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## 留言板

Investigation on the directed transport efficiency of feedback-control ratchet

## Investigation on the directed transport efficiency of feedback-control ratchet

Fan Li-Ming, Lü Ming-Tao, Huang Ren-Zhong, Gao Tian-Fu, Zheng Zhi-Gang
• #### Abstract

Brownian motion in the environment of the thermal fluctuations is a long-study issue in nonequilibrium statistical physics. In recent years, the directed transport properties of Brownian ratchets attract the widespread attention of scholars. When a ratchet system possesses the spatio-temporal symmetry-breaking feature, the directed transport can be produced. Although the breakthrough progress in the directed transport of the Brownian ratchet has been made, the energy conversion efficiency of feedback ratchet is not clear. Therefore, the center-of-mass mean velocity and the energy conversion efficiency of coupled ratchet under the influences of the time asymmetry of external force and the spatial asymmetry of external potential are discussed in detail. The overdamped coupled Brownian particles are investigated. Nevertheless, the optimized control of the coupled ratchet is the important for directed transport. Therefore, the closed-loop control which depends on the state of the system is adopted. The dynamic behavior of coupled particles can be described by the overdamped Langevin equation, and the equation is numerically solved by using the stochastic Runge-Kutta algorithm. Some properties of the directed transport can be obtained through this method, such as the center-of-mass mean velocity, the energy conversion efficiency, etc. It is interesting to find that the center-of-mass mean velocity can reach a maximum as the amplitude of external force increases. However, the mean velocity can show the quasi-periodic oscillations with the increase of the period of external force for different values of the spatial asymmetry of external potential. In addition, it can be found that the feedback ratchet needs strong noise to make the directed transport of the ratchet reach the maximum as the coupled strength increases. On the other hand, the energy conversion efficiencies of the feedback ratchet can achieve their corresponding maximum values with the increase of the amplitude of external force for different values of the time asymmetry, and the maximum increases as the time asymmetry increases. However, the efficiency can also show the quasi-periodic oscillations with the increase of the period of the external force for different values of the spatial asymmetry of external potential. Moreover, the energy conversion efficiency can achieve the maximum as the noise strength increases, but the maximum of the efficiency will decrease with the increase of coupling strength. From the discussion above, the optimal values of the time asymmetry, the spatial asymmetry, the period of the external force and the noise strength can promote the directed transport of the feedback coupled Brownian ratchet. These conclusions can provide some guidance in the enhancement of the energy conversion efficiency of a nanomachine.

#### Authors and contacts

###### Corresponding author: Gao Tian-Fu, tianfugao@synu.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China（Grand Nos. 11475022, 11347003) and the Scientific Research Funds of Huaqiao University and the Excellent Talents Program of Shenyang Normal University, China（Grand No. 91400114005).

#### References

 [1] Mateos J L 2000 Phys. Rev. Lett. 84 258 [2] Barbi M, Salerno M 2000 Phys. Rev. E 62 1988 [3] Sumithra K, Sintes T 2009 Rev. Mod. Phy. 81 387 [4] Zheng Z G 2004 Spantiotemporal Dynamics and Collective Behaviors in Coupled Nonlinear System（1st Ed.)（Beijing:Higher Education Press) pp278-340(in Chinese)[郑志刚2004耦合非线性动力系统的时空动力学与合作行为(第一版)（北京:高等教育出版社)第278–340页] [5] Machura L, Kostur M, Luczka J 2010 Chem. Phys. 375 445 [6] Mielke A 2000 Phys. Rev. Lett. 84 818 [7] Doering C R 1995 Nuovo Cimento 17 685 [8] Jlicher F, Ajdari A, Prost J 1997 Rev. Mod. Phys. 69 1269 [9] Astumian R D, Bier M 1994 Phys. Rev. Lett. 72 1766 [10] Xie P 2002 Phys. Rep. 361 57 [11] Rozenbaum V M, Yang D Y, Lin S H, Tsong T Y 2006 Physica A 363 211 [12] Wang H Y, Bao J D 2003 Physica A 323 197 [13] Linke H 2002 Appl. Phys. A:Mater. Sci. Process. 75 167 [14] van den Heuvel M G L, Dekker C 2007 Science 317 333 [15] Braun O M, Kivshar Y S 1998 Phys. Rep. 306 1 [16] Landa P S, McClintock P V E 2000 Phys. Rep. 323 1 [17] Ai B Q, He Y F, Zhong W R 2011 Phys. Rev. E 83 051106 [18] Li C P, Han Y R, Zhan Y, Hu J J, Zhang L G, Qu J 2011 Mod. Phys. Lett. B 25 1179 [19] Downton M T, Zuchermann M J, Craig E M, Plischke M, Linke H 2006 Phys. Rev. E 73 011909 [20] Wang H Y, Bao J D 2007 Physica A 374 33 [21] Feito M, Cao F J 2006 Phys. Rev. E 74 041109 [22] Feito M, Cao F J 2007 Eur. Phys. J. B 59 63 [23] Feito M, Cao F J 2007 Phys. Rev. E 76 061113 [24] Feito M, Cao F J 2008 Physica A 387 4553 [25] Gao T F, Chen J C 2009 J. Phys. A:Math. Theor. 42 065002 [26] Rousselet J, Salome L, Ajdari A, Prost J 1994 Nature 370 446 [27] Bier M 2007 Biosystems 88 301 [28] Zhang H W, Wen S T, Chen G R, Li Y X, Cao Z X, Li W 2012 Chin. Phys. B 21 038701 [29] Bustamante C, Chemla Y R, Forde N R, Izhaky D 2004 Annu. Rev. Biochem. 73 705 [30] Cao F J, Feito M, Touchette H 2009 Physica A 388 113 [31] 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] [32] Qin T Q, Wang F, Yang B, Luo M K 2015 Acta Phys. Sin. 64 120501 （in Chinese)[秦天齐, 王飞, 杨博, 罗懋康2015物理学报64 120501] [33] Wang H Y, Bao J D 2005 Physica A 357 373 [34] Zhao A K 2007 M. S. Dissertation（Zhengzhou:Zhengzhou University)（in Chinese)[赵阿可2007硕士学位论文(郑州:郑州大学)] [35] Derényi I, Astumian R D 1999 Phys. Rev. E 59 R6219 [36] Bao J D 2012 An Introduction to Anomalous Statisticl Dynamics （1st Ed.)（Beijing:Science Press) pp127-184(in Chinese)[包景东2012反常统计动力学导论第一版(北京:科学出版社)第127–184页] [37] Li G, Tu Z C 2016 Sci. China:Phys. Mech. Astron. 59 640501 [38] Zheng Z G, Cross M C, Hu G 2002 Phys. Rev. Lett. 89 154102

#### Cited By

•  [1] Mateos J L 2000 Phys. Rev. Lett. 84 258 [2] Barbi M, Salerno M 2000 Phys. Rev. E 62 1988 [3] Sumithra K, Sintes T 2009 Rev. Mod. Phy. 81 387 [4] Zheng Z G 2004 Spantiotemporal Dynamics and Collective Behaviors in Coupled Nonlinear System（1st Ed.)（Beijing:Higher Education Press) pp278-340(in Chinese)[郑志刚2004耦合非线性动力系统的时空动力学与合作行为(第一版)（北京:高等教育出版社)第278–340页] [5] Machura L, Kostur M, Luczka J 2010 Chem. Phys. 375 445 [6] Mielke A 2000 Phys. Rev. Lett. 84 818 [7] Doering C R 1995 Nuovo Cimento 17 685 [8] Jlicher F, Ajdari A, Prost J 1997 Rev. Mod. Phys. 69 1269 [9] Astumian R D, Bier M 1994 Phys. Rev. Lett. 72 1766 [10] Xie P 2002 Phys. Rep. 361 57 [11] Rozenbaum V M, Yang D Y, Lin S H, Tsong T Y 2006 Physica A 363 211 [12] Wang H Y, Bao J D 2003 Physica A 323 197 [13] Linke H 2002 Appl. Phys. A:Mater. Sci. Process. 75 167 [14] van den Heuvel M G L, Dekker C 2007 Science 317 333 [15] Braun O M, Kivshar Y S 1998 Phys. Rep. 306 1 [16] Landa P S, McClintock P V E 2000 Phys. Rep. 323 1 [17] Ai B Q, He Y F, Zhong W R 2011 Phys. Rev. E 83 051106 [18] Li C P, Han Y R, Zhan Y, Hu J J, Zhang L G, Qu J 2011 Mod. Phys. Lett. B 25 1179 [19] Downton M T, Zuchermann M J, Craig E M, Plischke M, Linke H 2006 Phys. Rev. E 73 011909 [20] Wang H Y, Bao J D 2007 Physica A 374 33 [21] Feito M, Cao F J 2006 Phys. Rev. E 74 041109 [22] Feito M, Cao F J 2007 Eur. Phys. J. B 59 63 [23] Feito M, Cao F J 2007 Phys. Rev. E 76 061113 [24] Feito M, Cao F J 2008 Physica A 387 4553 [25] Gao T F, Chen J C 2009 J. Phys. A:Math. Theor. 42 065002 [26] Rousselet J, Salome L, Ajdari A, Prost J 1994 Nature 370 446 [27] Bier M 2007 Biosystems 88 301 [28] Zhang H W, Wen S T, Chen G R, Li Y X, Cao Z X, Li W 2012 Chin. Phys. B 21 038701 [29] Bustamante C, Chemla Y R, Forde N R, Izhaky D 2004 Annu. Rev. Biochem. 73 705 [30] Cao F J, Feito M, Touchette H 2009 Physica A 388 113 [31] 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] [32] Qin T Q, Wang F, Yang B, Luo M K 2015 Acta Phys. Sin. 64 120501 （in Chinese)[秦天齐, 王飞, 杨博, 罗懋康2015物理学报64 120501] [33] Wang H Y, Bao J D 2005 Physica A 357 373 [34] Zhao A K 2007 M. S. Dissertation（Zhengzhou:Zhengzhou University)（in Chinese)[赵阿可2007硕士学位论文(郑州:郑州大学)] [35] Derényi I, Astumian R D 1999 Phys. Rev. E 59 R6219 [36] Bao J D 2012 An Introduction to Anomalous Statisticl Dynamics （1st Ed.)（Beijing:Science Press) pp127-184(in Chinese)[包景东2012反常统计动力学导论第一版(北京:科学出版社)第127–184页] [37] Li G, Tu Z C 2016 Sci. China:Phys. Mech. Astron. 59 640501 [38] Zheng Z G, Cross M C, Hu G 2002 Phys. Rev. Lett. 89 154102
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•  Citation:
##### Metrics
• Abstract views:  401
• Cited By: 0
##### Publishing process
• Received Date:  07 June 2016
• Accepted Date:  30 September 2016
• Published Online:  05 January 2017

## Investigation on the directed transport efficiency of feedback-control ratchet

###### Corresponding author: Gao Tian-Fu, tianfugao@synu.edu.cn;
• 1. College of Physical Science and Technology, Shenyang Normal University, Shenyang 110034, China;
• 2. College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
Fund Project:  Project supported by the National Natural Science Foundation of China（Grand Nos. 11475022, 11347003) and the Scientific Research Funds of Huaqiao University and the Excellent Talents Program of Shenyang Normal University, China（Grand No. 91400114005).

Abstract: Brownian motion in the environment of the thermal fluctuations is a long-study issue in nonequilibrium statistical physics. In recent years, the directed transport properties of Brownian ratchets attract the widespread attention of scholars. When a ratchet system possesses the spatio-temporal symmetry-breaking feature, the directed transport can be produced. Although the breakthrough progress in the directed transport of the Brownian ratchet has been made, the energy conversion efficiency of feedback ratchet is not clear. Therefore, the center-of-mass mean velocity and the energy conversion efficiency of coupled ratchet under the influences of the time asymmetry of external force and the spatial asymmetry of external potential are discussed in detail. The overdamped coupled Brownian particles are investigated. Nevertheless, the optimized control of the coupled ratchet is the important for directed transport. Therefore, the closed-loop control which depends on the state of the system is adopted. The dynamic behavior of coupled particles can be described by the overdamped Langevin equation, and the equation is numerically solved by using the stochastic Runge-Kutta algorithm. Some properties of the directed transport can be obtained through this method, such as the center-of-mass mean velocity, the energy conversion efficiency, etc. It is interesting to find that the center-of-mass mean velocity can reach a maximum as the amplitude of external force increases. However, the mean velocity can show the quasi-periodic oscillations with the increase of the period of external force for different values of the spatial asymmetry of external potential. In addition, it can be found that the feedback ratchet needs strong noise to make the directed transport of the ratchet reach the maximum as the coupled strength increases. On the other hand, the energy conversion efficiencies of the feedback ratchet can achieve their corresponding maximum values with the increase of the amplitude of external force for different values of the time asymmetry, and the maximum increases as the time asymmetry increases. However, the efficiency can also show the quasi-periodic oscillations with the increase of the period of the external force for different values of the spatial asymmetry of external potential. Moreover, the energy conversion efficiency can achieve the maximum as the noise strength increases, but the maximum of the efficiency will decrease with the increase of coupling strength. From the discussion above, the optimal values of the time asymmetry, the spatial asymmetry, the period of the external force and the noise strength can promote the directed transport of the feedback coupled Brownian ratchet. These conclusions can provide some guidance in the enhancement of the energy conversion efficiency of a nanomachine.

Reference (38)

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