Renyi entropy and fractional order mechanics for predicting complex mechanics of materials

Pahari, Basanta R.; Oates, William S.

doi:10.1117/12.2613356

Cited by 4 publications

(4 citation statements)

References 5 publications

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“…In addition to encompassing other entropy measures such as collision entropy R 2 , minimum entropy R ∞ , and maximum entropy R 0 , Renyi entropy also has applications in a variety of scientific disciplines. Recent research shows that Renyi entropy is useful in building constitutive models for multi-functional polymers 29 and describing the dynamics of material interactions such as charged particle diffusion. 27 In thermodynamics, Renyi entropy offers a method to derive free energy by conceiving the probability distribution as a Gibbs state-a state of thermal equilibrium for a specific Hamiltonian at temperature T .…”

Section: Entropy and Multifractalsmentioning

confidence: 99%

Multifractal analysis of heat transport in DLA structures

Carvajal,

Oates,

Pahari

2024

Behavior and Mechanics of Multifunctional Materials XVIII

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Multifractal analysis originated from the desire to obtain a deeper understanding of materials that follow random fractal structures. Complex systems display diverse degrees of irregularity and self-similarity on multiple scales that cannot be adequately described by a single fractal dimension due to uncertainties as well as deterministic changes in structure at different length and time scales. The multifractal spectrum measures the distribution of a specific property across different scales or levels within a system using a range of exponents. These properties can encompass anything from turbulence intensity in fluid flow to the distribution of heat flow through a solid. Each exponent in the spectrum corresponds to a particular level of irregularity, and a broad range of exponents indicates more complexity. Multifractal measures have a range of applications, from biology to mechanical engineering, among many others. In multifunctional materials used in adaptive systems, a more accurate description of material properties, such as viscoelasticity, heat transport, phase transformations, often involves considering information about their underlying fractal material structure. This study goes beyond fractals to include randomness in terms of multifractal measures. This involves analysis of the Renyi entropy. We evaluate this approach by examining heat transport in DLA structures which display multifractal properties. Renyi entropy is relevant here due to its entropy order parameter's close connection to the multifractal spectrum. In fact, the multifractal spectrum can be directly calculated through the generalized Renyi entropy dimension and a box-counting process. We investigate the multifractality of different DLA structures and leverage this insight to better understand heat diffusion processes using fractal-order diffusion equations; however, the methodology is more broadly applicable to a wide range of multiphysics problems involving energy transport over complex media.

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Section: Entropy and Multifractalsmentioning

confidence: 99%

Multifractal analysis of heat transport in DLA structures

Carvajal,

Oates,

Pahari

2024

Behavior and Mechanics of Multifunctional Materials XVIII

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“…Even though it may not seem apparently useful, this result can be exploited to establish the ideal gas law and has other applications in mechaincs. 4 Since uniform density maximizes unconstrained Renyi entropy for all α, it is natural to ask how is max entropy related to uniform density. We now establish a new relationship between uniform densities and max entropy by defining fixed points of the Renyi entropy for a given probability distribution.…”

Section: Renyi Entropy and Probability Densitiesmentioning

confidence: 99%

“…The molecular dynamics model we derive here relies on the entropy of a given system and our approach is similar to the relative Shannon entropy 6,7 and Renyi entropy formulation of the polymer deformation process. 4 We will only present a 1D dimensional case but the idea can be extended to higher dimensions. Now we define the Renyi divergence analog to Shannon's relative entropy of two different distributions.…”

Section: Entropy Dynamics Approach To Molecular Dynamicsmentioning

confidence: 99%

Game theoretic simulations and entropy dynamics framework for modeling complex material interactions

Pahari

Ogunlana

Oates

2023

Behavior and Mechanics of Multifunctional Materials XVII

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Molecular dynamics is a technique used to understand how the interactions between atoms control complex material behavior under internal and external forces. We use a game theoretical approach to simulate molecular interactions, build material surrogates, and study material nature as a function of the Coulomb potential. The electric Coulomb potential is applicable in studying electromagnetic properties, for instance, electrical conductivity and dielectric permittivity. We investigate how probability densities of material positions and charges influence the diffusion process induced by this potential. By exploiting the information hidden in the seemingly random molecular exchanges, we show that an information-theoretical measure of entropy can describe the dynamics of material interactions. In particular, we use the Renyi entropy to derive posterior density representing the most likely particle distribution after these exchanges and thus connect molecular dynamics to entropy dynamics.

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“…In general, the kappa/HCDT entropy is connected with the thermodynamics of particle systems, while the Rényi entropy is used in a different context, such as, in fractal dimension analysis [21][22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

Extensive entropy: the case of zero entropy defect

Livadiotis,

McComas

2023

Phys. Scr.

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This paper shows that the Rényi and Boltzmann-Gibbs (BG) extensive entropies share the same functional relationship with the nonextensive entropy associated with kappa distributions, which coincides with the well-known Havrda/Charvát/Daróczy/Tsallis (HCDT) entropy. We find that while the relationship between kappa/HCDT and Rényi entropies is merely a mathematical identity between their entropic statistical definitions, the relationship between kappa/HCDT and BG entropies is based on their thermodynamic connection. The latter connects the entropy between a system characterized by correlations among their all constituents (kappa/HCDT entropy) and the entropy of the same system but with no correlations among their constituents (BG entropy). The origin of this relationship, and its connection with thermodynamics, is examined using the concept of entropy defect, that is, the decrease in a system’s entropy caused by the presence of long-range correlations among its constituents; in the limiting case of zero correlations, the entropy defect vanishes, and the entropy becomes extensive and expressed by the BG formulation.

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Renyi entropy and fractional order mechanics for predicting complex mechanics of materials

Cited by 4 publications

References 5 publications

Multifractal analysis of heat transport in DLA structures

Multifractal analysis of heat transport in DLA structures

Game theoretic simulations and entropy dynamics framework for modeling complex material interactions

Extensive entropy: the case of zero entropy defect

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