Delays and Differential Delay Equations

Mathematically speaking, the most important tools used by the chemical kineticist to study chemical reactions like the ones we have been considering are sets of coupled, first-order, ordinary differential equations that describe the changes in time of the concentrations of species in the system, that is, the rate laws derived from the Law of Mass Action. In order to obtain equations of this type, one must make a number of key assumptions, some of which are usually explicit, others more hidden. We have treated only isothermal systems, thereby obtaining polynomial rate laws instead of the transcendental expressions that would result if the temperature were taken as a variable, a step that would be necessary if we were to consider thermochemical oscillators (Gray and Scott, 1990), for example, combustion reactions at metal surfaces. What is perhaps less obvious is that our equations constitute an average over quantum mechanical microstates, allowing us to employ a relatively small number of bulk concentrations as our dependent variables, rather than having to keep track of the populations of different states that react at different rates. Our treatment ignores fluctuations, so that we may utilize deterministic equations rather than a stochastic or a master equation formulation (Gardiner, 1990). Whenever we employ ordinary differential equations, we are making the approximation that the medium is well mixed, with all species uniformly distributed; any spatial gradients (and we see in several other chapters that these can play a key role) require the inclusion of diffusion terms and the use of partial differential equations. All of these assumptions or approximations are well known, and in all cases chemists have more elaborate techniques at their disposal for treating these effects more exactly, should that be desirable. Another, less widely appreciated idealization in chemical kinetics is that phenomena take place instantaneously—that a change in [A] at time t generates a change in [B] time t and not at some later time t + τ. On a microscopic level, it is clear that this state of affairs cannot hold.