the probability that an electron recombines with the positive ion before separating again to a large distance from the cation was determined. The value of the recombination rate constant was calculated by matching the net inward current of electrons... 

The force in eqn. (208) on the particle k is F(rfe) due to the motion of all the N particles with respect to the unperturbed solvent. In Chap. 3, the Debye—Smoluchowski equation was derived from thermodynamic arguments. It was pointed out that the spatial gradient of the chemical potential at some point is the force acting at that point. If the potential energy is U, then the chemical potential is [cf. eqn. (40)]... 

Morse and Feshbach [499] have discussed the variation approach to a description of equations of motion for diffusion. Their approach is straightforward and is generalised here to consider the cases where there is an energy of interaction, U, between the pair of particles, separated by a distance r at time t. It is relatively easy to extend this to a many-body situation. The usual Euler form of the equation of motion is the Debye— Smoluchowski equation, which has been discussed in much detail before, viz. 

As is well known, dynamic properties of polymer molecules in dilute solution are usually treated theoretically by Brownian motion methods. Tn particular, the standard approach is to use a Fokker-Planck (or Smoluchowski) equation for diffusion of the distribution function of the polymer molecule in its configuration space. 

The above-described pair problem is treated by the Smoluchowski equation [3, 19] - see Fig. 1.10. It operates with the probability densities (Fig. 1.11) and contains the recombination rate characterizing particle motion. Knowledge of the probability density to find a particle at a given point at time moment t gives us (by means of a trivial integration over reaction volume) the quantity of our primary interest - survival probability of a particle in the system with... 

New difficulties arise when we try to take into account the dynamical interaction of particles caused by pair potentials U(r) mutual attraction (repulsion) leads to the preferential drift of particles towards (outwards) sinks. This kind of motion is described by the generalization of the Smoluchowski equation shown in Fig. 1.10. In terms of our illustrative model of the chemical reaction A + B —> B the drift in the potential could be associated with a search of a toper by his smell (Fig. 1.12). An analogy between Schrodinger and Smoluchowski equations is more than appropriate indeed, it was used as a basis for a new branch of the chemical kinetics operating with the mathematical formalism of quantum field theory (see Chapter 2). 

Differentiating equation (3.2.5) and making use of the motion equation (3.2.4), we arrive at the well-known Smoluchowski equation [65-70]... 

Electrophoresis is the motion of charged particles relative to the electrolyte in response to an applied DC-electric field the field causes a shift in the particle counterion cloud, the counterion-diminished end of the particle attracts other counterions from the bulk fluid, counterions from the displaced cloud diffuse out into the bulk fluid, and the particle migrates. The particle velocity is predicted by the Smoluchowski equation. 

Diffusion can be considered as a stochastic or random process and described by the so-called Fokker-Planck equation adapted to Brownian motion. This equation is also known as the Smoluchowski equation. We consider the description of stochastic processes and Brownian motion in more detail in Section 11.1 and Appendix H. 

University, Krak6w [i]. He described Brownian molecular motion independently from Einstein considering the collisions explicitly between a particle and the surrounding solvent molecules [ii], worked on colloids [iv-v], and obtained an expression for the rate with which two particles diffuse together (-> Smoluchowski equation (b)) [iii-v]. He also derived an equation for the limiting velocity of electroosmotic flow through a capillary (-> Smoluchowski equation (a)). 

At first the results arising from the Einstein-Smoluchowski equation ( = 2Dt) may seem difficult to understand. Thus, the diffusion considered in the equation is random. Nevertheless, the equation tells us that there is net movement in one direction arising from this random motion. Furthermore, it allows us to calculate how far the diffusion front has traveled. Is there something curious about randomly moving particles covering distances in one direction Comment constructively on this apparently anomalous situation. 

The excellent review of Chandrasekhar provides a detailed account of the history of the subject, to which both Smoluchowski and Einstein made fundamental contributions. It is worth mentioning the well-known paper of Kramers, who provided a rigorous derivation of the Smoluchowski equation from the complete Fokker-Planck equation of a Brownian particle in an external potential. This problem allows us to explain what we mean by a systematic version of the AEP. We can state the problem as follows. Let us consider the motion of a free Brownian particle described by the one-dimensional counterpart of Eq. (1.2),... 

We begin with the following Smoluchowski equation describing the Brownian motion under a potential V(i) for the probability density p(x, t) ... 

For three-dimensional motion under a central potential V(r) where r is the radial distance, the Smoluchowski equation corresponding to Eq. (1) can be expressed by... 

As the shear rate increases, the numerical solutions of the Smoluchowski equation (11-3) begin to show deviations from the predictions of the simple Ericksen theory. Tn particular, the scalar order parameter S begins to oscillate during the tumbling motion of the director (for a discussion of tumbling, see Sections 11.4.4 and 10.2.6). The maxima in the order parameter occur when the director is in the first and third quadrants of the deformation plane i.e., 0 < 9 — nn < it j2, where n is an integer. Minima of S occur in the second and fourth quadrants. The amplitude of the oscillations in S increases as y increases, until S is reduced to only 0.25 or so over part of the tumbling cycle. 

Howarth (HI5) assumed a mechanism analogous to Brownian motion. But his derivation for collision rates is wrong because he introduced a variable particle diffusion coefficient in the Smoluchowski equation (06, S27) which had been derived on the basis of a constant coefficient. 

Debye extended the foregoing arguments in order to establish the Smoluchowski equation [Eq. (5)] for the rotational Brownian motion of a dipolar particle about a... 

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