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E-folding
In science, e-folding is the time interval in which an exponentially growing quantity increases or decreases 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, 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, ΔXi, will have decreased to ΔXi /e where the number e ≈ 2.71828.
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, homogeneous and separable differential equation, we may separate the terms such that the equation becomes:
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E-folding
In science, e-folding is the time interval in which an exponentially growing quantity increases or decreases 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, 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, ΔXi, will have decreased to ΔXi /e where the number e ≈ 2.71828.
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, homogeneous and separable differential equation, we may separate the terms such that the equation becomes: