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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Evolution of Thermal Diffusion Measurement by Statistical Mathematics

Discontinuity, Nonlinearity, and Complexity 11(2) (2022) 203--216 | DOI:10.5890/DNC.2022.06.002

J. Volkmann$^{1,2}$, N. Suedland$^3$, R. Rablbauer$^1$, A. Wollenberg$^1$, O. Schauerte$^1$, M. Prouvier$^1$, \newline A. Winkler$^1$, M. Frambourg$^1$, F. Klein$^3$, N. Migranov$^4$

$^1$ Group Innovation, Volkswagen AG, 38436 Wolfsburg, Germany

$^2$ International Laboratory of Theoretical and Mathematical Physics of Molecules and Crystals, Ufa Federal

Research Centre, Russian Academy of Sciences, Prospect Octyabrya, 71, Ufa 450054, Russia

$^3$ Aage GmbH, Roentgenstr. 24, 73431 Aalen, Germany

$^4$ Department of Medical Physics, Bashkir State Medical University, Lenin Str. 3, Ufa, 450008, Russia

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The variance theorem from the field of statistical analysis is used to evaluate the experimental data set of a classical heat diffusion experiment measuring temperatures as a function of time on a thermocouples prepared aluminium bar. The experimental data set during heating of the bar shows a nonlinear behavior. Known mathematical concepts to describe anomalous diffusion are shortly quoted and subsequent a novel approach of evaluating the data is presented. On the basis of the fundamental solution of the homogenous diffusion equation using finite boundaries defined by the discrete positions of the thermocouples, three different linearly independent momenta: expectation value, variance, and asymmetry are determined. For each momentum the parameter: thermal diffusivity $a$ of the solution equation is calculated. The three values are in the same range and coincide well to the known technical diffusivity of aluminium.


  1. [1]  Einstein, A. (1905), {\"U}ber die von der molekularkinetischen Theorie der W{\"a}rme geforderte Bewegung von in ruhenden Fl{\"u}ssigkeiten suspendierten Teilchen (About the movement of particles suspended in quiescent fluids required by the molecular-kinetic theory of heat), Ann. d. Phys., 322(8), 549-560.
  2. [2]  Smoluchowski, M.V. (1906), Zur kinetischen Theory der Brownschen Molekularbewegung und der Suspension (On the kinetic theory of Brownian motion and the suspension), Ann. d. Phys., 326(14), 756-780.
  3. [3]  Smoluchowski, M.V. (1916), {\"U}ber Brownsche Molekularbewegung unter Einwirkung {\"a}u{\ss}erer Kr{\"a}fte und deren Zusammenhang mit der verallgemeinerten Diffusionsgleichung (About Brownian motion under the influence of external forces and their relation to the generalized diffusion equation), Ann. d. Phys., 353(24), 1103-1112.
  4. [4]  Ibragimov, N.H. (2020), A Practical Course in Differential Equations and Mathematical Modelling, Higher Education Press / World Scientific Publishing, Beijing, Singapore 2010.
  5. [5]  S{\"u}dland, N. (2000), Fraktionale Diffusionsgleichungen und Foxsche H-Funktionen mit Beispielen aus der Physik (Fractional Diffusion Equations and Fox's H-Function with Examples of Physics), doctoral thesis, University of Ulm.
  6. [6]  Volkmann, J. and Suedland, N. (2019),The Variance Theorem for Finite Boundaries. Theory and Applications, WSEAS Transactions on Mathematics, 18, 394-406.
  7. [7]  Volkmann, J., Suedland, N., and Migranov, N. (2020), The Quality Gate and the Application of Momenta in Infinite Boundaries, WSEAS Transaction on Heat and Mass Transfer, 15, 138-150.
  8. [8]  Volkmann, J. and Suedland, N., The Variance Theorem and Real World Phenomena: The Heat Conduction Process, submitted to Chinese Physics B.
  9. [9]  Poznaniak, B., Kottowska, M., and Danielewicz-Ferchmin, I. (2003), Heat Conduction and Near-equilibrium Linear Regime, Revista Mexicana De Fisica, 49{(5)}, 477-481.
  10. [10]  Volkmann, J., Suedland, N., {{Transport Phenomena and the Variance Theorem for Infinite Boundaries}}, Proceedings of the Conference Differential Equations and Related Problems, Interdisciplinary Conference, Sterlitamak, 2018, 166 - 172
  11. [11]  Metzler, R., Chechkin, A.V., and Klafter, J., (2007), L{e}vy Statistics and Anomalous Transport: L{e}vy Flights and Subdiffusion, arXiv:0706.3553v1, $[$cond-mat. stat mech $]$.
  12. [12]  Bouchard, J.Ph. and Georges, A. (1990), Anormalous Diffusion in Disorderd Media: Statistical Mechanisms, Models and Physical Applications, Physics Reports, 195, 127-293.
  13. [13]  Klafter, J., Schlesinger, M.F., and Zumofen, G. (1996), Beyond Brownian Motion, Physics Today, 49, 33-39.
  14. [14]  Metzler, R. and Klafter, J. (2000), A Fractional Dynamics Approach, Physics Reports, 339, 1-77.
  15. [15]  Sokolov, I.M., Klafter, J., and Blumen, A. (2002), Fractional Kinetics, Physics Today, 55, 48-54.
  16. [16]  Zaslavsky, G.M. (2002), Chaos, Fractional Kinetics and Anomalous Transport, Physics Reports, 371, 461-580,
  17. [17]  Metzler, R. and Klafter, J. (2004), The Restaurant at the End of the Random Walk: Recent Developments in the Description of Anomalous Transport by Fractional Dynamics, Journal of Physics A: Mathematical and General, 37, R161-R208.
  18. [18]  Chechkin, A.V., Gonchar, V.Y., Klafter, J., and Metzler, R. (2006), Fundamental of Levy Flight Processes, Advances in Chemical Physics 133B, 439-496,
  19. [19]  West, B.J., Grigolini, P., Metzler, R., and Nonnenmacher, T.F. (1997) Fractionl Diffusion and Levy Stable Processes, Phys. Rev. E, 55, 99-106.
  20. [20]  S{\"u}dland, N., Volz, Chr., and Nonnenmacher, T.F. (2002), A Fractional Calculus Approach to Adsorbate Dynamics in Nanoporous Materials, Fractals in Biology and Medicine, Vol. III, Birkh{\"a}user, Basel, 325-332.
  21. [21]  Sokolov, I.M., Chechkin, A., and Klafter, Y. (2010), Generalized Diffusion Equations, Contribution of the 3rd Conference on Nonlinear Science and Complexity, Ankara.
  22. [22]  Schneider, W.R. and Wyss, W. (1989), Fractional Diffusion and Wave Equations, J. Math. Phys., 30(1), 134-144.
  23. [23]  Mathai, A.M., Saxena, R.K., and Hausbold, H.J. (2010), The H-function. Theory and Applications, Springer, New York.
  24. [24]  Kiryakova, V. (1994), Generalized fractional calculus and applications, Longman Scientific $\&$ Technical, Harlow.
  25. [25]  Samko, S.G., Kilbas, A.A., and Marichev, O.J. (1993), Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, S. A., Copyright by GPA (Amsterdam), P. V.
  26. [26]  Seshadri, V. and West, B.J. (1982), Fractal Dimensionality of Levy Processes, Proc. Natl. Acad. Sci., USA, 79, 4501.
  27. [27]  Sokolov, I.M., Schmidt, M.G.W., Schmidt-Martens, H., Froemberg, D., and Sagues, F. (2010), Theory of Reactions under Subdiffusion, Contribution of the 3rd Conference on Nonlinear Science and Complexity, Ankara.
  28. [28]  Wei, Q.H., Bechinger, C., and Leiderer, P. (2000), Single-file Diffusion of Colloids in Onedimensional Channels, Science, 287, 625-627.
  29. [29]  S{\"u}dland, N, Baumann, G., and Nonnenmacher, T.F. (1998), Who knows about the Aleph-Functions?, Fractional Calculus $\&$ Applied Analysis, 1(4), 401-402.
  30. [30]  Dixon, A.L. and Ferrar, W.L. (1936), A Class of Discontinuous Integrals, The Quarterly Journal of Mathematics, Oxford Series, 7, 81-96.
  31. [31]  Fox, Ch. (1961), The G and H Functions as Symmetrical Fourier Kernels, Transactions of the American Mathematical Society, 98, 395-429. %
  32. [32]  Braaksma, B.L.J. (1964), Asymptotic Expansions and Analytic Continuations for a Class of Barnesintegrals, Composition Mathematica, 15, 239-341.
  33. [33]  Oberhettinger, F. (1974), Tables of Mellin Transforms, Springer, Berlin.
  34. [34]  Poznaniak, B., Kottowska, M., and Danielewicz-Ferchmin, I. (2003), Heat Conduction and Near-equilibrium Linear Regime, Revista Mexicana De Fisica, 49(5), 477-481.
  35. [35]  Touloukian, Y.S., Powell, R.W., Ho, C.Y., and Nicolaou, M.C. (1974), Thermophysical Properties of Matter--TPRC Data Series, Thermal Diffusivity, 10, IFI Plenum.