Electric field buildup and relaxation for chemical diffusion

By applying Maxwell's equation (Poisson's law) to the flux equations for ionic diffusion it is shown that analytical solutions for the flux of charge and the compensating electrical field can be derived for the limits t→0 and t→∞. (Actually the long time limit becomes appropriate after a time equivalent to a small multiple of the mean time between successive jumps of all ions.) These expressions permit derivation of the Nernst-Planck equations and their generalization to multi-component systems without any self-contradictory assumptions. They also allow an approximate interpolation between the two limiting time periods so that a close approximation of the field and flux is obtained for the entire time interval. It should be noted that the flux of charge changes sign during the process and tends to zero much more quickly than the electric field. The scaling constants for time, energy and electrical flux are evaluated in general and for conditions relevant to alkali ion exchange in a silicate glass near the glass transition temperature. Values for the time constant, the maximum field, and the maximum flux are of the order of fractional micro-seconds, one megavolt/cm, and one ampere/cm², respectively.     

Novel perturbation expansion for the Langevin equation

We discuss the randomly driven systemdx/dt= -W(x) +f(t), wheref(t) is a Gaussian random function or stirring force withf(t)f(t)=2(t–t), andW(x) is of the formgx ¹⁺². The parameter is a measure of the nonlinearity of the equation. We show how to obtain the correlation functionsx(t)f(t)···x(t( n)) _f as a power series in. We obtain three terms in the expansion and show how to use Padé approximants to analytically continue the answer in the variable. By using scaling relations, we show how to get a uniform approximation to the equal-time correlation functions valid for allg and.