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Based on the theory of fractional integration, direct transport behaviors of coupled Brownian motors with feedback control in viscoelastic media are investigated. The mathematical model of fractional overdamped coupled Brownian motors is established by adopting the power function as damping kernel function of general Langevin equation due to the power-law memory characteristics of cytosol in biological cells. Numerical solution is observed by fractional difference method and the influence of model parameters on cooperative direct transport of the coupled Brownian motors is discussed in detail by numerical simulation. The research shows that the memory of the fractional dynamical system can affect the direct transport phenomenon of the coupled Brownian motors through changing the on-off switching frequency of the ratchet potential with feedback control. To be more specific, in a proper range of the fractional order, the memory of the dynamical system can increase the on-off switching frequency of the ratchet potential, which can lead to the velocity increase of the direct transport. Furthermore, in the case of small fractional order, since the coupled Brownian motors move under the competition between the damping force with memory and the potential force with feedback control, the resultant force exerted on the coupled particles is always positive when the ratchet potential with feedback control is on although the fractional damping force is large, which leads to the result that the coupled Brownian motors move in the positive direction in the mass. On the contrary, in the case of large fractional order, the on-off switching frequency of potential with feedback control becomes small, as a result of which the main influential factor of the direct transport becomes the potential depth. Therefore the coupled Brownian motors are more likely to stay in the potential wells for a long time because the probability that describes the possibility that the coupled Brownian motors surmount the potential barriers becomes small. Finally, with the parameters of the fractional dynamical system (e.g. potential depth, noise intensity) fixed, the direct transport velocity of the coupled Brownian motors shows the generalized stochastic resonant phenomenon while the fractional order varies.
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Keywords:
- feedback control /
- fractional Brownian motors /
- generalized stochastic resonance /
- directed transport
[1] Nishyama M, Muto E, Inoue Y, Yanagida T, Higuchi H 2001 Nature Cell Biology 3 425
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[3] Cordova N G, Ermentrout B, Oster G 1992 Proc. Natl. Acad. Sci. USA 89 339
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[5] Zeng C H, Wang H 2012 Chin. Phys. B 21 050502
[6] Ai B Q, He Y F, Zhong W R 2011 Phys. Rev. E 83 051106
[7] Dan D, Jayannavarar A M, Menon G I 2003 Physica A 318 40
[8] Rozenbaum V M, Yang D Y, Lin S H, Tsong T Y 2006 Physica A 363 211
[9] Dinis L, Parron do J M R, Cao F J 2005 Europhys. Lett. 71 536
[10] Lindén M, Tuohimaa T, Jonsson A B, Wallin M F 2006 Phys. Rev. E 74 021908
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[17] Feito M, Cao F J 2008 Physica A 387 4553
[18] Gao T F, Chen J C 2009 J. Phys. A: Math. Theor. 42 065002
[19] Zhao A K, Zhang H W, Li Y X 2010 Chin. Phys. B 19 110506
[20] Wang L F, Gao T F, Huang R Z, Zheng Y X 2013 Acta Phys. Sin. 62 070502 (in Chinese) [王莉芳, 高天附, 黄仁忠, 郑玉祥 2013 物理学报 62 070502]
[21] Evstigneev M, Gehlen S, Reimann P 2009
[22] Gao T F, Liu F S, Chen J C 2012 Chin. Phys. B 21 020502
[23] Bier M 2007 Biosystems 88 301
[24] Zhang H W, Wen S T, Chen G R, Li Y X, Cao Z X, Li W 2012 Chin. Phys. B 21 038701
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[26] Cao F J, Feito M, Touchette H 2009 Physica A 388 113
[27] Mathur A B, Collinsworth A M, Reichert W M, Kraus W E, Truskey G A 2001 J. Biomech. 34 1545
[28] Azuma N, Aysin S D, Ikeda M, Kito H, Akadaka N, Sasajima T, Sumpio B E 2000 J. Vasc. Surg. 32 789
[29] Guilak F, Tedrow J R, Burgkart R 2000 Biochem. Biophys. Res. Commun. 269 781
[30] Bao J D 2012 Introduction to Anomalous Statistics Dynamics (Beijing: Science Press) p196 (in Chinese) [包景东 2012 反常统计动力学导论 (北京: 科学出版社) 第196页]
[31] Widom A 1971 Phys. Rev. A 3 1394
[32] 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 物理学报 62 240501]
[33] Bai W S M, Peng H, Tu Z, Ma H 2012 Acta Phys. Sin. 61 210501 (in Chinese) [白文斯密, 彭皓, 屠浙, 马洪 2012 物理学报 61 210501]
[34] Gitterman M 2005 Phys. Stat. Mech. Appl. 352 309
[35] Oldham K B, Spanier J 1974 The Fractional Calculus (New York: Academic Press)
[36] Liu F, Anh V, Turner I, Zhuang P 2003 J. Appl. Math. Comput. 13 233
[37] Petrás I 2011 Fractional-Order Nonlinear Systerms Modeling, Analysis and Simulation (1st Ed.) (Beijing: Higher Education Press) p19
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[1] Nishyama M, Muto E, Inoue Y, Yanagida T, Higuchi H 2001 Nature Cell Biology 3 425
[2] Reimann P 2002 Phys. Rep. 361 57
[3] Cordova N G, Ermentrout B, Oster G 1992 Proc. Natl. Acad. Sci. USA 89 339
[4] Gao T F, Zhang Y, Chen J C 2009 Chin. Phys. B 18 3279
[5] Zeng C H, Wang H 2012 Chin. Phys. B 21 050502
[6] Ai B Q, He Y F, Zhong W R 2011 Phys. Rev. E 83 051106
[7] Dan D, Jayannavarar A M, Menon G I 2003 Physica A 318 40
[8] Rozenbaum V M, Yang D Y, Lin S H, Tsong T Y 2006 Physica A 363 211
[9] Dinis L, Parron do J M R, Cao F J 2005 Europhys. Lett. 71 536
[10] Lindén M, Tuohimaa T, Jonsson A B, Wallin M F 2006 Phys. Rev. E 74 021908
[11] Craig E M, Zuckermann M J, Linke H J 2006 Phys. Rev. E 73 051106
[12] Lattanzi G, Maritan A 2001 Phys. Rev. Lett. 86 1134
[13] Cao F J, Dinis L, Parrondo J M R 2004 Phys. Rev. Lett. 93 040603
[14] Feito M, Cao F J 2006 Phys. Rev. E 74 041109
[15] Feito M, Cao F J 2007 Eur. Phys. J. B 59 63
[16] Feito M, Cao F J 2007 Phys. Rev. E 76 061113
[17] Feito M, Cao F J 2008 Physica A 387 4553
[18] Gao T F, Chen J C 2009 J. Phys. A: Math. Theor. 42 065002
[19] Zhao A K, Zhang H W, Li Y X 2010 Chin. Phys. B 19 110506
[20] Wang L F, Gao T F, Huang R Z, Zheng Y X 2013 Acta Phys. Sin. 62 070502 (in Chinese) [王莉芳, 高天附, 黄仁忠, 郑玉祥 2013 物理学报 62 070502]
[21] Evstigneev M, Gehlen S, Reimann P 2009
[22] Gao T F, Liu F S, Chen J C 2012 Chin. Phys. B 21 020502
[23] Bier M 2007 Biosystems 88 301
[24] Zhang H W, Wen S T, Chen G R, Li Y X, Cao Z X, Li W 2012 Chin. Phys. B 21 038701
[25] Bustamante C, Chemla Y R, Forde N R, Izhaky D 2004 Annu. Rev. Biochem. 73 705
[26] Cao F J, Feito M, Touchette H 2009 Physica A 388 113
[27] Mathur A B, Collinsworth A M, Reichert W M, Kraus W E, Truskey G A 2001 J. Biomech. 34 1545
[28] Azuma N, Aysin S D, Ikeda M, Kito H, Akadaka N, Sasajima T, Sumpio B E 2000 J. Vasc. Surg. 32 789
[29] Guilak F, Tedrow J R, Burgkart R 2000 Biochem. Biophys. Res. Commun. 269 781
[30] Bao J D 2012 Introduction to Anomalous Statistics Dynamics (Beijing: Science Press) p196 (in Chinese) [包景东 2012 反常统计动力学导论 (北京: 科学出版社) 第196页]
[31] Widom A 1971 Phys. Rev. A 3 1394
[32] 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 物理学报 62 240501]
[33] Bai W S M, Peng H, Tu Z, Ma H 2012 Acta Phys. Sin. 61 210501 (in Chinese) [白文斯密, 彭皓, 屠浙, 马洪 2012 物理学报 61 210501]
[34] Gitterman M 2005 Phys. Stat. Mech. Appl. 352 309
[35] Oldham K B, Spanier J 1974 The Fractional Calculus (New York: Academic Press)
[36] Liu F, Anh V, Turner I, Zhuang P 2003 J. Appl. Math. Comput. 13 233
[37] Petrás I 2011 Fractional-Order Nonlinear Systerms Modeling, Analysis and Simulation (1st Ed.) (Beijing: Higher Education Press) p19
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