Please wait a minute...
Chin. Phys. B, 2021, Vol. 30(12): 128703    DOI: 10.1088/1674-1056/ac3c3f

Discontinuous and continuous transitions of collective behaviors in living systems

Xu Li(李旭)1,2,†, Tingting Xue(薛婷婷)1,†, Yu Sun(孙宇)2, Jingfang Fan(樊京芳)1,2, Hui Li(李辉)1,2, Maoxin Liu(刘卯鑫)3, Zhangang Han(韩战钢)1, Zengru Di(狄增如)1,2, and Xiaosong Chen(陈晓松)1,2,‡
1 School of Systems Science, Beijing Normal University, Beijing 100878, China;
2 Institute of Nonequilibrium Systems, Beijing Normal University, Beijing 100878, China;
3 School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
Abstract  Living systems are full of astonishing diversity and complexity of life. Despite differences in the length scales and cognitive abilities of these systems, collective motion of large groups of individuals can emerge. It is of great importance to seek for the fundamental principles of collective motion, such as phase transitions and their natures. Via an eigen microstate approach, we have found a discontinuous transition of density and a continuous transition of velocity in the Vicsek models of collective motion, which are identified by the finite-size scaling form of order-parameter. At strong noise, living systems behave like gas. With the decrease of noise, the interactions between the particles of a living system become stronger and make them come closer. The living system experiences then a discontinuous gas-liquid like transition of density. The even stronger interactions at smaller noise make the velocity directions of the particles become ordered and there is a continuous phase transition of collective motion in addition.
Keywords:  living systems      phase emergence      phase transitions      eigen mircostate  
Received:  16 November 2021      Revised:  22 November 2021      Accepted manuscript online:  23 November 2021
PACS:  87.10.-e (General theory and mathematical aspects)  
  05.70.Fh (Phase transitions: general studies)  
  64.60.Ht (Dynamic critical phenomena)  
  89.75.-k (Complex systems)  
Fund: Project supported by the Fundamental Research Funds for the Central Universities, China (Grant No. 2019XD-A10) and the National Natural Science Foundation of China (Grant No. 71731002).
Corresponding Authors:  Xiaosong Chen     E-mail:

Cite this article: 

Xu Li(李旭), Tingting Xue(薛婷婷), Yu Sun(孙宇), Jingfang Fan(樊京芳), Hui Li(李辉), Maoxin Liu(刘卯鑫), Zhangang Han(韩战钢), Zengru Di(狄增如), and Xiaosong Chen(陈晓松) Discontinuous and continuous transitions of collective behaviors in living systems 2021 Chin. Phys. B 30 128703

[1] Cavagna A, Cimarelli A, Giardina I, Parisi G, Santagati R, Stefanini F and Viale M 2010 Proc. Natl. Acad. Sci. USA 107 11865
[2] Bialek W, Cavagna A, Giardina I, Mora T, Silvestri E, Viale M and Walczak A M 2012 Proc. Natl. Acad. Sci. USA 109 4786
[3] Bialek W, Cavagna A, Giardina I, Mora T, Pohl O, Silvestri E, Viale M and Walczak A M 2014 Proc. Natl. Acad. Sci. USA 111 7212
[4] Chen X, Dong X, Be’er A, Swinney H L and Zhang H 2012 Phys. Rev. Lett. 108 148101
[5] Li H, Shi X q, Huang M, Chen X, Xiao M, Liu C, Chaté H and Zhang H 2019 Proc. Natl. Acad. Sci. USA 116 777
[6] Feinerman O, Pinkoviezky I, Gelblum A, Fonio E and Gov N S 2018 Nat. Phys. 14 683
[7] Altshuler E, Ramos O, Núñez Y, Fernández J, Batista-Leyva A and Noda C 2005 The American Naturalist 166 643
[8] Buhl J, Sumpter D J, Couzin I D, Hale J J, Despland E, Miller E R and Simpson S J 2006 Science 312 1402
[9] Attanasi A, Cavagna A, Del Castello L, et al. 2014 Phys. Rev. Lett. 113 238102
[10] Attanasi A, Cavagna A, Del Castello L, et al. 2014 PLoS Computational Biology 10 e1003697
[11] Ginelli F, Peruani F, Pillot M H, Chaté H, Theraulaz G and Bon R 2015 Proc. Natl. Acad. Sci. USA 112 12729
[12] Krause J, Ruxton G D, Ruxton G, et al. 2002 Living in Groups (Oxford University Press)
[13] Sumpter D J 2010 Collective Animal Behavior (Princeton University Press)
[14] Marchetti M C, Joanny J F, Ramaswamy S, Liverpool T B, Prost J, Rao M and Simha R A 2013 Rev. Mod. Phys. 85 1143
[15] Sethna J 2021 Statistical mechanics: entropy, order parameters, and complexity, volume 14 (Oxford University Press, USA)
[16] Anderson P W 1972 Science 177 393
[17] Wilson K G 1979 Scientific American 241 158
[18] Barmatz M, Hahn I, Lipa J and Duncan R 2007 Rev. Mod. Phys. 79 1
[19] Vicsek T, Czirók A, Ben-Jacob E, Cohen I and Shochet O 1995 Phys. Rev. Lett. 75 1226
[20] Grégoire G and Chaté H 2004 Phys. Rev. Lett. 92 025702
[21] Chaté H, Ginelli F, Grégoire G and Raynaud F 2008 Phys. Rev. E 77 046113
[22] Hu G, Liu T, Liu M, Chen W and Chen X 2019 Science China Physics, Mechanics & Astronomy 62 990511
[23] Sun Y, Hu G, Zhang Y, Lu B, Lu Z, Fan J, Li X, Deng Q and Chen X 2021 Commun. Theor. Phys. 73 065603
[24] Gibbs J W 1902 Elementary principles in statistical mechanics developed with especial reference to the rational foundation of thermodynamics (New York: C. Scribner)
[25] Griffin A, Snoke D W, and Stringari S 1996 Bose–Einstein Condensation (Cambridge University Press)
[26] Li X T and Chen X S 2016 Commun. Theor. Phys. 66 355
[27] Strang G, Strang G, Strang G and Strang G 1993 Introduction to Linear Algebra volume 3, (Wellesley: Wellesley-Cambridge Press)
[28] Li L, Chen W, Dong W and Chen X 2011 Euro. Phys. J. B 80 189
[29] Huepe C and Aldana M 2008 Physica A 387 2809
[30] Xue T, Li X, Grassberger P and Chen L 2020 Phys. Rev. Research 2 042017
[31] Lenton T M, Held H, Kriegler E, Hall J W, Lucht W, Rahmstorf S and Schellnhuber H J 2008 Proc. Natl. Acad. Sci. USA 105 1786
[32] Scheffer M, Carpenter S R, Lenton T M, et al. 2012 Science 338 344
[1] Emergent O(4) symmetry at the phase transition from plaquette-singlet to antiferromagnetic order in quasi-two-dimensional quantum magnets
Guangyu Sun(孙光宇), Nvsen Ma(马女森), Bowen Zhao(赵博文), Anders W. Sandvik, and Zi Yang Meng(孟子杨). Chin. Phys. B, 2021, 30(6): 067505.
[2] Dynamic phase transition of ferroelectric nanotube described by a spin-1/2 transverse Ising model
Chundong Wang(王春栋), Ying Wu(吴瑛), Yulin Cao(曹喻霖), and Xinying Xue(薛新英). Chin. Phys. B, 2021, 30(2): 020504.
[3] Tunable deconfined quantum criticality and interplay of different valence-bond solid phases
Bowen Zhao(赵博文), Jun Takahashi, Anders W. Sandvik. Chin. Phys. B, 2020, 29(5): 057506.
[4] Calculation of the infrared frequency and the damping constant (full width at half maximum) for metal organic frameworks
M Kurt, H Yurtseven, A Kurt, S Aksoy. Chin. Phys. B, 2019, 28(6): 066401.
[5] Heavy fermions in high magnetic fields
M Smidman, B Shen(沈斌), C Y Guo(郭春煜), L Jiao(焦琳), X Lu(路欣), H Q Yuan(袁辉球). Chin. Phys. B, 2019, 28(1): 017106.
[6] Monogamy quantum correlation near the quantum phase transitions in the two-dimensional XY spin systems
Meng Qin(秦猛), Zhongzhou Ren(任中洲), Xin Zhang(张欣). Chin. Phys. B, 2018, 27(6): 060301.
[7] Transport properties of mixing conduction in CaF2 nanocrystals under high pressure
Ting-Jing Hu(胡廷静), Xiao-Yan Cui(崔晓岩), Jing-Shu Wang(王婧姝), Jun-Kai Zhang(张俊凯), Xue-Fei Li(李雪飞), Jing-Hai Yang(杨景海), Chun-Xiao Gao(高春晓). Chin. Phys. B, 2018, 27(1): 016401.
[8] High pressure electrical transport behavior in SrF2 nanoplates
Xiao-Yan Cui(崔晓岩), Ting-Jing Hu(胡廷静), Jing-Shu Wang(王婧姝), Jun-Kai Zhang(张俊凯), Xue-Fei Li(李雪飞), Jing-Hai Yang(杨景海), Chun-Xiao Gao(高春晓). Chin. Phys. B, 2017, 26(4): 046401.
[9] Multiscale structures and phase transitions in metallic glasses: A scattering perspective
Si Lan(兰司), Zhenduo Wu(吴桢舵), Xun-Li Wang(王循理). Chin. Phys. B, 2017, 26(1): 017104.
[10] Fidelity spectrum: A tool to probe the property of a quantum phase
Wing Chi Yu, Shi-Jian Gu. Chin. Phys. B, 2016, 25(3): 030501.
[11] Carrier behavior of HgTe under high pressure revealed by Hall effect measurement
Hu Ting-Jing, Cui Xiao-Yan, Li Xue-Fei, Wang Jing-Shu, Lü Xiu-Mei, Wang Ling-Sheng, Yang Jing-Hai, Gao Chun-Xiao. Chin. Phys. B, 2015, 24(11): 116401.
[12] Phase behaviors of binary mixtures composed of banana-shaped and calamitic mesogens
M. Cvetinov, D. Ž. Obadović, M. Stojanović, A. Vajda, K. Fodor-Csorba, N. Eber, I. Ristić. Chin. Phys. B, 2014, 23(9): 096402.
[13] Characteristics of phase transitions via intervention in random networks
Jia Xiao, Hong Jin-Song, Yang Hong-Chun, Yang Chun, Shi Xiao-Hong, Hu Jian-Quan. Chin. Phys. B, 2014, 23(7): 076401.
[14] Nonequilibrium behavior of the kinetic metamagnetic spin-5/2 Blume-Capel model
Ümüt Temizer. Chin. Phys. B, 2014, 23(7): 070511.
[15] Dielectric and infrared properties of SrTiO3 single crystal doped by 3d (V, Mn, Fe, Ni) and 4f (Nd, Sm, Er) ions
S. Maletic, D. Maletic, I. Petronijevic, J. Dojcilovic, D. M. Popovic. Chin. Phys. B, 2014, 23(2): 026102.
No Suggested Reading articles found!