Abstract.

At the macroscopic scale, many important models of collective motion fall into the class of kinematic flows for which both velocity and diffusion terms depend only on particle density. When total particle numbers are fixed and finite, simulations of corresponding microscopic dynamics exhibit stochastic effects which can induce a variety of interesting behaviors not present in the large system limit. In this article we undertake a systematic examination of finite-size fluctuations in a general class of particle models whose statistics correspond to those of stochastic kinematic flows. Doing so, we are able to characterize phenomena including quasi-jams in models of traffic flow; stochastic pattern formation among spatially coupled oscillators; anomalous bulk subdiffusion in porous media; and travelling wave fluctuations in a model of bacterial swarming.

Keywords

  1. kinematic flows
  2. interacting particle systems
  3. traffic modelling
  4. organism swarming
  5. anomalous sub diffusion

MSC codes

  1. 60H15
  2. 60H30
  3. 58J65
  4. 92C35
  5. 76Z05

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
M. J. Lighthill and G. B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads, Proc. R. Soc. Lond. Ser. A Math. Phys. Sci., 229 (1955), pp. 317–345, https://doi.org/10.1098/rspa.1955.0089.
2.
H. P. McKean, Propagation of chaos for a class of non-linear parabolic equations, in Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), Air Force Office Sci. Res., Arlington, VA, 1967, pp. 41–57.
3.
L.-P. Chaintron and A. Diez, Propagation of Chaos: A Review of Models, Methods and Applications, preprint, https://arxiv.org/abs/2106.14812, 2021.
4.
M. Hitsuda and I. Mitoma, Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions, J. Multivariate Anal., 19 (1986), pp. 311–328, https://doi.org/10.1016/0047-259X(86)90035-7.
5.
A.-S. Sznitman, Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated, J. Funct. Anal., 56 (1984), pp. 311–336, https://doi.org/10.1016/0022-1236(84)90080-6.
6.
A.-S. Sznitman, A propagation of chaos result for Burgers’ equation, Probab. Theory Relat. Fields, 71 (1986), pp. 581–613, https://doi.org/10.1007/BF00699042.
7.
P.-H. Chavanis, The generalized stochastic Smoluchowski equation, Entropy, 21 (2019), https://doi.org/10.3390/e21101006.
8.
P. H. Chavanis and L. Delfini, Random transitions described by the stochastic Smoluchowski-Poisson system and by the stochastic Keller-Segel model, Phys. Rev. E, 89 (2014), 032139, https://doi.org/10.1103/PhysRevE.89.032139.
9.
S. H. Strogatz and R. E. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Stat. Phys., 63 (1991), pp. 613–635, https://doi.org/10.1007/BF01029202.
10.
J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), pp. 137–185, https://doi.org/10.1103/RevModPhys.77.137.
11.
D. Aikman and G. Hewitt, An experimental investigation of the rate and form of dispersal in grasshoppers, J. Appl. Ecol., 9 (1972), pp. 807–817, https://doi.org/10.2307/2401906.
12.
E. Luçon, Large population asymptotics for interacting diffusions in a quenched random environment, in From Particle Systems to Partial Differential Equations II, Springer Proc. Math. Stat. 129, Springer, 2015, pp. 231–251, https://doi.org/10.1007/978-3-319-16637-7_8.
13.
D. Grünbaum and A. Okubo, Modelling social animal aggregations, in Frontiers in Mathematical Biology, S. A. Levin, ed., Springer, Berlin, Heidelberg, 1994, pp. 296–325, https://doi.org/10.1007/978-3-642-50124-1_18.
14.
D. Grünbaum, Translating stochastic density-dependent individual behavior with sensory constraints to an Eulerian model of animal swarming, J. Math. Biol., 33 (1994), pp. 139–161, https://doi.org/10.1007/BF00160177.
15.
P. A. Milewski and X. Yang, A simple model for biological aggregation with asymmetric sensing, Commun. Math. Sci., 6 (2008), pp. 397–416, https://doi.org/10.4310/CMS.2008.v6.n2.a7.
16.
A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), pp. 534–570, https://doi.org/10.1007/s002850050158.
17.
A. Okubo and P. Kareiva, Some examples of animal diffusion, in Diffusion and Ecological Problems: Modern Perspectives, Interdiscip. Appl. Math. 14, Springer, 2001, pp. 170–196, https://doi.org/10.1007/978-1-4757-4978-6_6.
18.
D. S. Dean, Langevin equation for the density of a system of interacting Langevin processes, J. Phys. A, 29 (1996), pp. L613–L617, https://doi.org/10.1088/0305-4470/29/24/001.
19.
P. Blanchard, M. Röckner, and F. Russo, Probabilistic representation for solutions of an irregular porous media type equation, Ann. Probab., 38 (2010), pp. 1870–1900, https://doi.org/10.1214/10-AOP526.
20.
D. Godinho and C. Quiñinao, Propagation of chaos for a subcritical Keller-Segel model, Ann. Inst. Henri Poincar´e Probab. Stat., 51 (2015), pp. 965–992, https://doi.org/10.1214/14-AIHP606.
21.
F. Bolley, I. Gentil, and A. Guillin, Uniform convergence to equilibrium for granular media, Arch. Ration. Mech. Anal., 208 (2013), pp. 429–445, https://doi.org/10.1007/s00205-012-0599-z.
22.
A. J. McKane, T. Biancalani, and T. Rogers, Stochastic pattern formation and spontaneous polarisation: The linear noise approximation and beyond, Bull. Math. Biol., 76 (2014), pp. 895–921, https://doi.org/10.1007/s11538-013-9827-4.
23.
T. Biancalani, D. Fanelli, and F. Di Patti, Stochastic turing patterns in the Brusselator model, Phys. Rev. E, 81 (2010), 046215, https://doi.org/10.1103/PhysRevE.81.046215.
24.
E. Luçon, Quenched limits and fluctuations of the empirical measure for plane rotators in random media, Electron. J. Probab., 16 (2011), https://doi.org/10.1214/ejp.v16-874.
25.
N. Wiener, Generalized harmonic analysis, Acta Math., 55 (1930), pp. 117–258.
26.
D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys., 73 (2001), pp. 1067–1141, https://doi.org/10.1103/RevModPhys.73.1067.
27.
M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 51 (1995), pp. 1035–1042, https://doi.org/10.1103/PhysRevE.51.1035.
28.
L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and J. Soler, Exactly solvable phase oscillator models with synchronization dynamics, Phys. Rev. Lett., 81 (1998), pp. 3643–3646, https://doi.org/10.1103/PhysRevLett.81.3643.
29.
M. Bossy and D. Talay, A stochastic particle method for the McKean-Vlasov and the Burgers equation, Math. Comp., 66 (1997), pp. 157–192.
30.
C. F. Daganzo, A finite difference approximation of the kinematic wave model of traffic flow, Transp. Res. B Methodol., 29 (1995), pp. 261–276, https://doi.org/10.1016/0191-2615(95)00004-W.
31.
Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, H. Araki, ed., Springer, Berlin, Heidelberg, 1975, pp. 420–422, https://doi.org/10.1007/BFb0013365.
32.
B. Eckhardt, E. Ott, S. H. Strogatz, D. M. Abrams, and A. McRobie, Modeling walker synchronization on the Millennium Bridge, Phys. Rev. E, 75 (2007), 021110, https://doi.org/10.1103/PhysRevE.75.021110.
33.
B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), pp. 571–585, https://doi.org/10.1007/BF00164052.
34.
J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford Math. Mongr., Oxford University Press, Oxford, 2007, https://doi.org/10.1093/acprof:oso/9780198569039.001.0001.
35.
D. G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems, Springer, Berlin, Heidelberg, 1986, pp. 1–46, https://doi.org/10.1007/BFb0072687.
36.
O. A. Arqub and N. Shawagfeh, Application of reproducing kernel algorithm for solving Dirichlet time-fractional diffusion-Gordon types equations in porous media, J. Porous Media, 22 (2019), pp. 411–434, https://doi.org/10.1615/JPorMedia.2019028970.
37.
B. Gilding and L. Peletier, On a class of similarity solutions of the porous media equation, J. Math. Anal. Appl., 55 (1976), pp. 351–364, https://doi.org/10.1016/0022-247X(76)90166-9.
38.
J. U. Kim, On the stochastic porous medium equation, J. Differential Equations, 220 (2006), pp. 163–194, https://doi.org/10.1016/j.jde.2005.02.006.
39.
A. De Gregorio, Stochastic models associated to a nonlocal porous medium equation, Mod. Stoch. Theory Appl., 5 (2018), pp. 457–470, https://doi.org/10.15559/18-VMSTA112.
40.
C. Gardiner, Stochastic Methods. A Handbook for the Natural and Social Sciences, 4th ed., Springer, Berlin, 2009.
41.
C. M. Topaz, A. L. Bertozzi, and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), pp. 1601–1623, https://doi.org/10.1007/s11538-006-9088-6.
42.
F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), pp. 852–862, https://doi.org/10.1109/TAC.2007.895842.

Information & Authors

Information

Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1000 - 1024
ISSN (online): 1095-712X

History

Submitted: 3 May 2022
Accepted: 20 December 2022
Published online: 23 May 2023

Keywords

  1. kinematic flows
  2. interacting particle systems
  3. traffic modelling
  4. organism swarming
  5. anomalous sub diffusion

MSC codes

  1. 60H15
  2. 60H30
  3. 58J65
  4. 92C35
  5. 76Z05

Authors

Affiliations

Department of Mathematical Sciences, The University of Bath, Bath BA2 7AY, UK.
Department of Mathematical Sciences, The University of Bath, Bath BA2 7AY, UK.
Department of Mathematical Sciences, The University of Bath, Bath BA2 7AY, UK.

Funding Information

Funding: This work was funded by EPSRC.

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item

View Options

View options

PDF

View PDF

Full Text

View Full Text

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.