Why can the response to a small disturbance often be described by a first-order differential equation?

In the study of relaxation phenomena, the response of a system to a small disturbance is often described by a first-order differential equation because, for small displacements from equilibrium, the rate of return to equilibrium is approximately proportional to the magnitude of that displacement.1

This linear relationship allows scientists to apply a linear rate equation where terms denoting changes with time are to the first power. Specifically, the rate of disappearance of a small displacement ($\Delta X$) is expressed as:

$$\frac{d(\Delta X)}{dt}=-\frac{\Delta X}{\tau }$$

In this equation, $\tau$ represents the relaxation time, which is the reciprocal of the constant of proportionality. A significant advantage of using small disturbances is that this linear approximation holds true, simplifying the analysis of complex processes—such as energy redistribution among atoms or shifts in chemical reaction equilibria—into a mathematically manageable first-order model.1