S. Harris

Steady absorption of Brownian particles by a sphere

The problem of steady Brownian motion in the presence of an absorbing sphere is studied in the full Brownian particle position, velocity phase space. Approximate solutions are found for the Brownian particle distribution function by solving the Fokker–Planck equation subject to the boundary condition that the solution for emergent particles vanish on the surface of the sphere. Comparisons are made with results from the macroscopic theory based on the diffusion equation. We find that the macroscopic theory is not valid in a boundary layer of extent depending on the radius of the Brownian particle and absorber and the viscosity and temperature of the host fluid in a fairly complicated manner. Close to the boundary, Fick’s law only holds if the diffusion coefficient is taken to be space dependent. One significant consequence of our results is that the macroscopic theory of diffusion‐controlled rate constants is shown not to be generally valid. The corrected values for the rate constant that we find for aeros...

Dynamical Langevin Equation for a Simple Polar Fluid

We consider a dilute system of heavy (Brownian) polar molecules dissolved in a solvent of lighter molecules. Using a mass ratio expansion technique, equations of motion for the Brownian particle are obtained which have the form of coupled Langevin equations for the translational and rotational motion. Expressions for the friction coefficients in terms of the molecular properties are also obtained.

Brownian motion with an absorbing boundary

We consider the problem of determining the joint position–velocity distribution function for a Brownian particle confined by an absorbing boundary. The approach used here, which we believe to be novel in this context, involves an integral equation formulation in terms of the distribution function, for the unrestricted problem, of first turning points in the half‐space that does not contain the initial source. The advantage of this formulation is that it leads to an equation of the Wiener–Hopf type, so that in principle explicit solutions can be found by standard methods. An approximate solution is found for the case of large friction which reproduces the general features of the solution found in the theory based on the use of the diffusion equation; this result illustrates the potential of our formulation with regard to obtaining an exact solution to the absorbing boundary problem.

Perturbation solution of the one particle generalized Smoluchowski equation

The generalized Smoluchowski equation (GSE) can be used as the basis for a theory of the light scattering spectrum for a system of macromolecules in solution. We consider the solution of the one particle reduced GSE equation for the case of spherical macromolecules at low concentration through the use of a closure‐effecting ansatz that relates the one and two particle distribution functions and an expansion in the macromolecule concentration. The equations which result are well defined in the mathematical sense, and solutions are obtained for the first two terms in the expansion in a form suitable for application the light scattering problem. Although we restrict ourselves here to the usual model of the GSE with the hydrodynamic interactions given by the Oseen expression, more complicated forms for the hydrodynamic interaction could also be accomodated in the theory at the expense of considerably more lengthy calculations. A brief, formal application of the results obtained to the light scattering problem...

Convexity of the H Function: The Weak‐Coupled Master Equation

We show, for the weak‐coupled master equation, that the choice of H function which preserves the usual relationship with the thermodynamic entropy is convex as a function of time.

Structure of the H Function for a Hard‐Sphere Gas

We consider, for a spatially uniform hard‐sphere gas, the generalization of the connection between the H function and the thermodynamic entropy to that limit where gas imperfections are taken into account through the first virial correction term. Our choice of a generalized H function is based on the expansion of the canonical entropy density given by Nettleton and Green. We show that this choice allows us to give a physical significance to the condition that Boltzmanns H function be convex as a function of time.

Dynamical theory of light scattering in dilute macromolecular solutions: preliminary results based on oseen's model

Abstract A recently proposed method for solving the one-particle generalized Smoluchowski equation is used to determine the light scattering spectrum for a dilute solution of spherical, uncharged macromolecules. We find a frequency and wave number dependent effective diffusion coefficient which offers a possible explanation for the difference in sign found for this quantity at small and large frequencies in previous work. Our results are based on the use of Oseens model for the hydrodynamic interaction, and in this sense are considered as preliminary; an extension of these results using a higher order hydrodynamic theory is in progress.

Interface motion for mass redistribution at small supersaturation

We consider the problem of mass redistribution in a two‐phase system. Solutions are obtained for the interface motion utilizing both a model and the exact microscopic Fokker–Planck equation. In each case an initial linear growth is followed by a distinctive transition to the long time t1/2 diffusive growth.

Macromolecular self‐diffusion and momentum autocorrelation functions in dilute solutions

We use an improved version of the solution to the one particle generalized Smoluchowski equation found earlier to calculate the self‐diffusion coefficient D and the long time behavior of the momentum autocorrelation function φ (t) for a dilute system of uncharged, spherical macromolecules in solution. For D we find the infinite dilution result which is consistent with other theories based on Oseen’s model which we also use; our calculation does eliminate some other assumptions which are used in previous calculations. The asymptotic decay of φ (t) is found to be algebraic, as t−5/2; the −5/2 exponent is probably due to the use of Oseen’s model, and for a higher order hydrodynamic theory which gives a nonvanishing o (c) contribution to D we expect that φ (t) will decay as t−3/2.