Category: Glasses

Fractional glassy relaxation and convolution modules of distributions

Post author By R Hilfer
Post date September 30, 2020

T. Kleiner, R. Hilfer

Analysis and Mathematical Physics 11, 130 (2021)
https://doi.org/10.1007/s13324-021-00504-5

submitted on
Wednesday, September 30, 2020

Solving fractional relaxation equations requires precisely characterized domains of definition for applications of fractional differential and integral operators. Determining these domains has been a longstanding problem. Applications in physics and engineering typically require extension from domains of functions to domains of distributions. In this work convolution modules are constructed for given sets of distributions that generate distributional convolution algebras. Convolutional inversion of fractional equations leads to a broad class of multinomial Mittag-Leffler type distributions. A comprehensive asymptotic analysis of these is carried out. Combined with the module construction the asymptotic analysis yields domains of distributions, that guarantee existence and uniqueness of solutions to fractional differential equations. The mathematical results are applied to anomalous dielectric relaxation in glasses. An analytic expression for the frequency dependent dielectric susceptibility is applied to broadband spectra of glycerol. This application reveals a temperature independent and universal dynamical scaling exponent.

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dielectric relaxation Disordered Systems Glasses Transport Processes

Excess wing physics and nearly constant loss in glasses

Post author By R Hilfer
Post date May 31, 2019

R. Hilfer

Journal of Statistical Mechanics: Theory and Experiment 2019, 104007 (2019)
https://doi.org/10.1088/1742-5468/ab38bc

submitted on
Friday, May 31, 2019

Excess wings and nearly constant loss are almost universal nonequilibrium phenomena in glass formers. Both lack an accepted theoretical foundation. A model-free and unified theoretical description for these phenomena is presented that encompasses also fast β-processes, emergent Debye peaks, and the relaxation strength of the boson peak. The theory is model-free in the same way as the classical Debye relaxation equation for orientational polarisation. It is based on generalizing time flow from translation semigroups to composite time translation-convolution semigroups. Composite translation-convolution fits have less parameters than traditional fits. They need only one dynamic scaling exponent, while four are needed in Havriliak-Negami fits. For glycerol the single dynamic exponent in the translation-convolution fit is found to be temperature-independent.

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Tags 2019, Excess wing

dielectric relaxation Glasses

Excess wings in broadband dielectric spectroscopy

Post author By R Hilfer
Post date July 15, 2014

S. Candelaresi, R. Hilfer

AIP Conference Proceedings 1637, 1283 (2014)
https://doi.org/10.1063/1.4907293

submitted on
Tuesday, July 15, 2014

Analysis of excess wings in broadband dielectric spectroscopy data of glass forming materials provides evidence for anomalous time evolutions and fractional semigroups. Solutions of fractional evolution equations in frequency space are used to fit dielectric spectroscopy data of glass forming materials with a range between 4 and 10 decades in frequency. It is shown that with only three parameters (two relaxation times plus one exponent) excellent fits can be obtained for 5-methyl-2-hexanol and for methyl-m-toluate over up to 7 decades. The traditional Havriliak-Negami fit with three parameters (two exponents and one relaxation time) fits only 4-5 decades. Using a second exponent, as in Havriliak-Negami fits, the α-peak and the excess wing can be modeled perfectly with our theory for up to 10 decades for all materials at all temperatures considered here. Traditionally this can only be accomplished by combining two Havriliak-Negami functions with 7 parameters. The temperature dependent relaxation times are fitted with the Vogel-Tammann-Fulcher relation which provides the corresponding Vogel-Fulcher temperatures. The relaxation times turn out to obey almost perfectly the Vogel-Tammann-Fulcher law. Computable expressions of time dependent relaxation functions are also reported.

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dielectric relaxation Fractional Calculus Fractional Time Glasses

Experimental Evidence for Fractional Time Evolution in Glass Forming Materials

Post author By R Hilfer
Post date December 7, 2001

R. Hilfer

Chem.Phys. 284, 399 (2002)
https://doi.org/10.1016/S0301-0104(02)00670-5

submitted on
Friday, December 7, 2001

The infinitesimal generator of time evolution in the standard equation for exponential (Debye) relaxation is replaced with the infinitesimal generator of composite fractional translations. Composite fractional translations are defined as a combination of translation and the fractional time evolution introduced in [Physica A, 221 (1995) 89]. The fractional differential equation for composite fractional relaxation is solved. The resulting dynamical susceptibility is used to fit broad band dielectric spectroscopy data of glycerol. The composite fractional susceptibility function can exhibit an asymmetric relaxation peak and an excess wing at high frequencies in the imaginary part. Nevertheless it contains only a single stretching exponent. Qualitative and quantitative agreement with dielectric data for glycerol is found that extends into the excess wing. The fits require fewer parameters than traditional fit functions and can extend over up to 13 decades in frequency.

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dielectric relaxation Fractional Calculus Fractional Time Glasses

Fitting the excess wing in the dielectric α-relaxation of propylene carbonate

Post author By R Hilfer
Post date November 28, 2001

R. Hilfer

Journal of Physics: Condensed Matter 14, 2297 (2002)
https://doi.org/10.1088/0953-8984/14/9/318

submitted on
Wednesday, November 28, 2001

A novel fitting function for the complex frequency-dependent dielectric susceptibility is introduced and compared against other fitting functions for experimental broadband dielectric loss spectra of propylene carbonate taken from Schneider et al (Schneider U, Lunkenheimer P, Brand R and Loidl A 1999 Phys. Rev. E 59 6924). The fitting function contains a single stretching exponent similar to the familiar Cole–Davidson or Kohlrausch stretched exponential fits. It is compared to these traditional fits as well as to the Havriliak–Negami susceptibility and a susceptibility for a two-step Debye relaxation. The results for the novel fit are found to give superior agreement.

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dielectric relaxation Glasses Special Functions

H-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems

Post author By R Hilfer
Post date June 28, 2001

R. Hilfer

Physical Review E 65, 061510 (2002)
https://doi.org/10.1103/PhysRevE.65.061510

submitted on
Thursday, June 28, 2001

Analytical expressions in the time and frequency domains are derived for non-Debye relaxation processes. The complex frequency-dependent susceptibility function for the stretched exponential relaxation function is given for general values of the stretching exponent in terms of H-functions. The relaxation functions corresponding to the complex frequency-dependent Cole-Cole, Cole-Davidson, and Havriliak-Negami susceptibilities are given in the time domain in terms of H-functions. It is found that a commonly used correspondence between the stretching exponent of Kohlrausch functions and the stretching parameters of Havriliak-Negami susceptibilities are not generally valid.

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dielectric relaxation Glasses Nonequilibrium Special Functions

Analytical representations for relaxation functions of glasses

Post author By R Hilfer
Post date April 13, 2001

R. Hilfer

Journal of Non-Crystalline Solids 305, 122 (2002)
https://doi.org/10.1016/S0022-3093(02)01088-8

submitted on
Friday, April 13, 2001

Analytical representations in the time and frequency domains are derived for the most frequently used phenomenological fit functions for non-Debye relaxation processes. In the time domain the relaxation functions corresponding to the complex frequency dependent Cole–Cole, Cole–Davidson and Havriliak–Negami susceptibilities are also rep- resented in terms of H-functions. In the frequency domain the complex frequency dependent susceptibility function corresponding to the time dependent stretched exponential relaxation function is given in terms of H-functions. The new representations are useful for fitting to experiment.

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Fractional Calculus Glasses

On Fractional Relaxation

Post author By R Hilfer
Post date April 2, 2001

R. Hilfer

Fractals 11, 251 (2003)
https://doi.org/10.1142/S0218348X03001914

submitted on
Monday, April 2, 2001

Generalized fractional relaxation equations based on generalized Riemann-Liouville derivatives are combined with a simple short time regularization and solved exactly. The solution involves generalized Mittag-Leffler functions. The associated frequency dependent susceptibilities are related to symmetrically broadened Cole-Cole susceptibilities occurring as Johari Goldstein β -relaxation in many glass formers. The generalized susceptibilities exhibit a high frequency wing and strong minimum enhancement.

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Fractional Calculus Glasses

On Fractional Relaxation

Post author By R Hilfer
Post date April 2, 2001

R. Hilfer

in: Scaling and Disordered Systems
edited by: F. Family and M. Daoud and H. Herrmann and H.E. Stanley
World Scientific, Singapore, 251 (2002)
https://doi.org/10.1142/9789812778109_0026
ISBN: 978-981-02-4838-3

submitted on
Monday, April 2, 2001

Generalized fractional relaxation equations based on generalized Riemann-Liouville derivatives are combined with a simple short time regularization and solved exactly. The solution involves generalized Mittag-Leffler functions. The associated frequency dependent susceptibilities are related to symmetrically broadened Cole-Cole susceptibilities occurring as Johari Goldstein β-relaxation in many glass formers. The generalized susceptibilities exhibit a high frequency wing and strong minimum enhancement.

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Disordered Systems Glasses

Theoretical Aspects of Realistic Spin Glass Models

Post author By R Hilfer
Post date July 27, 1989

R. Hilfer

in: New Trends in Magnetism
edited by: M.D. Coutinho-Filho and S.M. Rezende
World Scientific Publ.Co., Singapore, 32 (1989)

submitted on
Thursday, July 27, 1989

This note investigates the universality of spin glass models by calculating the distribution of instantaneous local magnetic fields, p(h). It is found that short range Ising models with Gaussian bond disorder fall into a different universality class than realistic models with RKKY-interactions and randomly positioned spins. The result is obtained from an analysis of p(h) at high temperatures where thelocal fields are sums of independent random variables. It is found that for realistic models these sums are in general not governed by the central limit theorem. In three dimensions a cutoff Cauchy distribution is obtained for p(h) instead of a Gaussian distribution. In general p(h) is a cutoff stable law whose characteristic exponent depends strongly on the dimension and the decay of the interactions. As a consequence a new short range model is proposed for dilute metallic spin glasses in three dimensions in which the bond disorder is taken to be a cutoff Cauchy distribution instead of a Gaussian. Preliminary considerations indicate a much smoother specific heat for models in this universality class and suggest the existence of strong precursor effects in qualitative agreement with experiment.

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