Table 3
t"^ and g(fmi«) as a function of P, where g(t) is the inverse Laplace transform of e~ -Continued n t^ gCmax)
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The inverse transform, g(t) = L-script-1(e-s(β), 0<β<1, is a stable law that arises in a number of different applications in chemical physics, polymer physics, solid-state physics, and applied mathematics. Because of its important applications, a number of investigators have suggested approximations to g(t). However, there have so far been no accur...
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Citations
... When the parameter β decreases, the distribution of relaxation times broadens and indicates that one has a layered distribution of pore sizes or interfacial voids. P(s,β) has been tabulated numerically for a range of values of β in Ref. 24. It is also known analytically for particular values of β such as β = 1/2, as detailed in Ref. 23. ...
... It is also known analytically for particular values of β such as β = 1/2, as detailed in Ref. 23. By setting x = t/τ 0 in Eq. (11), the Laplace transform is given by 23,24 ...
... Analytical solutions of Eq. (13) are found only in a few cases like β = 0.5, and it is, therefore, most convenient to use numerical tables when extracting the distribution or arbitrary values of the coefficient in the range 0 < β < 1. 24 Using Eq. (11), the voltage over the supercapacitor becomes ...
A charging electrochemical double layer supercapacitor can usually be described by a single capacitance and a single resistance in parallel, wherein the latter represents the ohmic losses. Such an ideal behavior may occur if the supercapacitor consists of self-similar porous carbon micro- and nanostructures. However, if the electrochemical double layer supercapacitor consists of a sequence of slices with different relaxation times, a strong deviation from ideal charging curves may occur. Here, it is demonstrated how such charging curves can be interpreted in terms of a distribution of relaxation times. It is found that in the presence of a broad distribution of charge transfer resistances, the voltage initially appears to increase faster than normal during galvanostatic charging. Care should be taken to avoid misinterpretation of the capacitance under such circumstances.
... The first-order f(k)s were crosschecked from the numerical Laplace inversion of [1 -R(t = s)/R∞], using the methodology reported by Valsa and Brančik (1998). The normalized tables reported by Dishon et al. (1990) were also used to support the results. In the case of compressed exponentials (aRR > 1), the Rosin-Rammler model suggests a decaying mechanism that is faster than any exponential function at long flotation times (Andrews et al., 2018), which cannot be expressed by the distributed first-order model of Table 4.3. ...
... where the condition V(0)=V 0 requires that ∫ ∞ 0 P(s)ds = 1. It should be emphasized that tabulated tables are available for different values of β [55], thus making the extraction of P(s) computationally very simple. The distribution function P(s) is clearly linked to a distribution of rate constants. ...
Supercapacitors are prone to self-discharging, which is most often measured as a voltage decrease with time under open circuit conditions. It is of substantial interest to find simple and general methods to extract information about the processes going on in the supercapacitor during self-discharge. The current work fits a stretched exponential function to experimental data available in the literature, thus extracting parameters that allows one to probe the internal processes of the supercapacitor. In particular, experimental data related to charge holding time, charging rate before self-discharge and temperature dependence are investigated. It is demonstrated how the fitting data can be understood in terms of a kinetic model exhibiting a distribution of rate constants which are related to the fitting parameters. The current work therefore proposes a method that allows one to quickly map the internal processes of a self-discharging supercapacitors with only two variables, and might therefore become a useful tool.
... The first-order f (k)s were crosschecked from the numerical Laplace inversion of [1 − R(t = s)/R ∞ ], using the methodology reported by Valsa and Brančik [64]. The normalized tables reported by Dishon, et al. [74] were also used to support the results. In case of compressed exponentials (a RR > 1), the Rosin-Rammler model suggests a decaying mechanism that is faster than any exponential function at long flotation times [72], which cannot be expressed by the distributed first-order model of Table 5. Figure 5 illustrates four first-order f (k)s from the Rosin-Rammler R(t) together with the respective time-recovery curves for a RR < 1. ...
Four kinetic models are studied as first-order reactions with flotation rate distribution f (k): (i) deterministic nth-order reaction, (ii) second-order with Rectangular f (k), (iii) Rosin-Rammler, and (iv) Fractional kinetics. These models are studied because they are considered as alternatives to the first-order reactions. The first-order representation leads to the same recovery R(t) as in the original domain. The first-order R ∞-f (k) are obtained by inspection of the R(t) formulae or by inverse Laplace Transforms. The reaction orders of model (i) are related to the shape parameters of first-order Gamma f (k)s. Higher reaction orders imply rate concentrations at k ≈ 0 in the first-order domain. Model (ii) shows reverse J-shaped first-order f (k)s. Model (iii) under stretched exponentials presents mounded first-order f (k)s, whereas model (iv) with derivative orders lower than 1 shows from reverse J-shaped to mounded first-order f (k)s. Kinetic descriptions that lead to the same R(t) cannot be differentiated between each other. However, the first-order f (k)s can be studied in a comparable domain.
... The stretched exponential function, of the form − e , f t ( * ) β with β and f* constants, was introduced by Kolrausch to explain the discharge of a capacitor (Leyden jar), after finding that a simple exponential decay did not explain the experimental data [42]. Later, it was applied to numerous physical situations were a sum of exponential relaxations were present [43][44][45]. In Ref. [46] a stretched exponential function − t τ exp( / ) 2 was used, corresponding to β = 1/2 conjectured from diffusion dynamics, to explain long-time discharging of the supercapacitor. ...
... This can also be transformed into Fourier transform upon suitable change of variables, and in a few cases including β = 0.5, analytical solutions exist [44,45]. For arbitrary values of the coefficient in the range 0<β<1, there are several methods for evaluating it numerically [44]. ...
... This can also be transformed into Fourier transform upon suitable change of variables, and in a few cases including β = 0.5, analytical solutions exist [44,45]. For arbitrary values of the coefficient in the range 0<β<1, there are several methods for evaluating it numerically [44]. ...
Supercapacitors are often modelled using electrical equivalent circuits with a limited number of branches. However, the limited number of branches often cannot explain long-term dynamics, and one therefore has to resort to more computationally challenging basic models governing diffusion and drift of ions. Here, it is shown that consistent modelling of a supercapacitor can be done in a straightforward manner by introducing a dynamic equivalent circuit model that naturally allows a large number or a continuous distribution of time constants, both in time and frequency domains. Such a model can be used to explain the most common features of a supercapacitor in a consistent manner. In the time domain, it is shown that the time-dependent charging rate and the self-discharge of a supercapacitor can both be interpreted in this model with either a few or a continuous distribution of relaxation times. In the frequency domain, the impedance spectrum allows one to extract a distribution of relaxation times. The unified model presented here may help visualizing how the distribution of relaxation times or frequencies govern the behaviour of a supercapacitor under varying circumstances.
... For the general case of 0 Ͻ y Ͻ 1, f y ͑t͒ was first considered in Ref. 34 and later in Ref. 35 as an inverse Laplace transform. Rewriting Eq. ͑22͒ as a inverse Laplace transform yields ...
Frequency-dependent loss and dispersion are typically modeled with a power-law attenuation coefficient, where the power-law exponent ranges from 0 to 2. To facilitate analytical solution, a fractional partial differential equation is derived that exactly describes power-law attenuation and the Szabo wave equation ["Time domain wave-equations for lossy media obeying a frequency power-law," J. Acoust. Soc. Am. 96, 491-500 (1994)] is an approximation to this equation. This paper derives analytical time-domain Green's functions in power-law media for exponents in this range. To construct solutions, stable law probability distributions are utilized. For exponents equal to 0, 1/3, 1/2, 2/3, 3/2, and 2, the Green's function is expressed in terms of Dirac delta, exponential, Airy, hypergeometric, and Gaussian functions. For exponents strictly less than 1, the Green's functions are expressed as Fox functions and are causal. For exponents greater than or equal than 1, the Green's functions are expressed as Fox and Wright functions and are noncausal. However, numerical computations demonstrate that for observation points only one wavelength from the radiating source, the Green's function is effectively causal for power-law exponents greater than or equal to 1. The analytical time-domain Green's function is numerically verified against the material impulse response function, and the results demonstrate excellent agreement.
The Beck-Cohen superstatistics became an important theory in the scenario of complex systems because it generates distributions representing regions of a nonequilibrium system, characterized by different temperatures T≡β−1, leading to a probability distribution f(β). In superstatistics, some classes have been most frequently considered for f(β), like χ2, χ2 inverse, and log-normal ones. Herein we investigate the superstatistics resulting from a χη2 distribution through a modification of the usual χ2 by introducing a real index η (0<η≤1). In this way, one covers two common and relevant distributions as particular cases, proportional to the q-exponential (eq−βx=[1−(1−q)βx]11−q) and the stretched exponential (e−(βx)η). Furthermore, an associated generalized entropic form is found. Since these two particular-case distributions have been frequently found in the literature, we expect that the present results should be applicable to a wide range of classes of complex systems.
The agglomeration kinetics of magnetic nanoparticles under the magnetic field is studied by measuring the temporal change of the magnetic weight. The kinetics of stretched exponential is observed for the growth of the magnetic weight, which results from the distribution of the energy barrier. The agglomeration kinetics is simulated with the Boltzmann distribution function of the energy barrier. The Boltzmann distribution seems plausible for the agglomeration of nanoparticles where the diffusion of low energy barrier is dominant. The comparison with the experimental results indicates that the Boltzmann distribution is inappropriate for energy barrier distribution function even though the dynamics close to the stretched exponential is reproduced. The kinetic parameters deduced from the simulation results suggest that the complex structural relaxation of the agglomerate with higher energy barrier is more dominant than the diffusion in the growth of the magnetic weight. While the Laplace transformation of the Boltzmann distribution function fits to the stretched exponential, the analysis with the Boltzmann distribution function does not give the reasonable Arrhenius parameters for the agglomeration of magnetic nanoparticles, which indicates that the Boltzmann distribution function is not appropriate for the energy barrier distribution function for the agglomeration dynamics.
In this paper, we give a two-line proof of a long-standing conjecture of Ben-Akiva in his 1973 PhD thesis regarding the random utility representation of the nested logit model, thus providing a renewed and straightforward textbook treatment of that model. As an application, we provide a closed-form formula for the correlation between two Fréchet random variables coupled by a Gumbel copula.
Agglomeration of magnetite nanoparticles in the aqueous solution is studied at the low magnetic field gradients of 23–34 T/m by monitoring the temporal change of magnetic weight. A conventional electronic balance is used to measure the magnetic weight that is the magnetic force on the magnetic sample by the magnetic field gradient. The magnetic weight grows slowly following the stretched exponential after the instantaneous jump by the Neel and Brown relaxation. Magnetization of the magnetite nanoparticles is estimated from the magnetic weight and compared with the Langevin function. The magnetization is close to the saturation in the studied magnetic field range and the saturation magnetization of the agglomerate of nanoparticles is about 60% of that of the bulk magnetite. Kinetic parameters of the stretched exponential show little the magnetic field dependence in the investigated range. Complex energy landscape is involved in the agglomeration as the stretched exponential dynamics indicates. The half‐life of the response function for the magnetic weight change suggests that the pathways of low energy barriers are activated by magnetic field at the early stage of agglomeration.
