This term is often used in many areas of science, such as in atmospheric chemistry, medicine, theoretical physics, and cosmology.
In other words, if we examine an observable, X, associated with a system, (temperature or density, for example) then after a time, τ, the initial difference between the initial value of the observable and the equilibrium value, ΔXi, will have decreased to ΔXi /e where the number e ≈ 2.71828.
We could depict this as an equation: Let us assume that this reaction follows first order kinetics, meaning that the conversion of A into B depends only on the concentration of A, and the rate constant which dictates the velocity at which this happens, k. We could write the following reaction to describe this first order kinetic process: This ordinary differential equation states that a change (in this case the disappearance) of the concentration of A, d[A]/dt, is equal to the rate constant k multiplied by the concentration of A.
This would yield This states that after one lifetime (1/k), the ratio of final to initial concentrations is equal to about 0.37.
It is in this manner that e-folding lends us an easy way to describe the number of lifetimes that have passed.