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A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation , where N is the quantity and λ ( lambda ) is a positive rate called the exponential decay constant , disintegration constant , [ 1 ] rate constant , [ 2 ] or ...
First order LTI systems are characterized by the differential equation + = where τ represents the exponential decay constant and V is a function of time t = (). The right-hand side is the forcing function f(t) describing an external driving function of time, which can be regarded as the system input, to which V(t) is the response, or system output.
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
The term "half-life" is almost exclusively used for decay processes that are exponential (such as radioactive decay or the other examples above), or approximately exponential (such as biological half-life discussed below). In a decay process that is not even close to exponential, the half-life will change dramatically while the decay is happening.
Exponential growth or exponential decay—where the varaible change is proportional to the variable value—are thus modeled with exponential functions. Examples are unlimited population growth leading to Malthusian catastrophe , continuously compounded interest , and radioactive decay .
In semiconductor lasers, the carrier lifetime is the time it takes an electron before recombining via non-radiative processes in the laser cavity. In the frame of the rate equations model, carrier lifetime is used in the charge conservation equation as the time constant of the exponential decay of carriers.
The most common form of damping, which is usually assumed, is the form found in linear systems. This form is exponential damping, in which the outer envelope of the successive peaks is an exponential decay curve. That is, when you connect the maximum point of each successive curve, the result resembles an exponential decay function.
The constant decay rate of the golden rule follows. [8] As a constant, it underlies the exponential particle decay laws of radioactivity. (For excessively long times, however, the secular growth of the a k ( t ) terms invalidates lowest-order perturbation theory, which requires a k ≪ a i .)