Conciliating the nonadditive entropy approach and the fractional model formulation when describing subdiffusion

• Authors: Tadeusz Kosztołowicz , Katarzyna Lewandowska
• Languages of publication: EN

Abstracts

EN

We consider here two different models describing subdiffusion. One of them is derived from Continuous Time Random Walk formalism and utilizes a subdiffusion equation with a fractional time derivative. The second model is based on Sharma-Mittal nonadditive entropy formalism where the subdiffusive process is described by a nonlinear equation with ordinary derivatives. Using these two models we describe the process of a substance released from a thick membrane and we find functions which determine the time evolution of the amount of substance remaining inside this membrane. We then find ‘the agreement conditions’ under which these two models provide the same relation defining subdiffusion and give the same function characterizing the process of the released substance. These agreement conditions enable us to determine the relation between the parameters occuring in both models.

Keywords

EN

subdiffusion; nonadditive entropy; releasing substane; fractional calculus

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2012
• Volume: 10
• Issue: 3
• Pages: 645-651

Dates

published

1 - 6 - 2012

online

17 - 6 - 2012

Contributors

author

Tadeusz Kosztołowicz

• Institute of Physics, Jan Kochanowski University, ul. Świȩtokrzyska 15, 25-406, Kielce, Poland

author

Katarzyna Lewandowska

• Department of Radiological Informatics and Statistics, Medical University of Gdańsk, ul. Tuwima 15, 80-210, Gdańsk, Poland

References

• [1] J. P. Bouchaud, A. Georges, Phys. Rep. 195 127 (1990)
• [2] R. Metzler, J. Klafter, Phys. Rep. 339 1 (2000) http://dx.doi.org/10.1016/S0370-1573(00)00070-3[Crossref]
• [3] R. Metzler, J. Klafter, J. Phys. A 37 R161 (2007) http://dx.doi.org/10.1088/0305-4470/37/31/R01[Crossref]
• [4] T. D. Frank, Nonlinear Fokker-Planck Equations. Fundamental and Applications, (Springer, Berlin, 2005)
• [5] C. Tsallis, Introduction to Nonextensive Statistical Mechanics, (Springer, NY, 2009)
• [6] D. A. Stariolo, Phys. Lett. A 185, 262 (1994) http://dx.doi.org/10.1016/0375-9601(94)90613-0[Crossref]
• [7] A. R. Plastino, A. Plastino, Physica A 222, 347 (1995) http://dx.doi.org/10.1016/0378-4371(95)00211-1[Crossref]
• [8] A. Compte, D. Jou, J. Phys. A 29, 4321 (1996) http://dx.doi.org/10.1088/0305-4470/29/15/007[Crossref]
• [9] T. D. Frank, Physica A 310, 397 (2002) http://dx.doi.org/10.1016/S0378-4371(02)00821-X[Crossref]
• [10] C. Beck, Contemp. Phys. 50, 495 (2009) http://dx.doi.org/10.1080/00107510902823517[Crossref]
• [11] T. Kosztołowicz, K. D. Lewandowska, Acta Phys. Pol. B 43 1043 (2012) http://dx.doi.org/10.5506/APhysPolB.43.1043[Crossref]
• [12] T. Kosztołowicz, K. Dworecki, S. Mrówcznski, Phys. Rev. Lett. 94, 170602 (2005) http://dx.doi.org/10.1103/PhysRevLett.94.170602[Crossref]
• [13] T. Kosztołowicz, K. Dworecki, S. Mrówcznski, Phys. Rev. E 71, 041105 (2005) http://dx.doi.org/10.1103/PhysRevE.71.041105[Crossref]
• [14] T. Kosztołowicz, K. D. Lewandowska, Phys. Rev. E 78, 066103 (2008) http://dx.doi.org/10.1103/PhysRevE.78.066103[Crossref]
• [15] S. B. Yuste, L. Acedo, K. Lindenberg, Phys. Rev. E 69, 036126 (2004) http://dx.doi.org/10.1103/PhysRevE.69.036126[Crossref]
• [16] K. Dworecki, T. Kosztołowicz, K. D. Lewandowska, Acta Phys. Pol. B 39, 1221 (2008)
• [17] A. Klemm, R. Metzler, R. Kimmich, Phys. Rev. E 65, 021112 (2002) http://dx.doi.org/10.1103/PhysRevE.65.021112[Crossref]
• [18] T. Kosztołowicz, K. Dworecki, K. D. Lewandowska, arXiv:0809.1809v1 [cond-mat.soft]
• [19] I. Podlubny, Fractional differential equations, (Academic Press, San Diego, 1999)
• [20] T. Kosztołowicz, J. Phys. A 31, 1943 (1998) http://dx.doi.org/10.1088/0305-4470/31/8/007[Crossref]
• [21] T. D. Frank, Physica A 331, 391 (2004) http://dx.doi.org/10.1016/j.physa.2003.09.056[Crossref]
• [22] T. D. Frank, A. Daffertshofer, Physica A 285, 351 (2000) http://dx.doi.org/10.1016/S0378-4371(00)00178-3[Crossref]
• [23] T. Kosztołowicz, K. D. Lewandowska, Acta Phys. Pol. B 40, 1437 (2009)
• [24] T. Kosztołowicz, K. D. Lewandowska, arXiv:1008.1477 [cond-mat.stat-mech]

Identifiers

DOI

10.2478/s11534-012-0078-z