E-folding

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In science, e-folding is the time interval in which an exponentially growing quantity increases or decreases by a factor of e;^[1] it is the base-e analog of doubling time. This term is often used in many areas of science, such as in atmospheric chemistry, medicine, theoretical physics, and cosmology.

In cosmology the e-folding time scale is the proper time in which the length of a patch of space or spacetime increases by the factor e.

In finance, the logarithmic return or continuously compounded return, also known as force of interest, is the reciprocal of the e-folding time.

The process of evolving to equilibrium is often characterized by a time scale called the e-folding time, τ. This time is used for processes which evolve exponentially toward a final state (equilibrium). 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, ΔX_i, will have decreased to ΔX_i /e where the number e ≈ 2.71828.

${\displaystyle T_{e}={\frac {T_{d}}{\ln(2)}}={\frac {t}{\ln({N(t)}/{N(0)})}}={\frac {t}{\ln(1+r/100)}}}$

• T_e e-folding time
• N(t) amount at time t
• N(0) initial amount
• T_d doubling time
• ln(2) ≈ 0.693 natural logarithm of 2
• r% growth rate in time t