Einstein should be allowed his mistakes, like the rest of us, and Steven Weinberg understandably points out only the most newsworthy. I write to point out another misunderstanding—mistake, if you will—in Einstein’s work only because it is often found in the literature today.
Einstein described diffusion as the motion of neutral particles on atomic (Brownian) length and time scales. He used a stochastic differential equation—a Langevin equation—in the high-friction limit to describe diffusive trajectories. Einstein did not discuss how his treatment could accommodate macroscopic boundary conditions or produce macroscopic flow, which is, after all, what Fick’s law of diffusion is all about.
Langevin equations, in the spirit of Einstein’s work, are widely used today to describe the motion and fluctuations of density of charged particles in, for example, aqueous solutions. The electric force in those equations is usually described by a steady function. Fluctuations in number density of charged particles are allowed in Einstein’s treatment but fluctuations in net charge and electric potential are not. Traditional Langevin equations of Brownian motion seem inconsistent with the idea that charge creates electric force and so are unlikely to be helpful, at least in my view. It is hard to imagine systems in which the number density of ions can fluctuate while the number density of charge does not.
I believe Einstein’s description of Brownian motion must be coupled to equations describing the electric field when the diffusing particles have significant charge. An equation is needed to show how the charge on one particle creates force on another. The ink particles studied by Robert Brown were surely charged. The fluctuating electric field and stochastic flow can be computed from the density of ink particles, ions, and solvent molecules by solving Poisson’s or Maxwell’s equations together with flow equations. (Spatially inhomogeneous boundary conditions are needed to force the macroscopic flow described by Fick’s law.)
This so-called self-consistent treatment of diffusion and the electric field is used in computational electronics to design the transistors and integrated circuits of our electronic technology. 1 Diffusion and the electric field have not been treated self-consistently in most of computational chemistry and biology—for example, in simulations of molecular dynamics of ions or proteins—although such treatments are found in analyses of ionic motion through protein channels. 2–5