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List of Participants and Conference Schedule 3Organizing committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Conference Dinner 8
Abstracts 9Nevanlinna-Pick functions coming from the free probability (M. Bozejko) . . . . 9Brownian yet Non-Gaussian: Diffusing Diffusivity versus Heterogeneous Models
(A. Chechkin) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Analysis of complex dynamics in nonlinear reaction-diffusion systems with frac-
tional derivatives on basis of a combination of operational and numericalmethods (B. Datsko) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
One parameter fractional and real groups of operators and their combinatorics(G. H. E. Duchamp) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Conservative random walks in confining potentials: tails versus central parts(B. Dybiec) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Some convolution equations with completely monotone kernels (A. Hanyga) . . . 11Composite Continuous Time Random Walks (R. Hilfer) . . . . . . . . . . . . . . 12Biased continuous-time random walks for ordinary and equilibrium cases: fa-
cilitation of diffusion, ergodicity breaking and ageing (R. Hou, Andrey G.Cherstvy, Ralf Metzler and Takuma Akimoto) . . . . . . . . . . . . . . . . . 12
Levy walk with position dependent resting time (A. Kaminskaand T. Srokowski) 13Subdiffusion-absorption process in a layered system (T. Koszto lowicz) . . . . . . 13Umbral Fractional Calculus and Physical Applications (S. Licciardi) . . . . . . . 14On modifications of the exponential integral with the Mittag-Leffler function
(F. Mainardiand E. Masina) . . . . . . . . . . . . . . . . . . . . . . . . . . 14Long-lived luminescence I – measurements, material properties and applications
(E. Mandowskaand A. Mandowski) . . . . . . . . . . . . . . . . . . . . . . . 14Long-lived luminescence II – development of theoretical models (A. Mandowskiand
E. Mandowska ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Non-ergodicity and ageing in single particle trajectories (R. Metzler) . . . . . . . 17A family of Eulerian functions involved in regularization of divergent polyzetas
(H. N. Minh) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Narain integral transform and universal kernels in random matrix theory (M. Nowak) 17Optimal Leader-Follower Control for the Fractional Opinion Formation Model
(T. Odzijewicz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18First Passage Properties by Space-Fractional Diffusion Equation (A. Padash) . . 18Integer Ratios of Factorials as Hausdorff Moments Versus Algebraicity (K. A. Penson,
G. H. E. Duchamp, G. Koshevoy) . . . . . . . . . . . . . . . . . . . . . . . 18
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Quantum Motion on a Comb:an Example of a Fractional Schrodinger Equation (I. Petreska) . . . . . . . 19
Umbral Voigt Transforms and Special Functions (R. M. Pidatellaand S. Licciardi) 19On modified Bessel functions and the McKay Bessel distribution (T. K. Pogany) 20Diffusion-wave equation and diffusion waves (Y. Povstenko) . . . . . . . . . . . . 20Characteristic crossover between different diffusion regimes: tempered motions
and applications (T. Sandev) . . . . . . . . . . . . . . . . . . . . . . . . . . 21Fractional stable laws: a new kind of statistics in nanodynamics (R. T. Sibatovand
V. V. Uchaikin) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Brownian yet non-Gaussian diffusion in heterogeneous media: characterisation
and first passage statistics (V. Sposini) . . . . . . . . . . . . . . . . . . . . . 23Levy flights for migration problems: How to take the environment structure into
account? (T. Srokowski) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Bayesian statistics for diffusion models (S. Thapa) . . . . . . . . . . . . . . . . . 24
Author Index 25
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List of Participants andConference Schedule
Nicolas Behr, Institut de Recherche en Informatique Fondamentale (IRIF), UniversiteParis-Diderot (Paris 07), France;e-mail: [email protected]
Marek Bozejko, Polska Akademia Nauk, ul. Kopernika 18, 50-001 Wroc law, Poland;e-mail: [email protected]
C. Burdık, Department of Mathematics, Czech Technical University in Prague, Prague,Czech Republic;e-mail: [email protected]
A. V. Chechkin, Institute for Physics and Astronomy, University of Potsdam, Pots-dam, Germany; Akhiezer Institute for Theoretical Physics, National Science Center“Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;e-mail: [email protected]
B. Datsko, Rzeszow University of Technology, Rzeszow, Poland;e-mail: [email protected]
G. Dattoli, ENEA - Centro Ricerche Frascati, Frascati (Roma), Italy;e-mail: [email protected]
G. H. E. Duchamp, Universite Paris 13, Sorbonne Paris Cite, LIPN, CNRS UMR 7030,93430 Villetaneuse, France and Institut Henri Poincare - IHP, Paris, France;e-mail: [email protected]
B. Dybiec, Institute of Physics, Jagiellonian University, Krakow, Poland;e-mail: [email protected]
K. Gorska, Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland;e-mails: [email protected] , [email protected]
A. Hanyga ul. Bitwy Warszawskiej 1920 r 14/52 02-366 Warszawa, Poland;e-mail: [email protected]
R. Hilfer, Institut fur Computerphysik, Universitat Stuttgart, Stuttgart, Germany;e-mail: [email protected]
A. Horzela, Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland;e-mail: [email protected]
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R. Hou, School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000,China and Institute for Physics and Astronomy, University of Potsdam, Germany;e-mail: [email protected]
A. Kaminska - Tabor, Institute of Physical Chemistry Polish Academy of Sciences,Warsaw, Poland;e-mail: [email protected]
T. Koszto lowicz, Institute of Physics, Jan Kochanowski University, Kielce, Poland;e-mail: [email protected]
A. Lattanzi, Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland;e-mails: [email protected] , [email protected]
K. Lewandowska, Nicolaus Copernicus University in Torun, Faculty of Chemistry, Torun,Poland;e-mail: [email protected]
S. Licciardi, ENEA - Centro Ricerche Frascati, Frascati (Roma), Italy;e-mail: [email protected]
F. Mainardi, Department of Physics and Astronomy, University of Bologna, Bologna,Italy;e-mails: [email protected], [email protected]
A. Mandowski Institute of Physics, Jan Dlugosz University Czestochowaa, Poland;e-mail: [email protected]
E. Mandowska Institute of Physics, Jan Dlugosz University, Czestochowa, Poland;e-mail: [email protected]
R. Metzler, Chair for Theoretical Physics, Institute for Physics and Astronomy, Univer-sity of Potsdam, Potsdam-Golm, Germany;e-mail: [email protected]
H. N. Minh, Lille II University, 59024 Lille, France;e-mails: [email protected] , [email protected]
M. Nowak, M. Smoluchowski Institute of Physics and Mark Kac Complex Systems Re-search Centre, Jagiellonian University, Krakow, Poland;e-mail: [email protected]
T. Odzijewicz, , Department of Mathematics and Mathematical Economics, WarsawSchool of Economics Poland;e-mail: [email protected]
A. Padash, Faculty of Environment, University of Tehran, Tehran, Iran;e-mail: [email protected]
K. A. Penson, Sorbonne Universite, Univ. Paris VI, LPTMC, France;e-mail: [email protected]
I. Petreska, Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss Cyriland Methodius University, Skopje, Macedonia;e-mail: [email protected]
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R. M. Pidatella, Universita degli Studi di Catania, Catania, Italy;e-mail: [email protected]
T. K. Pogany, Faculty of Maritime Studies, University of Rijeka, Croatia; Institute ofApplied Mathematics, Obuda University, Budapest, Hungary;e-mail: [email protected]
Y. Povstenko, National Academy of Sciences of Ukraine, Lviv, Ukraine Institute ofMathematics and Computer Science, Faculty of Mathematics and Natural Sciences,Jan D lugosz University in Czestochowa, Poland;e-mail: [email protected]
T. Sandev, Radiation Safety Directorate, Skopje, Macedonia; Institute of Physics, Fac-ulty of Natural Sciences and Mathematics, Ss Cyril and Methodius University,Skopje, Macedonia; Research Center for Computer Science and Information Tech-nologies, Macedonian Academy of Sciences and Arts, Skopje, Macedonia;e-mail: [email protected]
R. T. Sibatov, Ulyanovsk State University, Ulyanovsk, Russia;e-mail: ren [email protected]
V. Sposini, Institute for Physics and Astronomy, University of Potsdam, Potsdam–Golm,Germany; BCAM–Basque Center for Applied Mathematics, Bilbao, Basque Coun-try, Spain;e-mail: [email protected]
T. Srokowski, Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland;e-mail: [email protected]
S. Thapa, Institute for Physics and Astronomy, University of Potsdam, Germany andMEMPHYS, Department of Physics, Chemistry and Pharmacy, University of South-ern Denmark, Denmark;e-mail: [email protected]
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Organizing committee
Katarzyna GorskaH.Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciencesul.Eliasza - Radzikowskiego 152, 31342 Krakow, Polandtel.+48 12 662 81 61,e-mail: [email protected], [email protected];mob: +48 517 684 930Andrzej HorzelaH.Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciencesul.Eliasza - Radzikowskiego 152, 31342 Krakow, Polande-mail: [email protected]: +48 605071814Ambra LattanziH.Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciencesul.Eliasza - Radzikowskiego 152, 31342 Krakow, Polande-mail: [email protected], [email protected];mob: +39 3318956546
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Conference Dinner
The Conference Dinner will take place the evening of Tuesday 13th of July at the SzaraGes Resturant. Take the opportunity to relax and enjoy meeting with colleagues outsidethe congress program. It starts at 18:00.
How to get the resturant from the conference place:
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Abstracts
Nevanlinna-Pick functions coming from the free probability
M. Bozejkoa
a Polska Akademia Nauk, ul. Kopernika 18, 50-001 Wroc law, Poland
We will present new class of Nevanlinna-Pick functions coming from free infinitely divisiblelaws (FID) in free additive convolution of probability measures on real line. Also Bercovici-Pata bijection between classical IF and (FID) will be done with many interesting examples.We will follow our papers with T.Hasebe- On free infinitely divisibility for classical and freeMeixner distribution , Probability and Math.Statistics,33(2),363-375. and the paper withW.Bryc,On a class of free Levy laws related to a regression problem, J.Func.Anal..236,59-77.
Brownian yet Non-Gaussian: Diffusing Diffusivity versus HeterogeneousModels
A. Chechkina
a Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
We discuss the situations under which Brownian yet non-Gaussian (BnG) diffusion can beobserved in the model of a particle’s motion in a random landscape of diffusion coeffcientsslowly varying in space. Our conclusion is that such behavior is extremely unlikely in thesituations when the particles, introduced into the system at random at t = 0, are observedfrom the preparation of the system on. However, it indeed may arise in the case whenthe diffusion (as described in Ito interpretation) is observed under equilibrated conditions.This paradigmatic situation can be translated into the model of the diffusion coeffcientfluctuating in time along a trajectory, i.e. into a kind of the ”diffusing diffusivity” model.
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Analysis of complex dynamics in nonlinear reaction-diffusion systemswith fractional derivatives on basis of a combination of operational and
numerical methods
B. Datskoa
a Rzeszow University of Technology, 8 Powstancow Warszawy St., 35-959 Rzeszow, Poland
We analyze conditions for different types of instabilities and complex dynamics that occurin nonlinear two-component activator-inhibitor fractional reaction-diffusion systems. It isshown that the stability of steady-state solutions and their possible evolution are mainlydetermined by the eigenvalue spectrum of a linearized system and the fractional derivativeorder. An approach based on Laplace-Fourier transform methods which are also conve-nient for finding space-time instability conditions is considered. The results of the linearstability analysis are confirmed by computer simulations of the basic two-component frac-tional reaction-diffusion mathematical models. On the basis of these models, it is demon-strated that the conditions of instability and the pattern formation dynamics in fractionalactivator-inhibitor systems are more complex than in the standard ones. As a result, aricher and a more complicated spatiotemporal dynamics take place. A common pictureof possible nonlinear solutions for nonlinear fractional incommensurate two-componentsystems, which can appear in supercritical and subcritical domains, is presented. Somegeneral properties of fractional auto-oscillation and auto-wave systems are established.
One parameter fractional and real groups of operators and theircombinatorics
G. H. E. Duchamp a,b
a Universite Paris 13, Sorbonne Paris Cite, LIPN, CNRS UMR 7030, 93430 Villetaneuse, Franceb Institut Henri Poincare - IHP, Paris (France)
We start from the problem of square roots anf fractional powers as stated in Hilbertfifth problem and describe one-parameter groups of [1] and their matrices. We end withSchutzenberger’s calculus [2, 3] and the description of one (and multi-) parameter groupsof characters involved in Dyson-like solutions of evolution equations and factorizations ofthose with Drinfel’d asymptotic condition [4]. Effective algorithms are also provided.
[1] G. Duchamp, K.A. Penson, A.I. Solomon, A. Horzela and P. Blasiak, One-ParameterGroups and Combinatorial Physics, Proceedings of the Symposium COPROMAPH3 :Contemporary Problems in Mathematical Physics, Cotonou, Benin, Scientific World Pub-lishing (2004). arXiv: quant-ph/0401126[1] G. Duchamp, K.A. Penson, A.I. Solomon, A. Horzela and P. Blasiak, One-ParameterGroups and Combinatorial Physics, Proceedings of the Symposium COPROMAPH3 :Contemporary Problems in Mathematical Physics, Cotonou, Benin, Scientific World Pub-lishing (2004). arXiv: quant-ph/0401126[2] G. H. E. Duchamp and C. Tollu, Sweedler’s duals and Schutzenberger’s calculus, In K.Ebrahimi-Fard, M. Marcolli and W. van Suijlekom (eds), Combinatorics and Physics, p. 67- 78, Amer. Math. Soc. (Contemporary Mathematics, vol. 539), 2011. arXiv:0712.0125v3[3] The algebra of Kleene stars of the plane and polylogarithms, Ngoc Hoang (LIPN),
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Gerard Duchamp (LIPN), Hoang Ngoc Minh (LIPN), arXiv:1602.02801v2[4] M. Deneufchatel, G. H. E. Duchamp, Hoang Ngoc Minh, A. I. Solomon, Indepen-dence of hyperlogarithms over function fields via algebraic combinatorics, Lecture Notesin Computer Science (2011), Volume 6742 (2011), 127-139.
Conservative random walks in confining potentials: tails versus centralparts
B. Dybieca
a Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center,Jagiellonian University, ul. S. Lojasiewicza 11, 30-348 Krakow, Poland
Levy walks are continuous time random walks with spatio-temporal coupling of jumplengths and waiting times. In the simplest version Levy walks move with a finite speed.Here, we present an extension of the Levy walk scenario for the case when external forcefields influence the motion. The resulting motion is a combination of the response to thedeterministic force acting on the particle, changing its velocity according to the principleof total energy conservation, and random velocity reversals governed by the distribution ofwaiting times. Due to the fact that the considered motion is conservative it is fundamen-tally different from thermal motion in the same external potentials. We present resultsfor the velocity and position distributions for single well potentials of different steepness.Finally, we discuss the role played by tails and central part of waiting time distributions.
[1] B. Dybiec, K. Capa la, A. Chechkin and R. Metzler, Conservative random walks inconfining potentials, arXiv preprint arXiv:1804.09166 (2018).
Some convolution equations with completely monotone kernels
A. Hanygaa,b
a
b
T.B.A.
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Composite Continuous Time Random Walks
R. Hilfera
a Institute for Computational Physics (ICP), Universitat Stuttgart, Allmandring 3, 70569Stuttgart, Germany
Random walks in composite continuous time have been introduced in [1]. Composite timeflow is the product of translational time flow and fractional time flow [2] and arises natu-rally from a two-scale limit. The continuum limit of composite continuous time randomwalks gives a diffusion equation where the infinitesimal generator of time flow is the sumof a first order and a fractional time derivative. The latter is specified as a generalizedRiemann-Liouville derivative. Generalized and binomial Mittag-Leffler functions are foundas the exact results for waiting time density and mean square displacement.
[1] R. Hilfer, Composite continuous time random walks,The European Physical Journal B90.12 (2017): 233[2] Chem. Phys. 84, 399 (2002).
Biased continuous-time random walks for ordinary and equilibriumcases: facilitation of diffusion, ergodicity breaking and ageing
R. Houa,b, Andrey G. Cherstvyb, Ralf Metzlerb and Takuma Akimotoc
a School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Chinab Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
c Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
We examine renewal processes with power-law waiting time distributions (WTDs) andnon-zero drift via computing analytically and by computer simulations their ensemble andtime averaged spreading characteristics. All possible values of the scaling exponent α areconsidered for the WTD ψ(t) ∼ 1
t1+α . We treat continuous-time random walks (CTRWs)with 0 < α < 1 for which the mean waiting time diverges, and investigate the behaviourof the process for both ordinary and equilibrium CTRWs for 1 < α < 2 and α > 2. Wedemonstrate that in the presence of a drift CTRWs with α < 1 are ageing and non-ergodicin the sense of the non-equivalence of their ensemble and time averaged displacement char-acteristics in the limit of lag times much shorter than the trajectory length. In the senseof the equivalence of ensemble and time averages, CTRW processes with 1 < α < 2 areergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW re-newal processes with α > 2—both for the equilibrium and ordinary situation—are alwaysergodic. For the situations 1 < α < 2 and α > 2 the variance of the diffusion process,however, depends on the initial ensemble. For biased CTRWs with α > 1 we also inves-tigate the behaviour of the ergodicity breaking parameter. In addition, we demonstratethat for biased CTRWs the Einstein relation is valid on the level of the ensemble and timeaveraged displacements, in the entire range of the WTD exponent α.
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Figure 1: Schematic representation of asymmetric particle jumps for biased CTRWs, withsome parameters indicated.
Levy walk with position dependent resting time
A. Kaminskaa and T. Srokowskib
a Laboratory of Geomatics, Forest Research Institute, Sekocin Stary, 3 Braci Lesnej Street,05-090 Raszyn, Poland
b nstitute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland
The Levy walk process with rests is considered. The waiting time between the jumpsis given by an exponential distribution with a position-dependent jumping rate (scaleparameter). This corresponds to the Levy walk in environment with inhomogeneouslydistributed traps. The time of flight for both ranges of α: lower (0, 1) and higher (1, 2), isdiscussed. The transition probability is constructed and the master equation is derived.This equation is solved for a special form of the jumping rate. The characteristics ofsuch defined process are explored: in particular, the diffusion properties and asymptoticbehavior of density distribution is found. It is demonstrated that presented Levy walkmodel possesses properties which qualitatively agree with features of human and animalmovements.
Subdiffusion-absorption process in a layered system
T. Koszto lowicza
a Institute of Physics, Jan Kochanowski University, ul. Swietokrzyska 15, 25-406 Kielce, Poland
We consider subdiffusion and normal diffusion with possible absorption in a three-partone-dimensional system. The process is described by the subdiffusion-absorption equa-tions with fractional Riemann-Liouville time derivative. In each part of the system sub-diffusion and absorption parameters can be different. At the borders between media thenew boundary conditions are used. We use the model to describe subdiffusion of antibioticthrough a bacterial biofilm. It will be shown that the temporary evolution of the amountof substance that has diffused through the biofilm depends on the antibiotic absorptioncoefficient. Absorption intensity is related to the degree of defense of bacteria againstantibiotic treatment. We will show that the theoretical results coincide with the empiricalones.
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Umbral Fractional Calculus and Physical Applications
S. Licciardia,b
a Department of Mathematics and Computer Science, University of Catania,Viale A. Doria 6, 95125, Catania, Italy
b ENEA - Frascati Research Center, Via Enrico Fermi 45, 00044, Frascati, Rome, Italy
Umbral Methods are innovative techniques belonging to Operator Theory and SymbolicMethods. They use a “different” mathematical language aimed to recast and simplifyvarious expressions which can be found in pure Mathematics, as the evaluation of nontrivial PDE-ODE, specially in fractional dynamics, high order derivatives of products ofspecial functions, Number Theory, Combinatorics and so on. During the past years, theyhave been widely developed to deal with different problems in applied math too, in almostevery different fields of application as Optics, Quantum Physics, Biology, Termodynamic,Electromagnetism, ... We will introduce the umbral technique and will show the flexi-bility of the methods through two important examples of applications: a time fractionalSchroedinger equation and the Free Electron Laser high gain equation.
On modifications of the exponential integral with the Mittag-Lefflerfunction
F. Mainardia and E. Masinaa
a Department of Physics and Astronomy University of Bologna and INFN Bologna Italy
In this paper we survey the properties of the Schelkunoff modification of the Exponentialintegral and we generalize it with the Mittag-Leffler function. So doing we get a new specialfunction (as far as we know) that may be relevant in linear viscoelasticity because of itscomplete monotonicity properties in the time domain. We also consider the generalizedsine and cosine integral functions.
Long-lived luminescence I – measurements, material properties andapplications
E. Mandowskaa and A. Mandowskib
a Institute of Physics, Jan Dlugosz University, ul. Armii Krajowej 13/15, 42-200 Czestochowa,Poland
Luminescence is the excess emission of light over and above the thermal emission back-ground. The emission of light is caused by the previous supply of energy, i.e. excitation.Characteristic lifetimes vary several orders of magnitude. Very long-lived luminescencecan occur from 0.1 ms to many years after excitation. It relates to several well-knownphysical phenomena – e.g. phosphorescence (PH), thermoluminescence (TL) and opti-cally stimulated luminescence OSL. PH, TL and OSL can be observed only in insulatorsand wide band-gap semiconductors.TL and OSL are two-stage luminescence phenomena. The excitation is usually done byhigh energy ionizing radiation. Irradiated samples can be stored for a long time – eventhousands of years. Luminescence is triggered by thermal (TL) or optical (OSL) stimula-tion. We observe TL while heating the sample usually with the constant rate. In OSL thesample is illuminated by a strong monochromatic light. The light emission is observed at
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shorter wavelengths than the stimulation wavelength.In most cases the emitted luminescence (TL and OSL) is proportional to the dose of ra-diation absorbed by the material. Hence, TL and OSL are frequently used in dosimetryof ionizing radiation. TL and OSL materials are passive detectors not requiring a powersupply. This feature has made them very popular in personal as well as environmentaldosimetry (Mandowska et. al., 2010). However to recover the stored information it isnecessary to use a special TL or OSL reader.The ability to store the information on absorbed radiation dose for thousands of years isused for determination the age of archeological and geological objects (Aitken, 1985). Forthis purpose various natural minerals are used – e.g. quartz, feldspars and halite. TLor OSL readout allows to determine the absorbed dose of natural radiation from the lastmoment of zeroing the signal – e.g. the date of firing the pottery.Standard TL and OSL measurements are performed using a photomultiplier in the photoncounting mode. It does not provide an information related to spectral distribution of theluminescence. Spectrally resolved measurements are very difficult as the luminescence isvery week. Nonetheless, it gives insight into fundamental trapping and recombinationphenomena (Mandowska et al., 2017a, Mandowska et al., 2017b).
[1] M. J. Aitken, and V. Mejdahl. Thermoluminescence dating, Vol. 359. London: Aca-demic press (1985)[2] E. Mandowska, et al. Spectrally resolved thermoluminescence of highly irradiated LiF:Mg, Cu, P detectors, Radiation Measurements 45.3-6 (2010): 579-582[3] E. Mandowska, R. Majgier, and A. Mandowski, Spectrally resolved thermoluminescenceof pure potassium chloride crystals., Applied Radiation and Isotopes 129 (2017): 171-179[4] E. Mandowska, Characteristic features of spectrally resolved luminescence in crystallinephosphors, Journal of Luminescence 188 (2017): 313-318.
Long-lived luminescence II – development of theoretical models
A. Mandowskia and E. Mandowskaa
a Institute of Physics, Jan Dlugosz University, ul. Armii Krajowej 13/15, 42-200 Czestochowa,Poland
Most applications of long-lived luminescence is based on two phenomena – thermolumi-nescence (TL) and optically stimulated luminescence (OSL). Early theoretical models, in30’s of the last century, were developed for the phosphorescence (PH) and TL. OSL fun-damentals were developed and applied since late 80’s. These phenomena occur only inhigh resistivity solids, i.e. insulators or wide band gap semiconductors. Each crystallineinsulator contains a number of imperfections related to lattice defects and/or dopantswhich produce additional energy states within the band gap. These states may act aselectron/hole traps and recombination centers (RCs). These localized states are responsi-ble for very long lifetimes of luminescence. To detect PH, TL or OSL the material understudy has to be irradiated. High energy ionizing radiation produces band-to-band tran-sitions filling empty traps and RCs. These metastable excited states may last for manyyears. Luminescence is triggered by thermal or optical stimulation. The stimulation re-leases trapped charge carriers and allows subsequent recombination. In most cases therecombination is radiative producing emission of light in visible or UV region. Theoretical
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models relate to all stages of these phenomena, i.e. excitation, storage and stimulation.First models were based on the concept of delocalized transitions via transport bands(i.e. conduction and/or valence band) between traps and recombination centers. This isso called the simple trap model (STM). Approximate solutions under quasi-equilibriumconditions were found by Randall and Wilkins (1945) and Garlick and Gibson (1948) forthe case of week and strong retrapping, respectively. A different concept of localized tran-sitions (LT) was developed by Halperin and Braner (1960) and Land (1969) under thesame quasi-equilibrium approximation. In all cases analytical solutions of the systems ofnonlinear equations were not possible. Various phenomenological models and approximatesolutions were proposed in 70’s and 80’s for a more general description of the phenomena.These models were based mainly on the kinetic theory of chemical reactions and turned outto be inadequate for description of long lived luminescence mechanisms. In recent years,the theory was developed to include trapping and recombination phenomena in clustersystems (Mandowski and Swiatek, 1992; Mandowski, 2006) as well as the analytical for-mulation of trapping and recombination kinetics in semi-localized systems (Mandowski,2005). These models relate to the most important case of crystalline insulators commonlyused in dosimetry. Other models relate to disordered solids with randomly distributedtraps and RCs. This type of recombination can be found e.g. in feldspars, i.e. mineralsoften used for dating (Jain et al. 2012; Pagonis et al., 2017).
[1] G. F. J. Garlick and A. F. Gibson, The electron trap mechanism of luminescence insulphide and silicate phosphors, Proceedings of the physical society 60.6 (1948): 574-590[2] A. Halperin, and A. A. Braner, Evaluation of thermal activation energies from glowcurves, Physical Review 117.2 (1960): 408-415[3] M. Jain, G. Benny and M. T. Andersen, Stimulated luminescence emission from local-ized recombination in randomly distributed defects, Journal of physics: Condensed matter24.38 (2012): 385402 [12pp][4] P. L. Land, New methods for determining electron trap parameters from thermolumi-nescent or conductivity ‘glow curves’, Journal of Physics and Chemistry of Solids 30.7(1969): 1681-1692[5] A. Mandowski, Semi-localized transitions model for thermoluminescence, Journal ofPhysics D: Applied Physics 38.1 (2005): 17-21[6] A. Mandowski, Topology-dependent thermoluminescence kinetics, Radiation protectiondosimetry 119.1-4 (2006): 23-28[7] Mandowski, A., and J. Swialtek, Monte Carlo simulation of thermally stimulated re-laxation kinetics of carrier trapping in microcrystalline and two-dimensional solids, Philo-sophical Magazine B 65.4 (1992): 729-732[8] V. Pagonis, et al., An overview of recent developments in luminescence models with afocus on localized transitions, Radiation Measurements 106 (2017): 3-12[9] J. T. Randall and M. H. F. Wilkins, Phosphorescence and electron traps-I. The studyof trap distributions, Proc. R. Soc. Lond. A 184.999 (1945): 365-389.
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Non-ergodicity and ageing in single particle trajectories
R. Metzlera
a Institute of Physics and Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany
In 1905 Einstein formulated the laws of diffusion, and in 1908 Perrin published his Nobel-prize winning studies determining Avogadro’s number from diffusion measurements. Withsimilar, more refined techniques the diffusion behaviour of submicron tracer particles oreven single molecules is now routinely measured in the complex environment of livingbiological cells and similarly complex systems. It is frequently observed that the passivediffusion of such particles deviates from Einstein’s laws.
This talk will discuss the basic mechanisms leading to such anomalous diffusion as wellas point out its physical consequences. In particular, it will be discussed how we canunderstand the experimental observation of weakly non-ergodic behaviour (time and en-semble averages of physical observables behave differently) and ageing (physical observ-ables depend on the time span between initial preparation of the system and start of themeasurement).
A family of Eulerian functions involved in regularization of divergentpolyzetas
H. N. Minh a
a Lille II University, 59024 Lille, France
Eulerian functions are most significant for analytic number-theory and are largely involvedin Probably and in Physical sciences (Gamma and Beta densities). In this work, we givean extension of these functions and their relationship with the several parameter zetafunction. In particular, starting with the Weierstrass factorization (and the Newton-Girard identity) for Gamma function, we are interested in the ratio of ζ(2k)/π2k and wewill obtain an analogue situation and draw some consequences about a structure of thealgebra of polyzetas. This will be done via the combinatorics of noncommutative rationalpower series.
Narain integral transform and universal kernels in random matrixtheory
M. Nowaka
a Mark Kac Complex Systems Research Center, Jagiellonian University, 30-348 Cracow, Lojasiewicza 11, Poland
Using the ideas of spectral projection suggested by Olshanski and Borodin and advocatedby Tao we derive the spectral properties of the complex Wishart ensembles; first, theMarcenko-Pastur distribution interpreted as a Bohr-Sommerfeld quantization conditionfor the hydrogen atom; second, hard (Bessel), soft (Airy) and bulk (sine) kernels fromproperly rescaled radial Schroedinger equation for the hydrogen atom. Then we suggestan extension of the ideas of spectral projections into the biorthogonal ensembles formedby the squared singular values of the product of Wishart matrices and Muttalib-Borodinmatrix ensembles. We demonstrate that Narain integral transform leads in both abovecases to painless reconstruction of universal kernels.
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Optimal Leader-Follower Control for the Fractional Opinion FormationModel
T. Odzijewicza
a Department of Mathematics and Mathematical EconomicsWarsaw School of Economics inWarsaw, Poland
This work deals with an opinion formation model, that obeys a nonlinear system offractional-order differential equations. We introduce a virtual leader in order to attaina consensus. Sufficient conditions are established to ensure that the opinions of all agentsglobally asymptotically approach the opinion of the leader. We also address the problemof designing optimal control strategies for the leader so that the followers tend to con-sensus in the most efficient way. A variational integrator scheme is applied to solve theleader-follower optimal control problem. Finally, in order to verify the theoretical analysis,several particular examples are presented.
First Passage Properties by Space-Fractional Diffusion Equation
A. Padasha
a Faculty of Environment, University of Tehran, Tehran, Iran
We study the first passage properties of the Levy Flights (LFs) in a bounded domainfor symmetric and asymmetric jump length distributions. By solving the space fractionaldiffusion equation for the probability density function of the LFs the survival probabilityis investigated for different values of the Levy index alpha and the skewness parameterbeta. Also, by using the Skorokhod theorem for processes with independent incrementsit is demonstrated that the numerical results are in a good agreement with the analyticalexpressions for the probability density function of the first passage time for symmetric andasymmetric LFs as well.
Integer Ratios of Factorials as Hausdorff Moments Versus Algebraicity
K. A. Pensona, G. H. E. Duchampb, G. Koshevoyc
a Sorbonne Universite, Univ. Paris VI, LPTMC, Franceb Universite Paris 13, Sorbonne Paris Cite, LIPN, CNRS UMR 7030, 93430 Villetaneuse, France,
c Interdisciplinary Scientific Center J.-V. Poncelet (ISCP)
Consider two series of positive integers: a = a1, a2, ...aK , and b = b1, b2...bK , b(K+1), with∑(ai, i = 1..K) =
∑(bi, i = 1..K + 1),K = 1, 2, ... . We form the following ratios of
factorials:
un(a, b) = ((a1)∗n)!∗((a2)∗n)!...∗((aK)∗n)!/[((b1)∗n)!∗((b2)∗n)!...∗((bK+1)∗n)!], n = 0, 1, ....(1)
It turns out that for many choices of a and b the ratios un(a, b) in (1) are themselvesintegers. In these cases we conceive un(a, b) as nth moments of the weight functionsW (a, b, x) in the Hausdorff moment problem un(a, b) = Integral(xn ∗ W (a, b, x), x =0..R(a, b)), where R(a, b) is the upper edge of the support (0, R(a, b)). We solve exactly andexplicitly the above Hausdorff moment problem via the inverse Mellin transform methodthus providing the analytic forms of R(a, b) as well as of W (a, b, x) in terms of Meijer
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G-functions and generalized hypergeometric functions. We prove formally the positivityof the weights W (a, b, x) which are all U-shaped and singular at both edges of the support;as such they are generalizations of the arcsin distributions. We discuss a potential linkbetween the proven algebraicity of the ordinary generating functions of un(a, b) and apossible algebraicity of corresponding weights W (a, b, x).
Quantum Motion on a Comb:an Example of a Fractional Schrodinger Equation
I. Petreskaa
a Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss Cyril and MethodiusUniversity, P.O. Box 162, 1001 Skopje, Macedonia
This work is focused on quantum systems modeling, giving another physical example wherefractional calculus emerges. In particular, we consider a time-dependent Schrodinger equa-tion on comb-like structures, investigating both, two- and three- dimensional cases. Ge-ometric constraints are achieved by introducing Dirac delta term in the kinetic energyoperator. We show how a fractional Schrodinger equation can be derived by projection ofthe three- and two-dimensional comb dynamics in the two- and one-dimensional configu-ration space, respectively. Closed-form solutions for the reduced probability density areobtained in terms of Fox H-functions, employing the Green’s function approach. Comb-like structures have been already successfully implemented to model anomalous diffusionbehavior in low-dimensional percolation clusters. The present work extends the area ofapplication of fractional dynamics models further, going beyond the classical picture andgrasping also the quantum-mechanical features of the particles.
[1] T. Sandev, I. Petreska and E. K. Lenzi, Generalized time-dependent Schrodinger equa-tion in two dimensions under constraints, J. Math. Phys. 59, 012104 (2018)[2] I. Petreska, A. S. M. de Castro, T. Sandev and E. K. Lenzi, Standard Schrodingerequation in three dimensions under geometric constraints, submitted (2018).
Umbral Voigt Transforms and Special Functions
R. M. Pidatellaa and S. Licciardia,b
a Universita degli Studi di Catania, Catania, Italyb ENEA - Frascati Research Center, Via Enrico Fermi 45, 00044, Frascati, Rome, Italy
One of the most appropriate tool to deal with analytical and numerical solutions of linearand non linear ordinary and partial differential equations are the Special Functions.In this work, it is illustrated how the used techniques, based on umbral operational calcu-lus, provide a straightforward derivation of the relevant properties and the generalizationsto negative and higher order functions. We propose further extension of the method andof the relevant concepts as well and obtain new families of integral transforms.
[1] M. Artioli, G. Dattoli, S. Licciardi, R. M. Pidatella, Hermite and Laguerre Functions:
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a unifying point of view, in preparation[2] D. Babusci, G. Dattoli, M. Del Franco, S. Licciardi, Mathematical Methods for Physics,invited Monograph by World Scientific, Singapore (2017), in press[3] S. Licciardi: PhD Thesis, Umbral Calculus, a Different Mathematical Language, Math-ematics and Computer Sciences, Dep. of Mathematics and Computer Sciences, XXIXcycle, University of Catania, arXiv:1803.03108 [math.CA] (2018).
On modified Bessel functions and the McKay Bessel distribution
T. K. Poganya,b
a Faculty od Maritime Studies, University of Rijeka, Studentska 2, 51000 Rijeka, Croatiab Applied Mathematics Institute, Obuda University, 1034 Budapest, Hungary
Motivated by recent results reported in [1] in which the authors introduced a new for-mula for cumulative distribution function FK of the McKay Bessel function distributioncontaining the modified Bessel function of the second kind Kν we aim to present newformulae for FK especially for certain values of parameters for which their formula failsto be calculated by in-built numerical routines. Next, a summation formula for the finiteNeumann series built by Kν is established as a by–product of our results. Finally, twocomputational series formulae are obtained for the CDF of McKay Bessel distributionwhich contains modified Bessel function of the first kind Iν ; one is given in terms of theHorn confluent hypergeometric function Φ2, and the other by incomplete Gamma functionand Exton’s double hypergeometric X–function.
[1] S. Nadarajah, H. M. Srivastava, A. K. Gupta. Skewed Bessel function distributionswith application to rainfall data, Statistics 41(4) 333–344 (2007).
Diffusion-wave equation and diffusion waves
Y. Povstenkoa
a Institute of Mathematics and Computer Science, Jan D lugosz University in Czestochowa, al.Armii Krajowej 13/15, 42-200 Czestochowa, Poland
The time-nonlocal generalization of the classical Fourier law with the “long-tail” powerkernel can be interpreted in terms of fractional integrals and derivatives and results in thetime-fractional diffusion-wave equation with the Caputo derivative of the order 0 < α ≤ 2.This equation interpolates the standard parabolic diffusion equation (α = 1) and thehyperbolic wave equation (α = 2), which substantiates the term “diffusion-wave equation”.Starting from the pioneering papers [1]–[3], considerable interest has been shown in findingsolutions to time-fractional diffusion-wave equation. The book [4] sums up investigations inthis field. It should be emphasized that the term “diffusion-wave”is also used in anothercontext. Angstrom [5] was the first to consider the classical diffusion equation underharmonic (wave) impact. In that case, the terms “oscillatory diffusion” and “diffusionwaves” are used [6]. We discuss the time-fractional diffusion-wave equation under harmonicimpact.
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[1] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation,Appl. Math. Lett. 9, 23–28 (1996)[2] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena,Chaos, Solitons Fractals 7, 1461–1477 (1996)[3] A. Hanyga, Multidimensional solutions of the time-fractional diffusion-wave equations,Proc. Roy. Soc. London A, 458, 933–957 (2002)[4] Y. Povstenko. Linear Fractional Diffusion-Wave Equation for Scientists and Engineers,Birkhauser, New York (2015)[5] A.J. Angstrom, Neue Methode, das Warmeleitungsvermogen der Korper zu bestimmen,Ann. Phys. Chem. 144, 513–530 (1861)[6] A. Mandelis, Diffusion-Wave Fields, Springer, New York (2001).
Characteristic crossover between different diffusion regimes: temperedmotions and applications
T. Sandeva,b,c
a Radiation Safety Directorate, Skopje, Macedoniab Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss Cyril and Methodius
University, Skopje, Macedoniac Research Center for Computer Science and Information Technologies, Macedonian Academy of
Sciences and Arts, Skopje, Macedonia
Characteristic crossover between different diffusion regimes (anomalous and normal) hasbeen observed in many complex systems. To describe this crossover dynamics we employthe so-called tempered fractional calculus approach. We use the tempered versions of thecontinuous time random walk (CTRW) model, generalized Langevin equation (GLE), andfractional Brownian motion (FBM). In many biological systems the observed anomalousdiffusion is either of CTRW type with trapping, or FBM and GLE motion type withpower-law correlations of the driving noise. At sufficiently long times, this anomalousdiffusion turns to normal, when the system’s temporal evolution exceeds some correla-tion time. I will show that these models may give same or similar results for the meansquare displacement, however, the corresponding processes described by them are totallydifferent. Excellent agreement of the theory with the results obtained by simulations andexperiments will be demonstrated.
[1] D. Molina-Garcia, T. Sandev, H. Safdari, G. Pagnini, A. Chechkin and R. Metzler,Crossover from anomalous to normal diffusion: truncated power-law noise correlationsand applications to dynamics in lipid bilayers, New J. Phys. 20 (2018): 103027[2] T. Sandev, W. Deng and P. Xu, Models for characterizing the transition among anoma-lous diffusions with different diffusion exponents, J. Phys. A: Math. Theor. 51 (2018):405002[3] T. Sandev, R. Metzler and A. Chechkin, From continuous time random walks to thegeneralized diffusion equation, Fract. Calc. Appl. Anal. 21 (2018): 10[4] T. Sandev, I. M. Sokolov, R. Metzler and A. Chechkin, Beyond monofractional kinetics,Chaos Solitons & Fractals 102 (2017):210[5] T. Sandev, Generalized Langevin equation and the Prabhakar derivative, Mathematics5.4 (2017): 66
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[6] A. Liemert, T. Sandev and H. Kantz, Generalized Langevin equation with temperedmemory kernel, Physica A 466 (2017): 356[7] T. Sandev, A. Chechkin, H. Kantz and R. Metzler, Diffusion and Fokker-Planck-Smoluchowski equations with generalized memory kernel, Fract. Calc. Appl. Anal. 18(2015) 1006.
Fractional stable laws: a new kind of statistics in nanodynamics
R. T. Sibatova,b and V. V. Uchaikina,b
a Ulyanovsk State Universityb Laboratory of Diffusion Processes
A new kind of statistics based on the fractional generalization of alpha-stable Levy lawsis expounded. These distributions appeared in [1] was partially investigated in work [2]where the class was named fractional stable distributions (FSDs). Fractional stable lawsplay an important role in the fractional kinetics of disordered media [3, 4]. In the presenttalk, FSDs are considered as limit distributions for subordinated processes. Three aspectsof the new statistics are described: self-similarity, generalized limit theorem, and fractionaldifferential equations. Fundamental solutions to classical problems of subordinated Levyflights, Levy walks without and with traps are expressed through FSD. New integralrepresentations and new Monte Carlo generation algorithms are proposed. New aspectsof applications to some nanophysical problems such as blinking quantum dot fluorescence,grain-boundary diffusion in nanostructured materials of Li-ion batteries and transmissionthrough the fractal quantum wire are discussed.The work is supported by the Russian Found. for Basic Res. (16-01-00556; 18-51-53018).
[1] M. Kotulski, Asymptotic distributions of continuous-time random walks: a probabilisticapproach, Journal of statistical physics 81.3-4 (1995): 777-792[2] V. Kolokoltsov, V. Korolev, and V. Uchaikin, Fractional stable distributions, Journalof Mathematical Sciences 105.6 (2001): 2569-2576[3] V. Uchaikin, V. Vasilevich, and R. Sibatov, Fractional kinetics in solids: anomalouscharge transport in semiconductors, dielectrics, and nanosystems, World Scientific (2013)[4] V. Uchaikin, V., Sibatov, R., Fractional Kinetics in Space, World Scientific (2018).
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Brownian yet non-Gaussian diffusion in heterogeneous media:characterisation and first passage statistics
V. Sposinia,b
a Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germanyb Basque Center for Applied Mathematics, 48009 Bilbao, Spain
A considerable number of systems have recently been reported in which Brownian yet non-Gaussian dynamics was observed. These are processes characterised by a linear growth intime of the mean squared displacement, yet the probability density function of the particledisplacement is distinctly non-Gaussian, and often of exponential (Laplace) shape. Thisbehaviour has been interpreted as resulting from diffusion in inhomogeneous environmentsand mathematically represented through a variable, stochastic diffusion coefficient. Indeeddifferent models describing a fluctuating diffusivity have been studied. In particular, wepropose the very generic class of the generalised Gamma distribution for the random dif-fusion coefficient. I will present two models for the particle spreading in such randomdiffusivity settings. The first belongs to the class of generalised grey Brownian motionwhile the second follows from the idea of diffusing diffusivities. The two processes exhibitsignificant characteristics which reproduce experimental results from different biologicaland physical systems. Finally, addressing the first passage problem for the two models,I will emphasize that even when the non-Gaussian character appears for certain regimesonly and in the tails of the distributions (thus with low probability), it may be essentialfor those systems in which rare events dominate triggered actions.
[1] V. Chechkin, F. Seno, R. Metzler, I. M. Sokolov, Brownian yet non-Gaussian diffu-sion: from superstatistics to subordination of diffusing diffusivities, Physical Review X 7.2(2017): 021002[2] V. Sposini, et al. Random diffusivity from stochastic equations: comparison of twomodels for Brownian yet non-Gaussian diffusion, New Journal of Physics 20.4 (2018):043044[3] V. Sposini, A. V. Chechkin, and R. Metzler, First passage statistics for diffusing diffu-sivity, arXiv preprint arXiv:1809.09186 (2018).
Levy flights for migration problems: How to take the environmentstructure into account?
T. Srokowskia
a Institute of Nuclear Physics, Polish Academy of Sciences, PL 31-342 Krakow, Poland
The environment structure may change a jumping-size distribution in the random walk,introducing a position dependence. Two cases can be considered: position before a jumpand after that. The second case corresponds to a natural behaviour of people and someanimals since they are usually focused on a target. The theory predicts qualitatively dif-ferent properties of both cases: in particular, the second case may be characterised bya finite variance. The problem is exactly solvable in the asymptotic limit for power-lawdependences; then all kinds of diffusion are observed. In the continuous limit, the corre-sponding Langevin equation possesses a multiplicative noise in the anti-Ito interpretation.
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Bayesian statistics for diffusion models
S. Thapaa,b
a Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germanyb MEMPHYS, Department of Physics, Chemistry and Pharmacy, University of Southern
Denmark, 5230 Odense M, Denmark
Particle diffusion in heterogeneous systems poses the following question: Can a singlemodel describe the entire dynamics of a particle in such complex biological/soft mattersystems? Isn’t it better, instead, to rank each possible model based on how well it explainsthe dynamics? I will talk about how this can be done within the Bayesian framework byassigning probabilities to each model. In particular, I will talk about the implementation ofthis powerful tool to compare - at the single trajectory level - models of Brownian motion,Fractional Brownian motion and Diffusing Diffusivity. Finally, I will discuss the results ofthe application of this method to experimental data of tracer diffusion in polymer-basedhydrogels.
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Author Index
Bozejko Marek, 9
Chechkin Aleksei , 9
Datsko Bohdan, 10Duchamp Gerard H. E. , 10Dybiec Bart lomiej , 11
Hanyga Andrzej , 11Hilfer Rudolf, 12Hoang Ngoc Minh , 17Hou Ru , 12
Kaminska - Tabor Agnieszka , 13Koszto lowicz Tadeusz , 13
Liccardi Silvia, 14
Mainardi Fransesco , 14Mandowska Ewa , 14
Mandowski Arkadiusz, 15Metzler Ralf, 17
Nowak Maciej, 17
Odzijewicz Tatiana , 18
Padash Amin , 18Penson Karol A. , 18Petreska Irina, 19Pidatella Rosa Maria , 19Pogany Tibor K. , 20Povstenko Yuri, 20
Sandev Trifce , 21Sibatov Renat T., 22Sposini Vittoria, 23Srokowski Tomasz, 23
Thapa Samdrajit, 24
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