In
science
Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earli ...
, ''e''-folding is the time interval in which an
exponentially growing quantity increases by a factor of
''e''; it is the base-''e'' analog of
doubling time. This term is often used in many areas of science, such as in
atmospheric chemistry
Atmospheric chemistry is a branch of atmospheric science in which the chemistry of the Earth's atmosphere and that of other planets is studied. It is a multidisciplinary approach of research and draws on environmental chemistry, physics, meteorol ...
,
medicine
Medicine is the science and practice of caring for a patient, managing the diagnosis, prognosis, prevention, treatment, palliation of their injury or disease, and promoting their health. Medicine encompasses a variety of health care pr ...
and
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, especially when
cosmic inflation is investigated. Physicists and chemists often talk about the ''e''-folding time scale that is determined by the
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
in which the length of a patch of
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...
or
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
increases by the factor ''e'' mentioned above.
In
finance, the logarithmic return or
continuously compounded return, also known as
force of interest, is the reciprocal of the ''e''-folding time.
The term ''e''-folding time is also sometimes used similarly in the case of
exponential decay, to refer to the timescale for a quantity to decrease to 1/''e'' of its previous value.
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
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer.
Thermometers are calibrated in various Conversion of units of temperature, temp ...
or
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
, for example) then after a time, ''τ'', the initial difference between the initial value of the observable and the equilibrium value, Δ''X''
''i'', will have decreased to Δ''X''
''i'' /''e'' where the number
''e'' ≈ 2.71828.
:
* ''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''
Example of lifetime as ''e''-folding time
The concept of ''e''-folding time may be used in the analysis of
kinetics. Consider a chemical species A, which decays into another chemical species, B. 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. Consider what the units of ''k'' would be. On the left hand side, we have a concentration divided by a unit of time. The units for ''k'' would need to allow for these to be replicated on the right hand side. For this reason, the units of ''k'', here, would be 1/time.
Because this is a
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
,
homogeneous and
separable differential equation, we may separate the terms such that the equation becomes:
:
We may then
integrate both sides of this equation, which results in the inclusion of the constant ''e'':
:
where
sub>''f'' and
sub>''i'' are the final and initial concentrations of A. Upon comparing the ratio on the left hand side to the equation on the right hand side, we conclude that the ratio between the final and initial concentrations follows an exponential function, of which ''e'' is the base.
As mentioned above, the units for ''k'' are inverse time. If we were to take the reciprocal of this, we would be left with units of time. For this reason, we often state that the lifetime of a species that undergoes first order decay is equal to the reciprocal of ''k''. Consider, now, what would happen if we were to set the time, ''t'', to the reciprocal of the rate constant, ''k'', i.e. ''t'' = 1/''k''. This would yield
:
This states that after one lifetime (1/''k''), the ratio of final to initial concentrations is equal to about 0.37. Stated another way, after one lifetime, we have
:
which means that we have lost (1 − 0.37 = 0.63) 63% of A, with only 37% left. With this, we now know that if we have 1 lifetime passed, we have gone through 1 "e-folding". What would 2 "e-foldings" look like? After two lifetimes, we would have , which would result in
:
which says that only about 14% of A remains. It is in this manner that ''e''-folding lends us an easy way to describe the number of lifetimes that have passed. After 1 lifetime, we have 1/''e'' remaining. After 2 lifetimes, we have 1/''e''
2 remaining. One lifetime, therefore, is one ''e''-folding time, which is the most descriptive way of stating the decay.
See also
*
Doubling time
{{DEFAULTSORT:E-Folding
Inflation (cosmology)
Temporal exponentials