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Fractional Calculus Functional analysis

Fractional Calculus for Distributions

R. Hilfer, T. Kleiner

Fractional Calculus and Applied Analysis , (2024)
https://doi.org/10.1007/s13540-024-00306-z

submitted on
Friday, March 29, 2024

Fractional derivatives and integrals for measures and distributions are reviewed. The focus is on domains and co-domains for translation invariant fractional operators. Fractional derivatives and integrals interpreted as D’-convolution operators with power law kernels are found to have the largest domains of definition. As a result, extending domains from functions to distributions via convolution operators contributes to far reaching unifications of many previously existing definitions of fractional integrals and derivatives. Weyl fractional operators are thereby extended to distributions using the method of adjoints. In addition, discretized fractional calculus and fractional calculus of periodic distributions can both be formulated and understood in terms of D’-convolution.



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Fractional Calculus Functional analysis Mathematics

Convolution on Distribution Spaces Characterized by Regularization

T. Kleiner, R. Hilfer

Mathematische Nachrichten 296, 1938-1963 (2023)
https://doi.org/10.1002/mana.202100330

submitted on
Friday, October 15, 2021

Locally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space of test functions into a given locally convex lattice of measures. The convolutor spaces are endowed with the topology of uniform convergence on bounded sets. Their locally convex structure is characterized via regularization and function-valued seminorms under mild structural assumptions on the space of measures. Many recent generalizations of classical distribution spaces turn out to be special cases of the general convolutor spaces introduced here. Recent topological characterizations of convolutor spaces via regularization are extended and improved. A valuable property of the convolutor spaces in applications is that convolution of distributions inherits continuity properties from those of bilinear convolution mappings between the locally convex latti\-ces of measures.



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dielectric relaxation Disordered Systems electrical conductivity Heterogeneous Materials Percolation Transport Processes

Effective transport coefficients of anisotropic disordered materials

R. Hilfer, J. Hauskrecht

European Physical Journal B 95, 117 (2022)
https://doi.org/10.1140/epjb/s10051-022-00338-5

submitted on
Tuesday, January 4, 2022

A novel effective medium theory for homogenized transport coefficients of anisotropic mixtures of possibly anisotropic materials is developed. Existing theories for isotropic systems cannot be easily extended, because that would require geometric characterizations of anisotropic connectivity. In this work anisotropic connectivity is characterized by introducing a tensor that is constructed from a histogram of local percolating directions. The construction is inspired by local porosity theory. A large number of known and unknown generalized effective medium approximations for anisotropic media are obtained as limiting special cases from the new theory. Among these limiting cases the limit of strong cylindrical anisotropy is of particular interest. The parameter space of the generalized theory is explored, and the advanced results are applied to experiment.



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Fractional Calculus Functional analysis Mathematics

Sequential generalized Riemann–Liouville derivatives based on distributional convolution

T. Kleiner, R. Hilfer

Fractional Calculus and Applied Analysis 25, 267-298 (2022)
https://doi.org/10.1007/s13540-021-00012-0

submitted on
Friday, October 15, 2021

Sequential generalized fractional Riemann-Liouville derivatives are introduced as composites of distributional derivatives on the right half axis and partially defined operators, called Dirac-function removers, that remove the component of singleton support at the origin of distributions that are of order zero on a neighborhood of the origin. The concept of Dirac-function removers allows to formulate generalized initial value problems with less restrictions on the orders and types than previous approaches to sequential fractional derivatives. The well-posedness of these initial value problems and the structure of their solutions are studied.



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fluid flow Porous Media review article Two-Phase Flow

A Brief Review of Capillary Number and its Use in Capillary Desaturation Curves

H. Guo, K. Song, R. Hilfer

Transport in Porous Media 144, 3-31 (2022)
https://doi.org/10.1007/s11242-021-01743-7

submitted on
Monday, August 9, 2021

Capillary number, understood as the ratio of viscous force to capillary force, is one of the most important parameters in enhanced oil recovery (EOR). It continues to attract the interest of scientists and engineers, because the nature and quantification of macroscopic capillary forces remains controversial. At least 41 different capillary numbers have been collected here from the literature. The ratio of viscous and capillary force enters crucially into capillary desaturation experiments. Although the ratio is length scale dependent, not all definitions of capillay number depend on length scale, indicating potential inconsistencies between various applications and publications. Recently, new numbers have appeared and the subject continues to be actively discussed. Therefore, a short review seems appropriate and pertinent.



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Equilibrium Lattice Models Nonequilibrium Statistical Physics

Foundations of statistical mechanics for unstable interactions

R. Hilfer

Physical Review E 105, 024142 (2022)
https://doi.org/10.1103/PhysRevE.105.024142

submitted on
Thursday, May 27, 2021

Traditional Boltzmann-Gibbs statistical mechanics does not apply to systems with unstable interactions, because for such systems the conventional thermodynamic limit does not exist. In unstable systems the ground state energy does not have an additive lower bound, i.e., no lower bound linearly proportional to the number N of particles or degrees of freedom. In this article unstable systems are studied whose ground state energy is bounded below by a regularly varying function of N with index \sigma\geq 1. The index \sigma\geq 1 of regular variation introduces a classification with respect to stability. Stable interactions correspond to σ = 1. A simple example for an unstable system with σ =2 is an ideal gas with a nonvanishing constant two-body potential. The foundations of statistical physics are revisited, and generalized ensembles are introduced for unstable interactions in such a way that the thermodynamic limit exists. The extended ensembles are derived by identifying and postulating three basic properties as extended foundations for statistical mechanics: first, extensivity of thermodynamic systems, second, divisibility of equilibrium states, and third, statistical independence of isolated systems. The traditional Boltzmann-Gibbs postulate, resp. the hypothesis of equal a priori probabilities, is identified as a special case of the extended ensembles. Systems with unstable interactions are found to be thermodynamically normal and extensive. The formalism is applied to ideal gases with constant many-body potentials. The results show that, contrary to claims in the literature, stability of the interaction is not a necessary condition for the existence of a thermodynamic limit. As a second example the formalism is applied to the Curie-Weiss-Ising model with strong coupling. This model has index of stability σ = 2. Its thermodynamic potentials [originally obtained in R. Hilfer, Physica A 320, 429 (2003)] are confirmed up to a trivial energy shift. The strong coupling model shows a thermodynamic phase transition of order 1 representing a novel mean-field universality class. The disordered high temperature phase collapses into the ground state of the system. The metastable extension of the high temperature free energy to low temperatures ends at absolute zero in a phase transition of order 1/2. Between absolute zero and the critical temperature of the first order transition all fluctuations are absent.



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Mathematical Physics Porous Media Two-Phase Flow

Existence and Uniqueness of Nonmonotone Solutions in Porous Media Flow

R. Steinle, T. Kleiner, P. Kumar, R. Hilfer

Axioms 11, 327 (2022)
https://doi.org/10.3390/axioms11070327

submitted on
Thursday, May 5, 2022

Existence and uniqueness of solutions for a simplified model of immiscible two-phase flow in porous media are obtained in this paper. The mathematical model is a simplified physical model with hysteresis in the flux functions. The resulting semilinear hyperbolic-parabolic equation is expected from numerical work to admit non-monotone imbibition-drainage fronts. We prove the local existence of imbibition-drainage fronts. The uniqueness, global existence, maximal regularity and boundedness of the solutions are also discussed. Methodically, the results are established by means of semigroup theory and fractional interpolation spaces.



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Disordered Systems Heterogeneous Materials Percolation Porous Media

Percolativity of Porous Media

R. Hilfer, J. Hauskrecht

Transport in Porous Media 145, 1-12 (2022)
https://doi.org/10.1007/s11242-021-01735-7

submitted on
Monday, April 19, 2021

Connectivity and connectedness are non-additive geometric functionals on the set of pore scale structures. They determine transport of mass, volume or momentum in porous media, because without connectivity there cannot be transport. Percolativity of porous media is introduced here as a geometric descriptor of connectivity, that can be computed from the pore scale and persists to the macroscale through a suitable upscaling limit. It is a measure that combines local percolation probabilities with a probability density of ratios of eigenvalues of the tensor of local percolating directions. Percolativity enters directly into generalized effective medium approximations. Predictions from these generalized effective medium approximations are found to be compatible with apparently anisotropic Archie correlations observed in experiment.



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Fractional Calculus Functional analysis Mathematics

On extremal domains and codomains for convolution of distributions and fractional calculus

T. Kleiner, R. Hilfer

Monatshefte für Mathematik 198, 122-152 (2022)
https://doi.org/10.1007/s00605-021-01646-1

submitted on
Wednesday, December 30, 2020

It is proved that the class of c-closed distribution spaces contains extremal domains and codomains to make convolution of distributions a well-defined bilinear mapping. The distribution spaces are systematically endowed with topologies and bornologies that make convolution hypocontinuous whenever defined. Largest modules and smallest algebras for convolution semigroups are constructed along the same lines. The fact that extremal domains and codomains for convolution exist within this class of spaces is fundamentally related to quantale theory. The quantale theoretic residual formed from two c-closed spaces is characterized as the largest c-closed subspace of the corresponding space of convolutors. The theory is applied to obtain maximal distributional domains for fractional integrals and derivatives, for fractional Laplacians, Riesz poten- tials and for the Hilbert transform. Further, maximal joint domains for families of these operators are obtained such that their composition laws are preserved.



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Categories
Fractional Calculus Functional analysis Glasses Mathematical Physics Mathematics Special Functions

Fractional glassy relaxation and convolution modules of distributions

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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fluid flow Porous Media Two-Phase Flow

A Critical Review of Capillary Number and its Application in Enhanced Oil Recovery

H. Guo, K. Song, R. Hilfer

SPE Conference Proceedings 2020, SPE-200419-MS (2020)
https://doi.org/10.2118/200419-MS

submitted on
Sunday, August 30, 2020

Capillary number (Ca), defined as dimensionless ratio of viscous force to capillary force, is one of the most important parameters in enhanced oil recovery (EOR). The ratio of viscous and capillary force is scale-dependent. At least 33 different Cas have been proposed, indicating inconsistencies between various applications and publications. The most concise definition containing velocity, interfacial tension and viscosity is most widely used in EOR. Many chemical EOR applications are thus based on the correlation between residual oil saturation (ROS) and Ca, which is also known as capillary desaturation curve (CDC). Various CDCs lead to a basic conclusion of using surfactant to reduce interfacial tension to ultra-low values to get a minimum ROS and maximum displacement efficiency. However, after a deep analysis of Ca and recent new experimental observations, the traditional definition of Ca was found to have many limitations and based on misunderstandings. First, the basic object in EOR is a capillary-trapped oil ganglia, thus Darcy’s law is only valid under certain conditions. Further, many recent tests reported results contradicting previous ones. It seems most Cas cannot account for mixed-wet CDC. The influence of wettability on two-phase flow is important but not reflected in the definition of the Ca. Then, it is certainly very peculiar that, when the viscous and capillary forces acting on a blob are equal, the current most widely used classic Ca is equal to 2.2* 10−3. Ideally, the condition Ca ∼ 1 marks the transition from capillary dominated to viscous-dominated flow, but most Cas cannot fulfill this expectation. These problems are caused by scale dependent flow characterization. It has been proved that the traditional Ca is of microscopic nature. Based on the dynamic characterization of the change of capillary force and viscous force on the macroscopic scale, a macroscopic Ca can well explain these complex results. The requirement of ultra-low IFT from microscopic Ca for surfactant flood is not supported by macroscopic Ca. The effect of increasing water viscosity to EOR is much higher than reducing IFT. Realizing the microscopic nature of the traditional Ca and using CDCs based on the more reasonable macroscopic Ca helps to update screening criteria for chemical flooding.



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Porous Media Two-Phase Flow

Capillary number correlations for two-phase flow in porous media

R. Hilfer

Physical Review E 102, 053103 (2020)
https://doi.org/10.1103/PhysRevE.102.053103

submitted on
Thursday, August 20, 2020

Relative permeabilities and capillary number correlations are widely used for quantitative estimates of enhanced water flood performance in porous media. They enter as essential parameters into reservoir simulations. Experimental capillary number correlations for seven different reservoir rocks and 21 pairs of wetting and nonwetting fluids are analyzed. The analysis introduces generalized local macroscopic capillary number correlations. It eliminates shortcomings of conventional capillary number correlations. Surprisingly, the use of capillary number correlations on reservoir scales may become inconsistent in the sense that the limits of applicability of the underlying generalized Darcy law are violated. The results show that local macroscopic capillary number correlations can distinguish between rock types. The experimental correlations are ordered systematically using a three-parameter fit function combined with a novel fluid pair based figure of merit.



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Categories
Fractional Calculus Functional analysis Uncategorized

Maximal Domains for Fractional Derivatives and Integrals

R. Hilfer, T. Kleiner

Mathematics 8, 1107 (2020)
https://doi.org/10.3390/math8071107

submitted on
Wednesday, March 11, 2020

The purpose of this short communication is to announce the existence of fractional calculi on precisely specified domains of distributions. The calculi satisfy desiderata proposed above in Mathematics 7, 149 (2019). For the desiderata (a)–(c) the examples are optimal in the sense of having maximal domains with respect to convolvability of distributions. The examples suggest to modify desideratum (f) in the original list.



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Categories
dielectric relaxation Disordered Systems Glasses Transport Processes

Excess wing physics and nearly constant loss in glasses

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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Functional analysis Mathematics

Weyl Integrals on Weighted Spaces

T. Kleiner, R. Hilfer

Fractional Calculus and Applied Analysis 22, 1225-1248 (2019)
DOI: 10.1515/fca-2019-0065

submitted on
Thursday, January 31, 2019

Weighted spaces of continuous functions are introduced such that Weyl fractional integrals with orders from any finite nonnegative interval define equicontinuous sets of continuous linear endomorphisms for which the semigroup law of fractional orders is valid. The result is obtained from studying continuity and boundedness of convolution as a bilinear operation on general weighted spaces of continuous functions and measures.



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Fractional Calculus Mathematics

Desiderata for Fractional Derivatives and Integrals

R. Hilfer, Yu. Luchko

Mathematics 7, 149 (2019)
https://doi.org/10.3390/math7020149

submitted on
Friday, January 11, 2019

The purpose of this brief article is to initiate discussions in this special issue by proposing desiderata for calling an operator a fractional derivative or a fractional integral. Our desiderata are neither axioms nor do they define fractional derivatives or integrals uniquely. Instead they intend to stimulate the field by providing guidelines based on a small number of time honoured and well established criteria.



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Functional analysis Mathematics

Convolution Operators on Weighted Spaces of Continuous Functions and Supremal Convolution

T. Kleiner, R. Hilfer

Annali di Matematica Pura ed Applicata 199, 1547-1569 (2020)
https://doi.org/10.1007/s10231-019-00931-z

submitted on
Tuesday, September 25, 2018

The convolution of two weighted balls of measures is proved to be contained in a third weighted ball if and only if the supremal convolution of the corresponding two weights is less than or equal to the third weight. Here supremal convolution is introduced as a type of convolution in which integration is replaced with supremum formation. Invoking duality the equivalence implies a characterization of equicontinuity of weight-bounded sets of convolution operators having weighted spaces of continuous functions as domain and range. The overall result is a constructive method to define weighted spaces on which a given set of convolution operators acts as an equicontinuous family of endomorphisms. The result is applied to linear combinations of fractional Weyl integrals and derivatives with orders and coefficients from a given bounded set.



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Fractional Calculus Functional analysis Mathematical Physics Mathematics Stochastic Processes

Mathematical and physical interpretations of fractional derivatives and integrals

R. Hilfer

in: Handbook of Fractional Calculus with Applications: Basic Theory, Vol. 1
edited by: A. Kochubei and Y. Luchko
Walter de Gruyter GmbH, Berlin, 47-86 (2019)
https://doi.org/10.1515/9783110571622
ISBN: 9783110571622

submitted on
Saturday, June 2, 2018

Brief descriptions of various mathematical and physical interpretations of fractional derivatives and integrals have been collected into this chapter as points of reference and departure for deeper studies. “Mathematical interpretation” in the title means a brief description of the basic mathematical idea underlying a precise definition. “Physical interpretation” means a brief description of the physical theory underlying an identification of the fractional order with a known physical quantity. Numerous interpretations had to be left out due to page limitations. Only a crude, rough and ready description is given for each interpretation. For precise theorems and proofs an extensive list of references can serve as a starting point.



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Categories
Ergodicity Fractional Time Irreversibility Statistical Physics Theory of Time

On Local Equilibrium and Ergodicity

R. Hilfer

Acta Physica Polonica B 49, 859 (2018)
DOI: 10.5506/APhysPolB.49.859

submitted on
Friday, April 27, 2018

The main mathematical argument of the universal framework for local equilibrium proposed in Analysis 36, 49 (2016) is condensed and formulated as a fundamental dichotomy between subsets of positive measure and subsets of zero measure in ergodic theory. The physical interpretation of the dichotomy in terms of local equilibria rests on the universality of time scale separation in an appropriate long-time limit.



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Equilibrium Statistical Physics

Ground state collapse at strong coupling

R. Hilfer

Journal MESA 8, 307-310 (2017)

submitted on
Friday, May 26, 2017

The infinite range Ising model is usually investigated in the weak coupling limit. Here the model is solved with ferromagnetic coupling at fixed and finite strength. Exact analytical expressions are found for the thermodynamic potentials as functions of enthalpy and external field. These results differ from the potentials for the weak coupling limit. The model shows a temperature driven first order phase transition from a paramagnetic phase at high temperatures into a low temperature phase from which thermal fluctuations are absent.



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