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Using the fractional calculus theory, we investigate the directional transport phenomenon in a fractional logarithm coupled system under the action of a non-periodic external force. When a Brownian particle moves in the media with memory such as viscoelastic media, the system should be modeled as a nonlinear fractional logarithm coupled one. Using the method of fractional difference, we can solve the model numerically and discuss the influences of the various system parameters on the average transport velocity of the particles. Numerical results show that: 1) The directional transport phenomenon in this fractional logarithmic coupled model appears only when the external force exists, and the value of the average transport velocity of the particles increases with increasing external force. 2) When the fractional order of the system is small enough, the damping memory has a significant impact on the average transport velocity of the particles. Furthermore, the average transport velocity of the particles has an upper bound (although it is very small), no matter how the external force, coupled force and the intensity of noise change, the average transport velocity of the particles is no more than the upper bound. When there is no external force and the damping force is big enough, the directional transport phenomenon disappears. 3) When the fractional order of the system and the external force are big enough, although the directional transport phenomenon appears, the coupled force and the intensity of noise have no impact on the system. 4) Only when the external force is small enough, could the coupled force and noise intensity influence the average transport velocity of the particles. In this situation, the directional transport phenomenon appears when the fractional order of the system is big enough, and the average transport velocity of the particles changes along with the change of the coupled force and the noise intensity.
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Keywords:
- fractional logarithm coupled system /
- media with memory /
- generalized stochastic resonance /
- directional transport phenomenon
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[2] Barbi M, Salerno M 2000 Phys. Rev. E 62 1988
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[4] Hanggi P, Marchesoni F 2009 Rev. Mod. Phy. 81 387
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[25] Bao J D 2009 Random Simulation Method of Classical and Quantum Dissipation (Beijing: Science Press) (in Chinese) [包景东 2009 经典和量子耗散系统的随机模拟方法 (北京:科学出版社)]
[26] Podlubny I 1998 Fractional Differential Equation (San Diego: Academic Press)
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[1] Mateos J L 2000 Phys. Rev. Lett. 84 258
[2] Barbi M, Salerno M 2000 Phys. Rev. E 62 1988
[3] Zheng Z G, Hu G, Hu B B 2001 Phys. Rev. Lett. 86 2273
[4] Hanggi P, Marchesoni F 2009 Rev. Mod. Phy. 81 387
[5] Zheng Z G 2004 Spantiotemporal Dynamics and Collective Behaviors in Coupled Nonlinear System (Beijing: Higher Education Press) [郑志刚 2004 耦合非线性动力系统的时空动力学与合作行为 (北京:高等教育出版社)]
[6] Machura L, Kostur M, Luczka J 2010 Chem Phys. 375 445
[7] Mielke A 2000 Phys. Rev. Lett. 84 818
[8] Guerin T, Prost J, Martin P 2010 Current Opinnion in Cell Biology 22 14
[9] Chen H B, Zheng Z G 2012 J. Univ. Shanghai Sci. Technol. 346 (in Chinese) [陈宏斌, 郑志刚 2012 上海理工大学学报 346]
[10] Lipowsky R, Klumpp S, Nieuwenhuizen T M 2001 Phys. Rev. Lett. 87 108101
[11] Downton M T, Zuchermann M J, Craig E M, Plischke M, Linke H 2006 Phys. Rev. E 73 011909
[12] Kumar K V, Ramaswamy S, Rao M 2008 Phys. Rev. E 77 020102
[13] Fendrik A J, Romanelli L, Reale M V 2012 Phys. Rev. E 85 041149
[14] Savel E S, Marchesoni F, Nori F 2003 Phys. Rev. Lett. 91 10601
[15] Veigel C, Schmidt C F 2011 Nat. Rev. Mol. Cel. Biol. 12 163
[16] Ai B Q, He Y F, Zhong W R 2010 Phys. Rev. E 82 061102
[17] Ernst D, Hellmann M, Kohler J, Weiss M 2012 Soft. Matter. Comput. 8 4886
[18] Tu Z, Lai L, Luo M K2014 Acta Phys. Sin. 63 120503 in Chinese 2014 63 120503 (in Chinese) [屠浙, 赖莉, 罗懋康 2014 63 120503]
[19] Wang F, Deng C, Tu Z, Ma H 2013 Acta Phys. Sin. 62 040501 (in Chinese) [王飞, 邓翠, 屠浙, 马洪 2013 62 040501]
[20] Liu F, Anh V V, Turner I, Zhuang p 2003 J. Appl. Math. Comp. 13 233
[21] Benson D A, Wheatcraft S W, Meerschaert M M 2000 Water Resour. Pes. 36 1403
[22] Zhang L, Deng K, Luo M K 2012 Chin. Phys. B 21 090505
[23] Lin L F, Zhou X W, Ma H 2013 Acta Phys. Sin. 62 240501 (in Chinese) [林丽烽, 周兴旺, 马洪 2013 62 240501]
[24] de Souza Silva C C, van de Vondel J, Morelle M, Moshchalkov V V 2006 Nature 440 651
[25] Bao J D 2009 Random Simulation Method of Classical and Quantum Dissipation (Beijing: Science Press) (in Chinese) [包景东 2009 经典和量子耗散系统的随机模拟方法 (北京:科学出版社)]
[26] Podlubny I 1998 Fractional Differential Equation (San Diego: Academic Press)
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