A luminescence decay function encompassing the stretched exponential and the compressed hyperbola
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Hyperbola
Dimensionless quantity
Exponential decay
Stretched exponential function
Constant (computer programming)
In this paper we study a nonlinear lattice with memory and show that the problem is globally well posed. Furthermore we find uniform rates of decay of the total energy. Our main result shows that the memory effect is strong enough to produce a uniform rate of decay. That is, if the relaxation function decays exponentially then, the corresponding solution also decays exponentially. When the relaxation kernel decays polynomially then, the solution also decays polynomially as time goes to infinity.
Lattice (music)
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Exponential decay
Logistic function
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The quantum-mechanical theory of the decay of unstable states is revisited. We show that the decay is non-exponential both in the short-time and long-time limits using a more physical definition of the decay rate than the one usually used. We report results of numerical studies based on Winter's model that may elucidate qualitative features of exponential and non-exponential decay more generally. The main exponential stage is related to the formation of a radiating state that maintains the shape of its wave function with exponentially diminishing normalization. We discuss situations where the radioactive decay displays several exponents. The transient stages between different regimes are typically accompanied by interference of various contributions and resulting oscillations in the decay curve. The decay curve can be fully oscillatory in a two-flavor generalization of Winter's model with some values of the parameters. We consider the implications of that result for models of the oscillations reported by GSI.
Exponential decay
Normalization
Radioactive decay
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Exponential decay
Stretched exponential function
Time constant
Crystal (programming language)
Porous Silicon
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Exponential decay
Feedback Control
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Hyperbola
Dimensionless quantity
Exponential decay
Stretched exponential function
Constant (computer programming)
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Eigenfunction
Exponential decay
Elliptic operator
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We consider the initial-value problem for the one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges for large times. We give explicit estimates of the rate of this exponential decay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The second one is based on the integral formulation of the problem and, under a more restrictive assumption on the variation of the coefficients, allows us to obtain improved decay rates.
Exponential decay
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The modeling of systems that exhibit near-exponential decay is most commonly done using a sum of exponentials or a stretched exponential. We note some drawbacks of these representations and present an alternative model, the stretched or compressed hyperbola, first described by E. Becquerel in the 1860s. This representation might be more helpful for interpolation, extrapolation, and classification of decays and requires only one additional parameter compared to simple exponential decay.
Hyperbola
Exponential decay
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Exponential decay, bi-exponential decay, and related decay processes are common in the physical world. Stretched exponential time dependence of the form e−(kt)c has been observed in connection with the discharge of electrophotographic photoconductors, luminescence in porous silicon, dielectric relaxation in glassy and polymeric materials, as well as in other systems. Exponential decay, the stretched exponential, the Kohlrausch-Williams-Watts function KWW, and the Buettner function satisfy a differential equation that depends on the exponent c and the entropy of the system. The form of the decay function determined by the exponent c can be shown to be consistent with cooperative events occurring during relaxation and can be related to the chemical potential of the system. This indicates that probabilistic, cooperative events may play a role in the dynamics of stretched exponential decay processes in addition to distributions of relaxation times and relaxation paths.
Exponential decay
Stretched exponential function
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