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带反馈的分数阶耦合布朗马达的定向输运

秦天奇 王飞 杨博 罗懋康

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带反馈的分数阶耦合布朗马达的定向输运

秦天奇, 王飞, 杨博, 罗懋康

Transport properties of fractional coupled Brownian motors in ratchet potential with feedback

Qin Tian-Qi, Wang Fei, Yang Bo, Luo Mao-Kang
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  • 研究具有幂律记忆性的带反馈耦合布朗马达的定向输运现象, 引入分数阶理论, 建立了带反馈的分数阶耦合布朗马达模型, 利用分数阶差分法求得模型数值解并分析了模型参数对合作定向输运性质的影响. 仿真结果表明, 系统的记忆性通过影响带反馈的棘齿势的打开和闭合而影响粒子的定向输运, 即当系统的阶数在较小的范围内, 系统的记忆性会使带反馈的棘齿势的开关频率增加, 从而增大定向流速; 当系统其他参数(势垒高度、噪声强度等)固定时, 输运速度随着阶数的变化出现广义随机共振现象.
    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.
    • 基金项目: 国家自然科学基金(批准号:11171238)和电子信息控制重点实验室基金(批准号:2013035)资助的课题.
    • Funds: Project supported by the National Natural Science Foundation of China (Grand No. 11171238) and the Foundation of Science and Technology on Electronic Information Control Laboratory, China (Grant No. 2013035).
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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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出版历程
  • 收稿日期:  2014-12-22
  • 修回日期:  2015-01-25
  • 刊出日期:  2015-06-05

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