Exponential decay

A quantity $E(t)$ undergoes exponential decay with rate $\gamma >0$ when

The time constant $\tau >0$ in exponential decay is the time over which the decaying quantity is multiplied by $e^{-1}$. A response approaching equilibrium as $q(t)=q_{\infty }+(q(0)-q_{\infty })e^{-t/\tau }$ has the same time constant. For a resistor-inductor circuit it is $L/R$.

A stretched exponential has the form $e^{-c{x}^{\alpha }}$ with $c>0$ and $0<\alpha <1$. A boundary cost proportional to the square root of an enclosed area naturally produces exponent $\alpha =1/2$.

The half-life is the time at which an exponentially decaying quantity reaches half its initial value. For decay rate $\gamma$,