Smoluchowski operator

The adjoint Smoluchowski operator was obtained using in the partial integrations over the particle positions the incompressibility condition, Trace K = 0, which should always holds for the solvents of interest in this review. It takes the explicit form (where boundary contributions are neglected throughout, simplifying the partial integrations) ... 

The SL operator is given simply by the sum of the FPK operator for subsystem 1 plus the Smoluchowski operator for subsystem 2. It is not complete, in the sense that it does not have a meaningful solution for r—> -f-co, which should be the equilibrium distribution. If we require that the total system tends to the Boltzmann distribution given by the total energy (including the interaction term Fj ,)... 

Here k x) is the sink function that invokes the contribution from the fast modes. The generalized Smoluchowski operator L,- is given by ... 

The Smoluchowski limit to Caldeira-Leggett s master equation is now readily obtained by replacing all those pB-dopendent operators in eqn (13.9) and eqn (13.11). In particular, the Fokker-Planck operator becomes the Smoluchowski operator, with a diffusion constant of yo = [Pg.344]

There are two general theories of the stabUity of lyophobic coUoids, or, more precisely, two general mechanisms controlling the dispersion and flocculation of these coUoids. Both theories regard adsorption of dissolved species as a key process in stabilization. However, one theory is based on a consideration of ionic forces near the interface, whereas the other is based on steric forces. The two theories complement each other and are in no sense contradictory. In some systems, one mechanism may be predominant, and in others both mechanisms may operate simultaneously. The fundamental kinetic considerations common to both theories are based on Smoluchowski s classical theory of the coagulation of coUoids. 

In the limit as ftact the rate of reaction of encounter pairs is very fast. The Collins and Kimball [4] expression, eqn. (25), reduces to the Smoluchowski rate coefficient, eqn. (19). Naqvi et al. [38a] have pointed out that this is not strictly correct within the limits of the classical picture of a random walk with finite jump size and times. They note the first jump of the random walk occurs at a finite rate, so that both diffusion and crossing of the encounter surface leads to finite rate of reaction. Consequently, they imply that the ratio kactj TxRD cannot be much larger than 10 (when the mean jump distance is comparable with the root mean square jump distance and both are approximately 0.05 nm). Practically, this means that the Reii of eqn. (27) is within 10% of R, which will be experimentally undetectable. A more severe criticism notes that the diffusion equation is not valid for times when only several jumps have occurred, as Naqvi et al. [38b] have acknowledged (typically several picoseconds in mobile solvents). This is discussed in Sect. 6.8, Chap. 8 Sect 2.1 and Chaps. 11 and 12. Their comments, though interesting, are hardly pertinent, because chemical reactions cannot occur at infinite rates (see Chap. 8 Sect. 2.4). The limit kact °°is usually taken for operational convenience. 

Rather than solve this equation by standard techniques and develop the connection between rate coefficient and density, p, as originally done by Smoluchowski, Northrup and Hynes used projection operator techniques to obtain the probability that a reactant pair survives at a time t after formation, P t) = /drp(r, t) as usual. They found that the survival probability satisfies an equation (which is derived in Appendix D)... 

We have in this way obtained a generalization of Einstein s theory of the interaction between matter and radiation including multiple photon processes and involving transition probabilities. But there is a basic difference. The operator definite positive. We no longer have a simple addition of transition probabilities. This corresponds exactly to the interference of probabilities discussed in Section IV. The process is not of the simple Chapman-Smoluchowski-Kolmogoroff type (Eq. (11)) the operator transition probability. As the result, the second of the two sequences discussed above may decrease the effect of the first one. It is very interesting that even in the limit of classical mechanics (which may be performed easily in the case of anharmonic oscillators) this interference of probabilities persists. This is in agreement with our conclusion in Section IV. 

Instead of the one-step process we now consider the generalized diffusion or Smoluchowski equation (1.9). Take a finite interval Lsplitting probabilities nL(y) and nR(y). They obey a differential equation with the adjoint operator... 

Exercise. Let W be the differential operator in the Smoluchowski equation (1.9), defined in the interval L y R on the space of functions P(y) that vanish at R and obey the reflecting boundary condition at L. Then the adjoint has the property that... 

J. Troe Professor Marcus, you were mentioning the 2D Sumi-Marcus model with two coordinates, an intra- and an intermolecu-lar coordinate, which can provide saddle-point avoidance. I would like to mention that we have proposed multidimensional intramolecular Kramers-Smoluchowski approaches that operate with highly nonparabolic saddles of potential-energy surface [Ch. Gehrke, J. Schroeder, D. Schwarzer, J. Troe, and F. Voss, J. Chem. Phys. 92, 4805 (1990)] these models also produce saddle-point avoidances, but of an intramolecular nature the consequence of this behavior is strongly non-Arrhenius temperature dependences of isomerization rates such as we have observed in the photoisomerization of diphenyl butadiene. 

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). 

This theory, as originated from the early work of Smoluchowski [20], nowadays has numerous applications in several branches of chemistry, such as colloidal chemistry, aerosol dynamics, catalysis and the physical chemistry of solutions as well as in the physics and chemistry of the condensed state [21-24]. Until recently, its branch called standard chemical kinetics [12, 15, 16] based on the law of mass action seemed to be quite a complete and universal theory. However, because of their entirely phenomenological character, theories of this kind always operate with the reaction rates K which are postulated to be time-independent parameters. 

The diffusion coefficient D has appeared in both the macroscopic (Section 4.2.2) and the atomistic (Section 4.2.6) views of diffusion. How does the diffusion coefficient depend on the structure of the medium and the interatomic forces that operate To answer this question, one should have a deeper understanding of this coefficient than that provided hy the empirical first law of Tick, in which D appeared simply as the proportionality constant relating the flux / and the concentration gradient dc/dx. Even the random-walk intapretation of the diffusion coefficient as embodied in the Einstein-Smoluchowski equation (4.27) is not fundamental enough because it is based on the mean square distance traversed by the ion after N stqis taken in a time t and does not probe into the laws governing each stq) taken by the random-walking ion. 

We suppose that a small probing held Fj, having been applied to the assembly of dipoles in the distant past (f = —oo) so that equilibrium conditions have been attained at time t = 0, is switched off at t = 0. Our starting point is the fractional Smoluchowski equation (172) for the evolution of the probability density function W(i), cp, t) for normal diffusion of dipole moment orientations on the unit sphere in configuration space (d and (p are the polar and azimuthal angles of the dipole, respectively), where the Fokker-Planck operator LFP for normal rotational diffusion in Eq. (8) is given by l j p — l j /> T L where... 

An equation with the form of the sink Smoluchowski equation may now be derived by introducing an operator that projects out the velocities of the solute molecules. This is just the pair version of the derivation presented in the previous section. 

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