Collection of Brownian Particles

The Smoluchowski equation (Equation 6.63) for the probability distribution function of a diffusing particle under the influence of an external force is also applicable for the time evolution of a collection of particles. Let c(r, t) be the local concentration of the particles at the location r at time t. In view of the conservation of the total number of particles in the system, the rate of change of local particle concentration in a volume element is given by the continuity equation 

In the presence of an external force f, the velocity of the particle is given in the friction-dominated situations as 

We have taken the potential and the free energy landscape acting on the particle to be time-independent. By substituting Equations 6.68 and 6.69 into Equation 6.67, we obtain 

In equilibrium, the flux must be zero, and the solution of Equation 6.71 must be the Boltzmann distribution function for the concentration of the particles at the location r where the effective potential is F(r), 

The stipulation that Equation 6.71 must be consistent with the equilibrium result of Equation 6.72 results in the Einsteinian result of Equation 6.43, 

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The rale of collection of Brownian particles under the influence of interaction forces between the collector surface and the particles is calculated by (a) incorporating the interaction forces in the rate constant of a virtual, first order, chemical reaction taking place on the surface of the collector, and by (b) solving the convective diffusion equation subject to that chemical reaction as a boundary condition. Several geometries (sphere, cylinder, rotating disc) are considered for the collector. 

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