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The **Curie–von Schweidler law** refers to the response of dielectric material to the step input of a direct current (DC) voltage first observed by Jacques Curie ^{ [1] } and Egon Ritter von Schweidler.^{ [2] }

A **dielectric** is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing **dielectric polarization**. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

According to this law, the current decays according to a power law:

where is the current at a given charging time, , and is the decay constant such that . Given that the dielectric has a finite conductance, the equation for current measured through a dielectric under a DC electrical field is:

where is a constant of proportionality, is the decay constant (i.e., ), and is the intrinsic conductance of the dielectric. This stands in contrast to the Debye formulation, which states that the current is proportional an exponential function with a time constant, , according to:

- .

The Curie–von Schweidler behavior has been observed in many instances such as those shown by Andrzej Ka Johnscher^{ [3] } and Jameson *et al*.^{ [4] } It has been interpreted as a many-body problem by Jonscher, but can also be formulated as an infinite number of resistor-capacitor circuits. This comes from the fact that the power law can be expressed as:

A **resistor–capacitor circuit**, or **RC filter** or **RC network**, is an electric circuit composed of resistors and capacitors driven by a voltage or current source. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.

where is the Gamma function. Effectively, this relationship shows the power law expression to be composed of an infinite *weighted sum* of Debye responses.

In mathematics, the **gamma function** is one of the extensions of the factorial function with its argument shifted down by 1, to real and complex numbers. Derived by Daniel Bernoulli, if *n* is a positive integer,

**Radioactive decay** is a condition and natural process in which subatomic particles within the atomic nucleus of a radioisotope get decayed due to the instability of the atom. This produces ionized radiation in the form of alpha particles, beta particles with neutrino or only a neutrino in the case of electron capture, or gamma rays. The rate of disintegration can be measured on a logarithmic scale. A material containing such unstable nuclei is considered **radioactive**. Certain highly excited short-lived nuclear states can decay through neutron emission, or more rarely, proton emission.

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**:

In mathematics, the **Dedekind eta function**, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive.

In mathematics, **theta functions** are special functions of several complex variables. They are important in many areas, including the theories of Abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory.

The **Drude model** of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials. The model, which is an application of kinetic theory, assumes that the microscopic behavior of electrons in a solid may be treated classically and looks much like a pinball machine, with a sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions.

**Bosonic string theory** is the original version of string theory, developed in the late 1960s. It is so called because it only contains bosons in the spectrum.

In mathematics, the **Hamilton–Jacobi equation** (**HJE**) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

**Variational Bayesian methods** are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables as well as unknown parameters and latent variables, with various sorts of relationships among the three types of random variables, as might be described by a graphical model. As is typical in Bayesian inference, the parameters and latent variables are grouped together as "unobserved variables". Variational Bayesian methods are primarily used for two purposes:

- To provide an analytical approximation to the posterior probability of the unobserved variables, in order to do statistical inference over these variables.
- To derive a lower bound for the marginal likelihood of the observed data. This is typically used for performing model selection, the general idea being that a higher marginal likelihood for a given model indicates a better fit of the data by that model and hence a greater probability that the model in question was the one that generated the data.

The **scaled inverse chi-squared distribution** is the distribution for *x* = 1/*s*^{2}, where *s*^{2} is a sample mean of the squares of ν independent normal random variables that have mean 0 and inverse variance 1/σ^{2} = τ^{2}. The distribution is therefore parametrised by the two quantities ν and τ^{2}, referred to as the *number of chi-squared degrees of freedom* and the *scaling parameter*, respectively.

The **Havriliak–Negami relaxation** is an empirical modification of the Debye relaxation model in electromagnetism. Unlike the Debye model, the Havriliak–Negami relaxation accounts for the asymmetry and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymers, by adding two exponential parameters to the Debye equation:

**Self-phase modulation** (SPM) is a nonlinear optical effect of light-matter interaction. An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variation in refractive index will produce a phase shift in the pulse, leading to a change of the pulse's frequency spectrum.

A **theoretical motivation for general relativity**, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation *a priori*. This provides a means to inform and verify the formalism.

The **Cole–Cole equation** is a relaxation model that is often used to describe dielectric relaxation in polymers.

The **stretched exponential function**

In mathematics, the theory of **optimal stopping** or **early stopping** is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance. A key example of an optimal stopping problem is the secretary problem. Optimal stopping problems can often be written in the form of a Bellman equation, and are therefore often solved using dynamic programming.

In probability theory and statistics, the **normal-gamma distribution** is a bivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and precision.

The **non-random two-liquid** model is an activity coefficient model that correlates the activity coefficients of a compound with its mole fractions in the liquid phase concerned. It is frequently applied in the field of chemical engineering to calculate phase equilibria. The concept of NRTL is based on the hypothesis of Wilson that the local concentration around a molecule is different from the bulk concentration. This difference is due to a difference between the interaction energy of the central molecule with the molecules of its own kind and that with the molecules of the other kind . The energy difference also introduces a non-randomness at the local molecular level. The NRTL model belongs to the so-called local-composition models. Other models of this type are the Wilson model, the UNIQUAC model, and the group contribution model UNIFAC. These local-composition models are not thermodynamically consistent for a one-fluid model for a real mixture due to the assumption that the local composition around molecule i is independent of the local composition around molecule j. This assumption is not true, as was shown by Flemr in 1976. However, they are consistent if a hypothetical two-liquid model is used.

The **Herschel–Bulkley fluid** is a generalized model of a non-Newtonian fluid, in which the strain experienced by the fluid is related to the stress in a complicated, non-linear way. Three parameters characterize this relationship: the consistency *k*, the flow index *n*, and the yield shear stress . The consistency is a simple constant of proportionality, while the flow index measures the degree to which the fluid is shear-thinning or shear-thickening. Ordinary paint is one example of a shear-thinning fluid, while oobleck provides one realization of a shear-thickening fluid. Finally, the yield stress quantifies the amount of stress that the fluid may experience before it yields and begins to flow.

**Metal-mesh optical filters** are optical filters made from stacks of metal meshes and dielectric. They are used as part of an optical path to filter the incoming light to allow frequencies of interest to pass while reflecting other frequencies of light.

In mathematics, the **Weil–Brezin map**, named after André Weil and Jonathan Brezin, is a unitary transformation that maps a Schwartz function on the real line to a smooth function on the Heisenberg manifold. The Weil–Brezin map gives a geometric interpretation of the Fourier transform, the Plancherel theorem and the Poisson summation formula. The image of Gaussian functions under the Weil–Brezin map are nil-theta functions, which are related to theta functions. The Weil–Brezin map is sometimes referred to as the Zak transform, which is widely applied in the field of physics and signal processing; however, the Weil–Brezin Map is defined via Heisenberg group geometrically, whereas there is no direct geometric or group theoretic interpretation from the Zak transform.

- ↑ Curie, Jaques (1889). "Recherches sur le pouvoir inducteur specifique et la conductibilite des corps cristallises".
*Annales de Chimie et de Physique*.**17**: 384–434. - ↑ Schweidler, Ergon Ritter von (1907). "Studien über die Anomalien im Verhalten der Dielektrika (Studies on the anomalous behaviour of dielectrics)".
*Annalen der Physik*.**329**(14): 711–770. Bibcode:1907AnP...329..711S. doi:10.1002/andp.19073291407. - ↑ Jonscher, Andrzej Ka (1983),
*Dielectric Relaxation in Solids*, Chelsea Dielectrics Press Limited, ISBN 978-0-9508711-0-3 - ↑ Jameson, N. Jordan; Azarian, Michael H.; Pecht, Michael (2017).
*Thermal Degradation of Polyimide Insulation and its Effect on Electromagnetic Coil Impedance*. Proceedings of the Society for Machinery Failure Prevention Technology 2017 Annual Conference.

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