Lambert function and a new non-extensive form of entropy

Shafee, Fariel

doi:10.1093/imamat/hxm039

Cited by 55 publications

(44 citation statements)

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“…[52] Interestingly, Linhart's m k has just the same functional form as those derived for the Tsallis entropy expressions using the Jaynes MAXENT approach. [4,53] As we have not employed the MAXENT approach here, Equations (14)- (16) prove that the Linhart theory should be consistent with the Tsallis-like generalization of the Boltzmann-Gibbs statistical mechanics.…”

Section: Thermodynamic Parameters Derived From Experimentally Measurementioning

confidence: 96%

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Many Faces of Entropy or Bayesian Statistical Mechanics

Starikov

2010

ChemPhysChem

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Some 80-90 years ago, George A. Linhart, unlike A. Einstein, P. Debye, M. Planck and W. Nernst, managed to derive a very simple, but ultimately general mathematical formula for heat capacity versus temperature from fundamental thermodynamic principles, using what we would nowadays dub a "Bayesian approach to probability". Moreover, he successfully applied his result to fit the experimental data for diverse substances in their solid state over a rather broad temperature range. Nevertheless, Linhart's work was undeservedly forgotten, although it represents a valid and fresh standpoint on thermodynamics and statistical physics, which may have a significant implication for academic and applied science.

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Section: Thermodynamic Parameters Derived From Experimentally Measurementioning

confidence: 96%

“…[3] In accordance with this, new functional definitions of entropy are still being suggested in the literature (e.g., see ref. [4] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Many Faces of Entropy or Bayesian Statistical Mechanics

Starikov

2010

ChemPhysChem

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“…Turnings to the discussion of the high-temperature resonances achieved at the small negative extrapolation length, one notices that, similar to the canonical ensemble, they can be mathematically explained with the help of formulas modified from their asymmetric counterparts; for example, first righthand side items of Equations (39), (40), (48)-(51) at → −0 have to be multiplied by two what, for the case of one fermion, leads to a decrease of the maximum and its shift to the warmer temperatures as compared to the asymmetric configuration what was also the case for the canonical distribution. Situation changes for the larger number of particles in the well; say, for N = 2, both corpuscles reside on the split-off orbitals whereas for − = − + , the second electron occupies lowest positive-energy state and, accordingly, its contribution to the total specific heat is much smaller than for its symmetric fellow.…”

Section: Fermionsmentioning

confidence: 99%

Thermodynamic Properties of the 1D Robin Quantum Well

Olendski

2018

Annalen der Physik

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The thermodynamic properties of the Robin quantum well with extrapolation length are analyzed theoretically both for the canonical and two grand canonical ensembles, with special attention being paid to the situation when the energies of one or two lowest-lying states are split off from the rest of the spectrum by the large gap that is controlled by the varying . For the single split-off level, which exists for the geometry with the equal magnitudes but opposite signs of the Robin distances on the confining interfaces, the heat capacity c V of the canonical averaging is a nonmonotonic function of the temperature T with its salient maximum growing to infinity as ln 2 for the decreasing to zero extrapolation length and its position being proportional to 1/( 2 ln ). The specific heat per particle c N of the Fermi-Dirac ensemble depends nonmonotonically on the temperature too, with its pronounced extremum being foregone on the T axis by the plateau, whose value at the dying is (N − 1)/(2N)k B , with N being the number of fermions. The maximum of c N , similar to the canonical averaging, unrestrictedly increases as goes to zero and is largest for one particle. The most essential property of the Bose-Einstein ensemble is the formation, for a growing number of bosons, of the sharp asymmetric shape on the c N −T characteristics, which is more protrusive at the smaller Robin distances. This cusp-like structure is a manifestation of the phase transition to the condensate state. For two split-off orbitals, one additional maximum emerges whose position is shifted to colder temperatures with the increase of the energy gap between these two states and their higher-lying counterparts and whose magnitude approaches a -independent value. All these physical phenomena are qualitatively and quantitatively explained by the variation of the energy spectrum by the Robin distance. Parallels with other structures are drawn and similarities and differences between them are highlighted. Generalization to higher dimensions is also provided.

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“…For a generalized wavelet with a variable fractional value u, the central frequency and the frequency band can be expressed analytically, using a special function, the Lambert W function (Lambert 1758, 1772; Euler 1779; Corless et al 1996;Banwell & Jayakumar 2000;Valluri et al 2000;Packel & Yuen 2004;Shafee 2007;Wang 2015a,b). The mean frequency and its deviation can also be derived analytically in terms of the Gamma function.…”

Section: Introductionmentioning

confidence: 99%

“…This paper will prove that the power spectrum of a generalized wavelet is close to a Gaussian distribution, and thus the mean frequency is approximately equal to the central frequency. The degree of similarity between a generalized wavelet spectrum and a Gaussian distribution depends upon the fractional value.For a generalized wavelet with a variable fractional value u, the central frequency and the frequency band can be expressed analytically, using a special function, the Lambert W function (Lambert 1758(Lambert , 1772Euler 1779;Corless et al 1996;Banwell & Jayakumar 2000;Valluri et al 2000;Packel & Yuen 2004;Shafee 2007; Wang 2015a,b). The mean frequency and its deviation can also be derived analytically in terms of the Gamma function.…”

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confidence: 99%

Generalized seismic wavelets

Wang¹

2015

Geophys. J. Int.

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S U M M A R YThe Ricker wavelet, which is often employed in seismic analysis, has a symmetrical form. Seismic wavelets observed from field data, however, are commonly asymmetric with respect to the time variation. In order to better represent seismic signals, asymmetrical wavelets are defined systematically as fractional derivatives of a Gaussian function in which the Ricker wavelet becomes just a special case with the integer derivative of order 2. The fractional value and a reference frequency are two key parameters in the generalization. Frequency characteristics, such as the central frequency, the bandwidth, the mean frequency and the deviation, may be expressed analytically in closed forms. In practice, once the statistical properties (the mean frequency and deviation) are numerically evaluated from the discrete Fourier spectra of seismic data, these analytical expressions can be used to uniquely determine the fractional value and the reference frequency, and subsequently to derive various frequency quantities needed for the wavelet analysis. It is demonstrated that field seismic signals, recorded at various depths in a vertical borehole, can be closely approximated by generalized wavelets, defined in terms of fractional values and reference frequencies.Key words: Time-series analysis; Numerical solutions; Computational seismology; Wave propagation. I N T RO D U C T I O NThe Ricker wavelet is a well-known symmetrical waveform in the time domain (Ricker 1953). In order to better represent practically observed non-Ricker forms of seismic signals (Hosken 1988), the symmetric Ricker wavelet is generalized to be asymmetrical.While the Ricker wavelet is the second derivative of a Gaussian function, generalization is achieved by modifying the derivative order from the integer '2' to a fractional value. For mathematical convenience, the base function is the same Gaussian function, rather than other alternative forms. Therefore, generalized wavelets are systematically defined by fractional derivatives of a Gaussian function.Generalized wavelets have similar Fourier spectra because they are derived from the same Gaussian function. Their spectra differ from each other only in a frequency-related factor (iω) u , where ω is the angular frequency and u is the fractional order of the time derivative. Although there are possibly different definitions for field seismic wavelets, the current paper provides a systematic definition of non-Ricker wavelets and meanwhile can use the Ricker wavelet as a benchmark. This paper will prove that the power spectrum of a generalized wavelet is close to a Gaussian distribution, and thus the mean frequency is approximately equal to the central frequency. The degree of similarity between a generalized wavelet spectrum and a Gaussian distribution depends upon the fractional value.For a generalized wavelet with a variable fractional value u, the central frequency and the frequency band can be expressed analytically, using a special function, the Lambert W function (Lambert 1758(Lambert , 1772...

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Lambert function and a new non-extensive form of entropy

Cited by 55 publications

References 3 publications

Many Faces of Entropy or Bayesian Statistical Mechanics

Many Faces of Entropy or Bayesian Statistical Mechanics

Thermodynamic Properties of the 1D Robin Quantum Well

Generalized seismic wavelets

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