Diffusion through a Membrane
This Demonstration shows the effect of time delay on the dynamics of a simple diffusion model.[more]
Consider a system of two compartments with lengths and containing the same species separated by a permeable membrane of unit area. Each compartment is assumed to be well stirred and homogeneous, so each can be represented by a single concentration variable, and .
Fick's first law of diffusion gives
, with ,
where represents time, is the diffusion coefficient of , and is the thickness of the membrane. Implicit in this equation is the assumption that the time required for diffusion through the membrane is zero; in reality, the time is not zero, in fact there is a distribution of times for molecules to cross the membrane. If we consider a model  in which all molecules take the same time to transverse the membrane, then Fick's law becomes
with initial history functions
This system of delayed differential equations is solved with . The solution shows remarkable features of the model. If the compartments have equal volumes and different initial concentrations, the system approaches equilibrium with damped oscillations. Further, if the volumes of the compartments are different, we can start with equal concentrations and the system will approach equilibrium in a damped oscillatory fashion. Even more surprising, we can start with equal concentrations and equal volumes and equilibrium will be approached with oscillations.[less]
What is the meaning of the highly counterintuitive and apparently nonphysical behavior? The key lies in the fact that the final equilibrium value of lies significantly below the starting concentrations in the two compartments. The oscillations occur while the initially empty membrane is filling up. Consider the case of equal initial concentrations with cell much longer than cell . In the first interval , the same number of molecules leaves each cell, but, because cell is smaller, its concentration drops further. At time , when molecules begin to arrive, the effect is reversed, and the concentration in cell rises more rapidly than that in cell . Since molecules are now flowing in both directions, the effect will soon damp out. Readers may find the above result more convincing if they try to picture how the initially equal densities of people in two rooms of very different sizes connected by identical doors through an anteroom will change in time if, when the doors are opened, people move from room to room with equal probability and equal velocity without collisions. It is also reassuring to observe that when the results are averaged over a realistic distribution of (which results, among other things, from molecular collisions within the membrane), the oscillations disappear and the concentrations behave monotonically as expected.
 I. R. Epstein, "Delay Effects and Differential Delay Equations in Chemical Kinetics," International Reviews in Physical Chemistry, 11(1), 1992 pp. 135–160.