Equilibrium and non-equilibrium statistical mechanics with generalized fractal derivatives: A review

Golmankhaneh, Alireza Khalili; Welch, Kerri

doi:10.1142/s0217732321400022

Cited by 33 publications

(6 citation statements)

References 86 publications

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“…Several phenomena involving fractal time are very interesting, and it is important to study them [51,52]. Equilibrium and non-equilibrium statistical mechanics involving generalized fractal derivatives were reviewed [53].…”

Section: Fractional Brownian Motion On Fractal Setsmentioning

confidence: 99%

Fractal Stochastic Processes on Thin Cantor-Like Sets

Golmankhaneh

Sibatov

2021

Mathematics

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We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal.

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Section: Fractional Brownian Motion On Fractal Setsmentioning

confidence: 99%

Fractal Stochastic Processes on Thin Cantor-Like Sets

Golmankhaneh

Sibatov

2021

Mathematics

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“…This type of calculus is algorithmic in nature and offers a more streamlined approach when compared to alternative methodologies [22][23][24]. Fractal calculus found generalization and application within the realm of physics, extending to stochastic equations and the establishment of Fourier and Laplace transforms [25][26][27][28][29][30][31][32]. The resolution of fractal delayed, neutral, and renewal delay differential equations with constant coefficients was accomplished using the method of steps and Laplace transforms [33].…”

Section: Introductionmentioning

confidence: 99%

Fractal Schrödinger equation: implications for fractal sets

Khalili Golmankhaneh,

Pellis,

Zingales

2024

J. Phys. A: Math. Theor.

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This paper delves into the world of fractal calculus, investigating its implications for fractal sets. It introduces the Fractal Schrödinger Equation and provides insights into its consequences. The study presents a General Solution for the Time-Dependent Schrödinger Equation, unveiling its core aspects. Exploring quantum mechanics in the context of fractals, the paper analyzes the Probability Density of the Radial Hydrogen Atom, unveiling its behavior within fractal dimensions. The investigation extends to deciphering the intricate Energy Levels of the Hydrogen Atom, uncovering the interplay of quantum mechanics and fractal geometry. Innovatively, the research applies the Fractal Schrödinger Equation to Simple Harmonic Motion, leading to the introduction of the Fractal Probability Density Function for the Harmonic Oscillator. The paper employs a series of illustrative ﬁgures that enhance the comprehension of the ﬁndings. By intertwining quantum mechanics and fractal mathematics, this research paves the way for deeper insights into their relationship.

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“…The practical applicability of the variational method in describing dissipative dynamical systems was showcased, and the Hamiltonian approach produced auxiliary constraints without reliance on Dirac auxiliary functions [44]. Furthermore, fractal stochastic differential equations have been defined, with categorizations for processes like fractional Brownian motion and diffusion occurring within mediums with fractal structures [45,46,47,48,49]. Local vector calculus within fractional-dimensional spaces, on fractals, and in fractal continua was developed.…”

Section: Introductionmentioning

confidence: 99%

Non-standard analysis for fractal calculus

Golmankhaneh

Welch

Serpa

et al. 2023

J Anal

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In this paper, we delve into the fascinating realm of fractal calculus applied to fractal sets and fractal curves. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for solving α-order differential equations. Notably, we extend our analysis to solve Fractal Bernoulli differential equations. The applications of our findings are then showcased through the solutions of problems such as fractal compound interest, the escape velocity of the earth in fractal space and time, and the estimation of time of death incorporating fractal time. Visual representations of our results are also provided to enhance understanding.

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Equilibrium and non-equilibrium statistical mechanics with generalized fractal derivatives: A review

Cited by 33 publications

References 86 publications

Fractal Stochastic Processes on Thin Cantor-Like Sets

Fractal Stochastic Processes on Thin Cantor-Like Sets

Fractal Schrödinger equation: implications for fractal sets

Non-standard analysis for fractal calculus

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