Why is the relaxation time defined using a decrease to 1/e of the initial displacement?

In the study of relaxation phenomena, the response of a system to a small disturbance is typically described by a first-order differential equation. When a system is displaced from equilibrium, the rate at which it returns is proportional to the magnitude of that displacement.1

Mathematically, this process follows an exponential decay function. The use of $1/e$ (approximately 0.368) as the benchmark for defining the relaxation time ($\tau$) is a standard convention for characterizing exponential processes. Because the decay is exponential, the displacement $\Delta X$ at time $t$ can be expressed as:2

$$\Delta X(t)=\Delta X_0{e}^{-t/\tau }$$

When the elapsed time $t$ equals the relaxation time $\tau$, the exponent becomes $-1$, and the displacement reduces to $1/e$ of its initial value ($\Delta X_0$). This specific value provides a consistent, universal measure to quantify how quickly a system re-equilibrates following a physical or chemical change.132