Abstract
Nonlinear equations generalizing the continuous-time random walk model to the case of finite concentrations have been derived. The equations take into account two factors responsible for the emergence of anomalous diffusion: nonlinearity and local equilibrium breaking. In locally equilibrium conditions, these equations are reduced to the nonlinear Fokker–Plank equation that can be interpreted as the transport equation for fermions with multiple energy levels. As a consequence of the nonlinear equations, two linear non-Markov equations with concentration-dependent memory functions have been obtained. One of these equations describes diffusion of a small deviation from the equilibrium state, while the other describes diffusion of tagged particles in the equilibrium system. It is shown that the emergence of anomalous diffusion is favored by low concentrations.
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REFERENCES
E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167 (1965).
H. Scher and M. Lax, Phys. Rev. B 7, 4491 (1973).
H. Scher and M. Lax, Phys. Rev. B 7, 4502 (1973).
H. Scher and E. W. Montroll, Phys. Rev. B 12, 2455 (1975).
J. P. Bouchaud and A. Georges, Phys. Rep. 195, 127 (1990).
R. Metzler and J. Klafter, J. Phys. A 37, R61 (2004).
R. Kutner and J. Masoliver, Eur. Phys. J. B 90, 50 (2017).
I. M. Sokolov, Soft Matter 8, 9043 (2012).
F. Höfling and T. Franosch, Rep. Prog. Phys. 76, 046602 (2013).
V. Uchaikin and R. Sibatov, Fractional Kinetics in Solids. Anomalous Transport in Semiconductors, Dielectric and Nanosystems (World Scientific, Singapore, 2013).
M. Bologna, C. Tsallis, and P. Grigolini, Phys. Rev. E 62, 2213 (2000).
D. Schertzer, M. Larcheveque, J. Duan, V. V. Yanovsky, and S. Lovejoy, J. Math. Phys. 42, 200 (2001).
C. Tsallis and E. K. Lenzi, Chem. Phys. 284, 341 (2002).
P. Straka and S. Fedotov, J. Theor. Biol. 366, 71 (2015).
P. Tan, Y. Liang, Q. Xu, E. Mamontov, J. Li, X. Xing, and L. Hong, Phys. Rev. Lett. 120, 248101 (2018).
V. P. Shkilev, Khim. Fiz. 24, 85 (2005).
V. P. Shkilev, J. Exp. Theor. Phys. 105, 1068 (2007).
V. P. Shkilev, Russ. J. Electrochem. 44, 1212 (2008).
V. P. Shkilev, Russ. J. Phys. Chem. B 27, 302 (2008).
J. Bisquert, Phys. Chem. Chem. Phys. 10, 1 (2008).
V. P. Shkilev, J. Exp. Theor. Phys. 101, 562 (2005).
G. Kaniadakis, Phys. A (Amsterdam, Neth.) 296, 405 (2001).
G. Kaniadakis and D. T. Hristopulos, Entropy 20, 426 (2018).
V. P. Zhdanov, Elementary Physical and Chemical Processes on the Surface (Nauka, Novosibirsk, 1988) [in Russian].
V. Pereyra, G. Zgrablich, and V. P. Zhdanov, Langmuir 6, 691 (1990).
P. H. Chavanis, Eur. Phys. J. B 62, 179 (2008).
D. V. Sivukhin, General Course of Physics (Nauka, Moscow, 1990), Vol. 2 [in Russian].
C. Tsallis and D. J. Bukman, Phys. Rev. E 54, R2197 (1996).
A. I. Saichev and S. G. Utkin, J. Exp. Theor. Phys. 99, 443 (2004).
J. Gajda and M. Magdziarz, Phys. Rev. E 82, 011117 (2010).
T. Miyaguchi and T. Akimoto, Phys. Rev. E 87, 032130 (2013).
B. L. Sprague, R. L. Pego, D. A. Stavreva, and J. G. McNally, Biophys. J. 86, 3473 (2004).
T. Ala-Nissila, R. Ferrando, and S. C. Ying, Adv. Phys. 51, 949 (2002).
I. Goychuk, Phys. Rev. E 86, 021113 (2012).
M. O. Vlad and J. Ross, Phys. Rev. E 66, 061908 (2002).
A. Yadav and W. Horsthemke, Phys. Rev. E 74, 066118 (2006).
V. P. Shkilev, J. Exp. Theor. Phys. 109, 852 (2009).
S. B. Yuste, E. Abad, and K. Lindenberg, Phys. Rev. E 82, 061123 (2010).
A. Plastino and A. Plastino, Phys. A (Amsterdam, Neth.) 222, 347 (1995).
T. D. Frank and A. Daffertshofer, Phys. A (Amsterdam, Neth.) 272, 497 (1999).
V. Schwämmle, F. D. Nobre, and E. M. F. Curado, Phys. Rev. E 76, 041123 (2007).
J. S. Andrade, G. F. T. da Silva, A. A. Moreira, F. D. Nobre, and E. M. F. Curado, Phys. Rev. Lett. 105, 260601 (2010).
M. S. Ribeiro, F. D. Nobre, and C. Tsallis, Phys. Rev. E 89, 052135 (2014).
W. Schirmacher, Solid State Commun. 39, 893 (1981).
B. Movaghar, M. Grünewald, B. Pohlmann, D. Würtz, and W. Schirmacher, J. Stat. Phys. 30, 315 (1983).
K. Godzik and W. Schirmacher, J. Phys. (Paris) 42, 127 (1981).
V. P. Shkilev, J. Exp. Theor. Phys. 133, 88 (2021).
J. van de Lagemaat, N. Kopidakis, N. R. Neale, and A. J. Frank, Phys. Rev. B 71, 035304 (2005).
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Shkilev, V.P. Continuous-Time Random Walks under Finite Concentrations. J. Exp. Theor. Phys. 134, 85–94 (2022). https://doi.org/10.1134/S1063776122010034
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DOI: https://doi.org/10.1134/S1063776122010034