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Exponential decay. A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constant ( λ) of 25, 5, 1, 1/5, and 1/25 for x from 0 to 5. A quantity is subject to exponential decay if it decreases at a rate proportional to its current value.
Half-life (symbol t½) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable atoms survive. The term is also used more generally to characterize any type of exponential (or, rarely ...
Note that if the term were replaced with , the problem would be linear (the exponential decay problem). Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered.
The stretched exponential function. is obtained by inserting a fractional power law into the exponential function . In most applications, it is meaningful only for arguments t between 0 and +∞. With β = 1, the usual exponential function is recovered. With a stretching exponent β between 0 and 1, the graph of log f versus t is ...
Quasinormal modes (QNM) are the modes of energy dissipation of a perturbed object or field, i.e. they describe perturbations of a field that decay in time. Example [ edit ] A familiar example is the perturbation (gentle tap) of a wine glass with a knife: the glass begins to ring, it rings with a set, or superposition, of its natural frequencies ...
Exponential stability is a form of asymptotic stability, valid for more general dynamical systems. Practical consequences [ edit ] An exponentially stable LTI system is one that will not "blow up" (i.e., give an unbounded output) when given a finite input or non-zero initial condition.
The Poisson–Boltzmann equation describes the electrochemical potential of ions in the diffuse layer. The three-dimensional potential distribution can be described by the Poisson equation [4] ψ is the electric potential. The freedom of movement of ions in solution can be accounted for by Boltzmann statistics.
This exponential decay is related to the motion of the particles, specifically to the diffusion coefficient. To fit the decay (i.e., the autocorrelation function), numerical methods are used, based on calculations of assumed distributions. If the sample is monodisperse (uniform) then the decay is