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͑ a ͒ Correspondence between Cole–Davidson exponent ␥ and Kohlrausch exponent  resulting from equating the second logarithmic moment ͑ continuous curve ͒ . For comparison the ‘‘least-squares’’ correspondence established by Lindsey and Patterson ͑ Ref. 18 ͒ ͑ dashed curve ͒ and the correspondence from asymptotic behavior ͑ dotted curve ͒ . ͑ b ͒ The performance of the three approximations demonstrated by comparing ⑀ Љ ͑͒ for  ϭ 1/2. The
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In this paper a novel way to quantify "nonexponential" relaxations is described. So far, this has been done in two ways: (1) by fitting empirical functions with a small number of parameters, (2) by calculation of the underlying distribution function g(ln tau) of (exponential) relaxations using regularization methods. The method described here is in...
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Context 1
... this method the logarithmic moments of g (ln ) can be calculated for the Kohlrausch function 1 ͑ row 11 of Table I ͒ . For  ϭ 1/2 the logarithmic moments are the same as for the Rajagopal–Ngai distribution. This is because the latter is constructed to be the exact distribution of relaxation times of the Kohlrausch function in the special case  ϭ 1/2 and a reasonable approximation for other values of  . 14 For the most commonly used empirical relaxation functions with one adjustable parameter, the Kohlrausch, Cole– Davidson, and Cole–Cole expressions, Fig. 1 shows the first and second logarithmic moments. As expected the logarithmic width of the corresponding relaxation time distribution increases with decreasing parameter  , ␣ , or ␥ . If the parameters have the value one all three functions reduce to the Debye/exponential case; thus the width of the distribution vanishes. If the parameters approach zero the width diverges as ln ϳ Ϫ 1 , ␣ Ϫ 1 , ␥ Ϫ 1 . For the asymmetric ones among these functions the absolute value of the skewness increases with increasing parameter  or ␥ ; they even diverge as ͉ ␥ 1 ln ͉ ϳ (1 Ϫ  ) Ϫ 1/2 , (1 Ϫ ␥ ) Ϫ 1/2 . This is a counterintuitive consequence of normalizing the third moment by the cubed width in definition ͑ 10 ͒ . Figure 2 shows a graphical representation of the results for the two-parameter Havriliak-Negami function. As abscissa an alternative definition of the skewness, 3 ln / ln 2 , ͑ 23 ͒ has been chosen to avoid the divergence of definition ͑ 10 ͒ in the limit ␣ → 1. It can be seen that the Havriliak–Negami function is able to represent any width of the relaxation but only negative skewness ͑ as has been pointed out by Havriliak and Havriliak 16 this deficiency can be resolved by allowing ␥ Ͼ 1 with the weaker restriction ␣␥ Ͻ 1 but only a few authors actually use this generalization ͒ in a certain range delimited by the Cole–Davidson case labeled by ␣ ϭ 1 in the plot. From Fig. 2 and the equivalent plots for the Jonscher and Matsumoto–Higasi functions it seems that the combination ‘‘large skewness, small width’’ cannot be attained. But this is not a mathematical property because distributions with any combination of the first three moments can be constructed. It is rather a general experimental outcome than an a priori limitation that such highly skewed distributions do not occur in reality. It is clear that all the empirical functions listed in Table I are distinct. They only coincide if one is a special case of another, e.g., the Cole–Davidson function and the Havriliak–Negami function. Especially, the Kohlrausch function has no equivalent in the frequency domain. Its Fourier transform is only known for special cases where  is a rational number, 17 e.g., for  ϭ 1/2: ⑀ ͑ ⑀ 0 Ϫ ͒ Ϫ ⑀ ρ ⑀ ρ ϭ ͱ 4 i K exp ͩ 4 i 1 K ͪ erfc ͩ ͱ 4 i 1 K ͪ . ͑ 24 ͒ This expression clearly cannot be represented exactly by any of the empirical functions in frequency domain. Therefore, considerable effort has been made in the literature to establish approximate relationships between the Kohlrausch function and frequency domain expressions. Starting from the asymptotic properties of the Kohlrausch function that ⑀ Љ ͑͒ϳ for low frequencies and ⑀ Љ ( ) ϳ Ϫ  for high frequencies the Cole–Davidson function with ␥ ϭ  seems to be the appropriate choice. But Fig. 3 ͑ b ͒ shows that the high- and low-frequency wings have different levels and will not coincide despite having the same slope. Therefore, it may be a better compromise for represent- ing actual data to choose a mediatory value ␥ Ͻ  . Such an attempt was first described by Lindsey and Patterson ͑ LP ͒ . 18 On grounds of a least-squares fit they propose the relation  ϭ ͭ 0.970 0.683 ␥ ␥ ϩ ϩ 0.144 0.316 for for 0.2 0.6 р ␥ р р ␥ 0.6, . ͑ 25 ͒ Using the logarithmic moments derived here an alternative correspondence can be established by imposing that the moments up to the second should coincide. This results in the relation . 26 ͱ 6 Ј ͑ ␥ ͒ Figure 3 ͑ a ͒ shows a comparison of the correspondences. It can be seen that the one based on the logarithmic moments lies in between the LP formula and the ‘‘asymptotic’’ one. Concerning the agreement the LP correspondence performs slightly better in the peak while the logarithmic moments approach works better on the high frequency side. In both cases the low frequency wing is poorly represented. In order to overcome this deficiency one can use a two- parameter function, namely Havriliak–Negami, as was first described by Alvarez, Alegr ́a, and Colmenero ͑ AAC ͒ . 19 They use a numerical algorithm to invert ͑ 6 ͒ and thus obtain the distribution function of the Kohlrausch relaxation. Sub- sequently, they use this distribution to calculate ⑀ K Љ ( ) from ͑ 7 ͒ . Finally, the Havriliak–Negami function is determined with the parameters for which ⑀ HN Љ ( ) fits best to the previ- ously calculated function. From the tabulated values of cor- responding exponent parameters, the following approximate relation to the Kohlrausch  is established for given Havriliak–Negami parameters:  ϭ ͑ ␣␥ ͒ 0.813 . ͑ 27 ͒ For a given Kohlrausch  , the additional relation ␥ ϭ 1 Ϫ 0.8121 ͑ 1 Ϫ ␣ ͒ 0.387 ͑ 28 ͒ together with ͑ 27 ͒ fixes the optimal ␣ , ␥ pair to be used. The characteristic times are related ...
Context 2
... this method the logarithmic moments of g (ln ) can be calculated for the Kohlrausch function 1 ͑ row 11 of Table I ͒ . For  ϭ 1/2 the logarithmic moments are the same as for the Rajagopal–Ngai distribution. This is because the latter is constructed to be the exact distribution of relaxation times of the Kohlrausch function in the special case  ϭ 1/2 and a reasonable approximation for other values of  . 14 For the most commonly used empirical relaxation functions with one adjustable parameter, the Kohlrausch, Cole– Davidson, and Cole–Cole expressions, Fig. 1 shows the first and second logarithmic moments. As expected the logarithmic width of the corresponding relaxation time distribution increases with decreasing parameter  , ␣ , or ␥ . If the parameters have the value one all three functions reduce to the Debye/exponential case; thus the width of the distribution vanishes. If the parameters approach zero the width diverges as ln ϳ Ϫ 1 , ␣ Ϫ 1 , ␥ Ϫ 1 . For the asymmetric ones among these functions the absolute value of the skewness increases with increasing parameter  or ␥ ; they even diverge as ͉ ␥ 1 ln ͉ ϳ (1 Ϫ  ) Ϫ 1/2 , (1 Ϫ ␥ ) Ϫ 1/2 . This is a counterintuitive consequence of normalizing the third moment by the cubed width in definition ͑ 10 ͒ . Figure 2 shows a graphical representation of the results for the two-parameter Havriliak-Negami function. As abscissa an alternative definition of the skewness, 3 ln / ln 2 , ͑ 23 ͒ has been chosen to avoid the divergence of definition ͑ 10 ͒ in the limit ␣ → 1. It can be seen that the Havriliak–Negami function is able to represent any width of the relaxation but only negative skewness ͑ as has been pointed out by Havriliak and Havriliak 16 this deficiency can be resolved by allowing ␥ Ͼ 1 with the weaker restriction ␣␥ Ͻ 1 but only a few authors actually use this generalization ͒ in a certain range delimited by the Cole–Davidson case labeled by ␣ ϭ 1 in the plot. From Fig. 2 and the equivalent plots for the Jonscher and Matsumoto–Higasi functions it seems that the combination ‘‘large skewness, small width’’ cannot be attained. But this is not a mathematical property because distributions with any combination of the first three moments can be constructed. It is rather a general experimental outcome than an a priori limitation that such highly skewed distributions do not occur in reality. It is clear that all the empirical functions listed in Table I are distinct. They only coincide if one is a special case of another, e.g., the Cole–Davidson function and the Havriliak–Negami function. Especially, the Kohlrausch function has no equivalent in the frequency domain. Its Fourier transform is only known for special cases where  is a rational number, 17 e.g., for  ϭ 1/2: ⑀ ͑ ⑀ 0 Ϫ ͒ Ϫ ⑀ ρ ⑀ ρ ϭ ͱ 4 i K exp ͩ 4 i 1 K ͪ erfc ͩ ͱ 4 i 1 K ͪ . ͑ 24 ͒ This expression clearly cannot be represented exactly by any of the empirical functions in frequency domain. Therefore, considerable effort has been made in the literature to establish approximate relationships between the Kohlrausch function and frequency domain expressions. Starting from the asymptotic properties of the Kohlrausch function that ⑀ Љ ͑͒ϳ for low frequencies and ⑀ Љ ( ) ϳ Ϫ  for high frequencies the Cole–Davidson function with ␥ ϭ  seems to be the appropriate choice. But Fig. 3 ͑ b ͒ shows that the high- and low-frequency wings have different levels and will not coincide despite having the same slope. Therefore, it may be a better compromise for represent- ing actual data to choose a mediatory value ␥ Ͻ  . Such an attempt was first described by Lindsey and Patterson ͑ LP ͒ . 18 On grounds of a least-squares fit they propose the relation  ϭ ͭ 0.970 0.683 ␥ ␥ ϩ ϩ 0.144 0.316 for for 0.2 0.6 р ␥ р р ␥ 0.6, . ͑ 25 ͒ Using the logarithmic moments derived here an alternative correspondence can be established by imposing that the moments up to the second should coincide. This results in the relation . 26 ͱ 6 Ј ͑ ␥ ͒ Figure 3 ͑ a ͒ shows a comparison of the correspondences. It can be seen that the one based on the logarithmic moments lies in between the LP formula and the ‘‘asymptotic’’ one. Concerning the agreement the LP correspondence performs slightly better in the peak while the logarithmic moments approach works better on the high frequency side. In both cases the low frequency wing is poorly represented. In order to overcome this deficiency one can use a two- parameter function, namely Havriliak–Negami, as was first described by Alvarez, Alegr ́a, and Colmenero ͑ AAC ͒ . 19 They use a numerical algorithm to invert ͑ 6 ͒ and thus obtain the distribution function of the Kohlrausch relaxation. Sub- sequently, they use this distribution to calculate ⑀ K Љ ( ) from ͑ 7 ͒ . Finally, the Havriliak–Negami function is determined with the parameters for which ⑀ HN Љ ( ) fits best to the previ- ously calculated function. From the tabulated values of cor- responding exponent parameters, the following approximate relation to the Kohlrausch  is established for given Havriliak–Negami parameters:  ϭ ͑ ␣␥ ͒ 0.813 . ͑ 27 ͒ For a given Kohlrausch  , the additional relation ␥ ϭ 1 Ϫ 0.8121 ͑ 1 Ϫ ␣ ͒ 0.387 ͑ 28 ͒ together with ͑ 27 ͒ fixes the optimal ␣ , ␥ pair to be used. The characteristic times are related ...
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Citations
... Both the fitted parameters and the computed total correlation time τ G shown in Table II obtained for various system sizes do not indicate any system size dependence, which is in accordance with the finding of Celebi et al. 53 who noticed that the finite size correction for the rotational diffusion scales with the inverse box volume and is, therefore, much smaller than the one for translational diffusion. Moreover, we would like to point out that the fitted exponents shown in Table II with β ≈ 0.9 indicate a rather narrow distribution of relaxation times with a distribution width στ with στ ≈ τ K , 55 suggesting a shape of the corresponding spectral densities close to that of a Lorentzian. Next, we want to compute the frequency-dependent spectral densities and thus the frequency-dependent relaxation rates. ...
We present a computational framework for reliably determining the frequency-dependent intermolecular and intramolecular nuclear magnetic resonance (NMR) dipole–dipole relaxation rates of spin 1/2 nuclei from Molecular Dynamics (MD) simulations. This approach avoids the alterations caused by the well-known finite-size effects of translational diffusion. Moreover, a procedure is derived to control and correct for effects caused by fixed distance-sampling cutoffs and periodic boundary conditions. By construction, this approach is capable of accurately predicting the correct low-frequency scaling behavior of the intermolecular NMR dipole–dipole relaxation rate and thus allows for the reliable calculation of the frequency-dependent relaxation rate over many orders of magnitude. Our approach is based on the utilization of the theory of Hwang and Freed for the intermolecular dipole–dipole correlation function and its corresponding spectral density [L.-P. Hwang and J. H. Freed, J. Chem. Phys. 63, 4017–4025 (1975)] and its combination with data from MD simulations. The deviations from the Hwang and Freed theory caused by periodic boundary conditions and sampling distance cutoffs are quantified by means of random walker Monte Carlo simulations. An expression based on the Hwang and Freed theory is also suggested for correcting those effects. As a proof of principle, our approach is demonstrated by computing the frequency-dependent intermolecular and intramolecular dipolar NMR relaxation rates of 1H nuclei in liquid water at 273 and 298 K based on the simulations of the TIP4P/2005 model. Our calculations are suggesting that the intermolecular contribution to the 1H NMR relaxation rate of the TIP4P/2005 model in the extreme narrowing limit has previously been substantially underestimated.
... First, the finite-difference error in estimating the reaction coefficient will be quantified. The finite-difference derivative approximation to the reaction coefficient in Eq. (23) will always overestimate the exact analytical solution by a small offset since the odd derivatives of a decay function are always negative [21], whereby P (t), P (t) < 0 in the Taylor expamsion: ...
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... (eqn (8) 15 ), e Àhln[t]i (eqn (9) 26 ). ...
... Previously we used numerically-approximated ESDs for the Generalised Debye model. 10 Logarithmic moments have been shown to be an appropriate metric for comparing different distributions, 26 ...
... 31 It may also occur that the distribution of rates in a sample is significantly skewed, and the waveform data are not well represented by a symmetric Generalised Debye distribution: one alternative is to use the Havriliak-Negami model 7 which also has a known expectation value and variance. 26 For compounds that are required to be modelled like this then we expect that e hln[t]i will provide the best measure to compare relaxation times in both cases. ...
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... 7), ⟨τ⟩ −1 (Eq. 8 15 ), e −⟨ln [τ]⟩ (Eq. 9 26 ). ...
... 31 It may also occur that the distribution of rates in a sample is significantly skewed, and the waveform data are not well represented by a symmetric Generalised Debye distribution: one alternative is to use the Havriliak-Negami model 7 which also has a known expectation value and variance. 26 For compounds that are required to be modelled like this then we expect that e ⟨ln [τ]⟩ will provide the best measure to compare relaxation times in both cases. ...
The use of magnetisation decay measurements to characterise very slow relaxation of the magnetisation in single-molecule magnets are becoming increasingly prevalent as relaxation times move to longer timescales outside of the AC susceptibility range. However, experimental limitations and a poor understanding of the distribution underlying the stretched exponential function, commonly used to model the data, may be leading to misinterpretation of the results. Herein we develop guidelines on the experimental design, data fitting, and analysis required to accurately interpret magnetisation decay measurements. Different measures of the magnetic relaxation rate extracted from magnetisation decay measurements of [Dy(Dtp)2][Al{OC(CF3)3}4], fitted using different combinations of fixing or freely fitting different parameters, are compared to those obtained using the innovative square-wave “waveform” technique of Hilgar et al, which is comparable to AC susceptometry for measurement of relaxation rates on long timescales. The most reliable measure of the relaxation time for magnetisation decays is found to be the the average logarithmic relaxation time, $e^{⟨ln [τ]⟩}$, obtained via a fit of the decay trace using a stretched exponential function, where the initial and equilibrium magnetisation are fixed to first measured point and target values respectively. This new definition has causes the largest differences to traditional approaches in the presence of large distributions or relaxation rates, with differences up to 50% with β = 0.45, and hence could have a significant impact on the chemical interpretation of magnetic relaxation rates. A necessary step in progressing towards chemical control of magnetic relaxation is the accurate determination of relaxation times, and such large variations in experimental measures stress the need for consistency in fitting and interpretation of magnetisation decays.
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... This union has brought new formulations for fractional diffusive models, since they render the Riemann-Liouville-Caputo fractional derivatives as individual cases [8,9,24]. In addition, the fractional Prabhakar derivative has been the most complete and sophisticated tool for describing complexity in physical systems such as, in the description of relaxation and response in anomalous dielectrics of Havriliak-Negami type [2,28], in dynamical models of spherical stellar systems [4] and in the fractional Viscoelasticity [10]. For a practical guide to Prabhakar fractional calculus refer to [11]. ...
... In addition, assume that the matrix I − K C T γ ,ρ,α,ω is nonsingular and f k,M is the approximate solution of the problem (1) obtained using Eq. (28). Then ...
... where F is the approximate solution of the linear algebraic system (28). ...
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... The separation of -process is clearly seen in the spectra of shrimps after 3 days of starvation and in the group regenerated by 14 days. Although there are no clear peaks visible in the spectra, the dielectric loss curves from the 3-day and regenerated groups have been fitted by Negami dielectric functions [21][22][23] which are given by the equation: ...
In this work, dielectric studies on Neocaridina davidi shrimps have been presented. The effect of starvation on dielectric properties such as conductivity or permittivity have been shown. It was found that the onset frequency of electrode polarization depends on starvation period, which is probably related to the cytoplasm viscosity. In the dielectric spectra of shrimps two relaxation processes have been identified i.e. α and β process. The α process is probably related to the counter ion polarization while β process to the mobility of macromolecules present in the body of a shrimp, mainly to the amount of lipids. It was also found that there is a difference in dielectric response between control group and the group regenerated after 14 day starvation period. Basing on dielectric response, one can conclude that the viscosity of cytoplasm of regenerated shrimps is higher and the cells are rich in the lipid droplets when compared to control group.
