August  2018, 11(4): 859-889. doi: 10.3934/krm.2018034

Local sensitivity analysis for the Cucker-Smale model with random inputs

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

2. 

Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea

3. 

Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA

Received  December 2017 Revised  January 2018 Published  April 2018

Fund Project: The work of S.-Y. Ha was supported by National Research Foundation of Korea(NRF-2017R1A2B2001864), and the work of S. Jin was supported by NSF grants DMS-1522184 and DMS-1107291, and the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin.

We present pathwise flocking dynamics and local sensitivity analysis for the Cucker-Smale(C-S) model with random communications and initial data. For the deterministic communications, it is well known that the C-S model can model emergent local and global flocking dynamics depending on initial data and integrability of communication function. However, the communication mechanism between agents is not a priori clear and needs to be figured out from observed phenomena and data. Thus, uncertainty in communication is an intrinsic component in the flocking modeling of the C-S model. In this paper, we provide a class of admissible random uncertainties which allows us to perform the local sensitivity analysis for flocking and establish stability to the random C-S model with uncertain communication.

Citation: Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034
References:
[1]

S. M. AhnH. ChoiS.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10.  Google Scholar

[2]

S. M. Ahn and S. -Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895.  Google Scholar

[3]

G. Albi, L. Pareschi and M. Zanella, Uncertain quantification in control problems for flocking models, Math. Probl. Eng., 2015 (2015), Art. ID 850124, 14pp. doi: 10.1155/2015/850124.  Google Scholar

[4]

J. A. CarrilloM. FornasierJ. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.  Google Scholar

[5]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming. Mathematical modeling of collective behavior in socio-economic and life sciences, in Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Inc., Boston, MA, (2010), 297-336. Google Scholar

[6]

J. A. Carrillo, L. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Preprint. Google Scholar

[7]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models and Methods in Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

[8]

Y. -P. Choi, S. -Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles Vol. Ⅰ -Theory, Models, Applications, Series: Modeling and Simulation in Science and Technology (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhauser-Springer, 1 (2017), 299-331.  Google Scholar

[9]

F. Cucker and J.-G Dong, On flocks influenced by closest neighbors, Math. Models Methods Appl. Sci., 26 (2016), 2685-2708.  doi: 10.1142/S0218202516500639.  Google Scholar

[10]

F. Cucker and J.-G Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.  Google Scholar

[11]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.  Google Scholar

[12]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[13]

P. Degond and S. Motsch, Large-scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1022.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[14]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Commun. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.  Google Scholar

[15]

S. -Y. Ha, J. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, To appear in Kinetic Relat. Models. Google Scholar

[16]

S.-Y. HaB. Kwon and M.-J. Kang, A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Mod. Meth. Appl. Sci., 24 (2014), 2311-2359.  doi: 10.1142/S0218202514500225.  Google Scholar

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S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[18]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[19]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinetic Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[20]

J. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys., 315 (2016), 150-168.  doi: 10.1016/j.jcp.2016.03.047.  Google Scholar

[21]

J. Hu and S. Jin, Uncertainty quantification for kinetic equations, in Uncertainty Quantification for Kinetic and Hyperbolic Equations, (eds S. Jin and L. Pareschi), 14 (2018), 193-229. doi: 10.1007/978-3-319-67110-9_6.  Google Scholar

[22]

J. HuS. Jin and D. Xiu, A stochastic Galerkin method for Hamiltonian-Jacobi equations with uncertainty, SIAM. J. Sci. Comput., 37 (2015), 2246-2269.  doi: 10.1137/140990930.  Google Scholar

[23]

H. -N. Huang, S. A. M. Marcantognini and N. J. Young, Chain rules for higher derivatives, The Mathematical Intelligencer, 28 (2006), 61-69, http://ambio1.leeds.ac.uk/~nicholas/abstracts/FaadiBruno3.pdf. doi: 10.1007/BF02987158.  Google Scholar

[24]

S. Jin, J. -G. Liu and Z. Ma, Uniform spectral convergence of the stochastic Galerkin method for the linear transport equations with random inputs in diffusive regime and a micro-macro decomposition based asymptotic preserving method, Research in Math. Sci., 4 (2017), Paper No. 15, 25 pp. doi: 10.1186/s40687-017-0105-1.  Google Scholar

[25]

S. Jin and L. Liu, An asymptotic-preserving stochastic Galerkin method for the semicondutor Boltzmann equation with random inputs and diffusive scalings, Multiscale Model. Simu., 15 (2017), 157-183.  doi: 10.1137/15M1053463.  Google Scholar

[26]

S. Jin, M. -B. Tran and E. Zuazua, A local sensivity analysis for a damped wave equation with random initial input, Preprint. Google Scholar

[27]

S. JinD. Xiu and X. Zhu, Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, J. Comput. Phys., 289 (2015), 35-52.  doi: 10.1016/j.jcp.2015.02.023.  Google Scholar

[28]

S. JinD. Xiu and X. Zhu, A well-balanced stochastic Galerkin method for scalar hyperbolic balance laws with random inputs, J. Sci. Comput., 67 (2016), 1198-1218.  doi: 10.1007/s10915-015-0124-2.  Google Scholar

[29]

S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple Scales, To appear in SIAM J. Math. Anal. Google Scholar

[30]

M.-J. Kang and A. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci., 25 (2015), 2153-2173.  doi: 10.1142/S0218202515500542.  Google Scholar

[31]

T. KarperA. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker-Smale model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.  doi: 10.1142/S0218202515500050.  Google Scholar

[32]

T. KarperA. Mellet and K. Trivisa, Existence of weak solutions to kinetic flocking models, SIAM J. Math. Anal., 45 (2013), 215-243.  doi: 10.1137/120866828.  Google Scholar

[33]

T. Karper, A. Mellet and K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, Springer Proc. Math. Stat., Springer, Heidelberg, 49 (2014), 227-242. doi: 10.1007/978-3-642-39007-4_11.  Google Scholar

[34]

N. E. LeonardD. A. PaleyF. LekienR. SepulchreD. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.  doi: 10.1109/JPROC.2006.887295.  Google Scholar

[35]

Q. Li and L. Wang, Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA J. Uncertainty Quantification, 5 (2017), 1193-1219.  doi: 10.1137/16M1106675.  Google Scholar

[36]

L. Liu and S. Jin, Hypocoercivity based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs, Preprint. Google Scholar

[37]

S. Motsch and E. Tadmor, Heterophilious dynamics: Enhanced Consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[38]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

[39]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems Magazine, 27 (2007), 89-105.   Google Scholar

[40]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. of Guidance, Control and Dynamics, 32 (2009), 527-537.   Google Scholar

[41]

A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana and S. Tarantola, Introduction to sensivity analysis, Global sensivity analysis, The Primer, (2008), 1-51.  Google Scholar

[42]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719.  doi: 10.1137/060673254.  Google Scholar

[43]

R. Shu and S. Jin, Uniform regularity in the random space and spectral accuracy of the stochastic Galerkin method for a kinetic-fluid two-phase flow model with random initial inputs in the light particle regime, Preprint. Google Scholar

[44]

E. Tadmor, Mathematical aspects of self-organized dynamics: consensus, emergence of leaders, and social hydrodynamics, SIAM News, 48 (2015). Google Scholar

[45]

J. Toner and Y. Tu, Flocks, herds, and Schools: A quantitative theory of flocking, Physical Review E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[46]

T. VicsekE. Ben-Jacob CzirókI. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[47]

D. Xiu, Numerical Methods fo Stochastic Computations, Princeton University Presss, 2010.  Google Scholar

[48]

D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM. J. Scientific Computiong, 24 (2002), 619-644.  doi: 10.1137/S1064827501387826.  Google Scholar

show all references

References:
[1]

S. M. AhnH. ChoiS.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10.  Google Scholar

[2]

S. M. Ahn and S. -Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895.  Google Scholar

[3]

G. Albi, L. Pareschi and M. Zanella, Uncertain quantification in control problems for flocking models, Math. Probl. Eng., 2015 (2015), Art. ID 850124, 14pp. doi: 10.1155/2015/850124.  Google Scholar

[4]

J. A. CarrilloM. FornasierJ. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.  doi: 10.1137/090757290.  Google Scholar

[5]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming. Mathematical modeling of collective behavior in socio-economic and life sciences, in Mathematical modeling of collective behavior in socio-economic and life sciences, Model. Simul. Sci. Eng. Technol., Birkhauser Boston, Inc., Boston, MA, (2010), 297-336. Google Scholar

[6]

J. A. Carrillo, L. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Preprint. Google Scholar

[7]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models and Methods in Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

[8]

Y. -P. Choi, S. -Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles Vol. Ⅰ -Theory, Models, Applications, Series: Modeling and Simulation in Science and Technology (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkhauser-Springer, 1 (2017), 299-331.  Google Scholar

[9]

F. Cucker and J.-G Dong, On flocks influenced by closest neighbors, Math. Models Methods Appl. Sci., 26 (2016), 2685-2708.  doi: 10.1142/S0218202516500639.  Google Scholar

[10]

F. Cucker and J.-G Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.  doi: 10.1109/TAC.2011.2107113.  Google Scholar

[11]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.  Google Scholar

[12]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[13]

P. Degond and S. Motsch, Large-scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1022.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[14]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Commun. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.  Google Scholar

[15]

S. -Y. Ha, J. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, To appear in Kinetic Relat. Models. Google Scholar

[16]

S.-Y. HaB. Kwon and M.-J. Kang, A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid, Math. Mod. Meth. Appl. Sci., 24 (2014), 2311-2359.  doi: 10.1142/S0218202514500225.  Google Scholar

[17]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[18]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[19]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinetic Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[20]

J. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty, J. Comput. Phys., 315 (2016), 150-168.  doi: 10.1016/j.jcp.2016.03.047.  Google Scholar

[21]

J. Hu and S. Jin, Uncertainty quantification for kinetic equations, in Uncertainty Quantification for Kinetic and Hyperbolic Equations, (eds S. Jin and L. Pareschi), 14 (2018), 193-229. doi: 10.1007/978-3-319-67110-9_6.  Google Scholar

[22]

J. HuS. Jin and D. Xiu, A stochastic Galerkin method for Hamiltonian-Jacobi equations with uncertainty, SIAM. J. Sci. Comput., 37 (2015), 2246-2269.  doi: 10.1137/140990930.  Google Scholar

[23]

H. -N. Huang, S. A. M. Marcantognini and N. J. Young, Chain rules for higher derivatives, The Mathematical Intelligencer, 28 (2006), 61-69, http://ambio1.leeds.ac.uk/~nicholas/abstracts/FaadiBruno3.pdf. doi: 10.1007/BF02987158.  Google Scholar

[24]

S. Jin, J. -G. Liu and Z. Ma, Uniform spectral convergence of the stochastic Galerkin method for the linear transport equations with random inputs in diffusive regime and a micro-macro decomposition based asymptotic preserving method, Research in Math. Sci., 4 (2017), Paper No. 15, 25 pp. doi: 10.1186/s40687-017-0105-1.  Google Scholar

[25]

S. Jin and L. Liu, An asymptotic-preserving stochastic Galerkin method for the semicondutor Boltzmann equation with random inputs and diffusive scalings, Multiscale Model. Simu., 15 (2017), 157-183.  doi: 10.1137/15M1053463.  Google Scholar

[26]

S. Jin, M. -B. Tran and E. Zuazua, A local sensivity analysis for a damped wave equation with random initial input, Preprint. Google Scholar

[27]

S. JinD. Xiu and X. Zhu, Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, J. Comput. Phys., 289 (2015), 35-52.  doi: 10.1016/j.jcp.2015.02.023.  Google Scholar

[28]

S. JinD. Xiu and X. Zhu, A well-balanced stochastic Galerkin method for scalar hyperbolic balance laws with random inputs, J. Sci. Comput., 67 (2016), 1198-1218.  doi: 10.1007/s10915-015-0124-2.  Google Scholar

[29]

S. Jin and Y. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple Scales, To appear in SIAM J. Math. Anal. Google Scholar

[30]

M.-J. Kang and A. Vasseur, Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Models Methods Appl. Sci., 25 (2015), 2153-2173.  doi: 10.1142/S0218202515500542.  Google Scholar

[31]

T. KarperA. Mellet and K. Trivisa, Hydrodynamic limit of the kinetic Cucker-Smale model, Math. Models Methods Appl. Sci., 25 (2015), 131-163.  doi: 10.1142/S0218202515500050.  Google Scholar

[32]

T. KarperA. Mellet and K. Trivisa, Existence of weak solutions to kinetic flocking models, SIAM J. Math. Anal., 45 (2013), 215-243.  doi: 10.1137/120866828.  Google Scholar

[33]

T. Karper, A. Mellet and K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, Springer Proc. Math. Stat., Springer, Heidelberg, 49 (2014), 227-242. doi: 10.1007/978-3-642-39007-4_11.  Google Scholar

[34]

N. E. LeonardD. A. PaleyF. LekienR. SepulchreD. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.  doi: 10.1109/JPROC.2006.887295.  Google Scholar

[35]

Q. Li and L. Wang, Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA J. Uncertainty Quantification, 5 (2017), 1193-1219.  doi: 10.1137/16M1106675.  Google Scholar

[36]

L. Liu and S. Jin, Hypocoercivity based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs, Preprint. Google Scholar

[37]

S. Motsch and E. Tadmor, Heterophilious dynamics: Enhanced Consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[38]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

[39]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems Magazine, 27 (2007), 89-105.   Google Scholar

[40]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. of Guidance, Control and Dynamics, 32 (2009), 527-537.   Google Scholar

[41]

A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana and S. Tarantola, Introduction to sensivity analysis, Global sensivity analysis, The Primer, (2008), 1-51.  Google Scholar

[42]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694-719.  doi: 10.1137/060673254.  Google Scholar

[43]

R. Shu and S. Jin, Uniform regularity in the random space and spectral accuracy of the stochastic Galerkin method for a kinetic-fluid two-phase flow model with random initial inputs in the light particle regime, Preprint. Google Scholar

[44]

E. Tadmor, Mathematical aspects of self-organized dynamics: consensus, emergence of leaders, and social hydrodynamics, SIAM News, 48 (2015). Google Scholar

[45]

J. Toner and Y. Tu, Flocks, herds, and Schools: A quantitative theory of flocking, Physical Review E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[46]

T. VicsekE. Ben-Jacob CzirókI. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[47]

D. Xiu, Numerical Methods fo Stochastic Computations, Princeton University Presss, 2010.  Google Scholar

[48]

D. Xiu and G. E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM. J. Scientific Computiong, 24 (2002), 619-644.  doi: 10.1137/S1064827501387826.  Google Scholar

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