In this paper we consider systems of weakly interacting particles driven by colored noise in a bistable potential, and we study the effect of the correlation time of the noise on the bifurcation diagram for the equilibrium states. We accomplish this by solving the corresponding McKean--Vlasov equation using a Hermite spectral method, and we verify our findings using Monte Carlo simulations of the particle system. We consider both Gaussian and non-Gaussian noise processes, and for each model of the noise we also study the behavior of the system in the small correlation time regime using perturbation theory. The spectral method that we develop in this paper can be used for solving linear and nonlinear, local and nonlocal (mean field) Fokker--Planck equations, without requiring that they have a gradient structure.


  1. McKean--Vlasov PDEs
  2. nonlocal Fokker--Planck equations
  3. interacting particles
  4. Desai--Zwanzig model
  5. colored noise
  6. Hermite spectral methods
  7. phase transitions

MSC codes

  1. 35Q70
  2. 35Q83
  3. 35Q84
  4. 65N35
  5. 65M70
  6. 82B26

Get full access to this article

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


A. Abdulle, G. A. Pavliotis, and U. Vaes, Spectral methods for multiscale stochastic differential equations, SIAM/ASA J. Uncertain. Quantif., 5 (2017), pp. 720--761, https://doi.org/10.1137/16M1094117.
S. Agmon, Lectures on Exponential Decay of Solutions of Second-order Elliptic Equations: Bounds on Eigenfunctions of $N$-body Schrödinger Operators, Math. Notes 29, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982.
E. L. Allgower and K. Georg, Introduction to numerical continuation methods, Classics Appl. Math. 45, SIAM, Philadelphia, 2003, https://doi.org/10.1137/1.9780898719154.
N. Bain and D. Bartolo, Critical mingling and universal correlations in model binary active liquids, Nat. Commun., 8 (2017), 15969.
J. Binney and S. Tremaine, Galactic Dynamics, Princeton Series in Astrophysics 20, Princeton University Press, Princeton, NJ, 2011.
G. Blankenship and G. C. Papanicolaou, Stability and control of stochastic systems with wide-band noise disturbances, \textupI, SIAM J. Appl. Math., 34 (1978), pp. 437--476, https://doi.org/10.1137/0134036.
J. A. Carrillo, A. Chertock, and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015), pp. 233--258, https://doi.org/10.4208/cicp.160214.010814a.
D. A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Statist. Phys., 31 (1983), pp. 29--85, https://doi.org/10.1007/BF01010922.
A. Dhooge, W. Govaerts, Y. A. Kuznetsov, W. Mestrom, A. M. Riet, and B. Sautois, MATCONT and CL MATCONT: Continuation toolboxes in MATLAB, Universiteit Gent, Belgium and Utrecht University, The Netherlands, 2006.
M. H. Duong and G. A. Pavliotis, Mean field limits for non-Markovian interacting particles: convergence to equilibrium, GENERIC formalism, asymptotic limits and phase transitions, Commun. Math. Sci., 16 (2018), pp. 2199--2230, https://doi.org/10.4310/CMS.2018.v16.n8.a7.
A. Durmus, A. Eberle, A. Guillin, and R. Zimmer, An Elementary Approach to Uniform in Time Propagation of Chaos, preprint, https://arxiv.org/abs/1805.11387, 2018.
F. Farkhooi and W. Stannat, A Complete Mean-field Theory for Dynamics of Binary Recurrent Neural Networks, preprint, https://arxiv.org/abs/1701.07128, 2017.
J. C. M. Fok, B. Guo, and T. Tang, Combined Hermite spectral-finite difference method for the Fokker-Planck equation, Math. Comp., 71 (2002), pp. 1497--1528, https://doi.org/10.1090/S0025-5718-01-01365-5.
J. Gagelman and H. Yserentant, A spectral method for Schrödinger equations with smooth confinement potentials, Numer. Math., 122 (2012), pp. 383--398, https://doi.org/10.1007/s00211-012-0458-8.
J. Garnier, G. Papanicolaou, and T.-W. Yang, Large deviations for a mean field model of systemic risk, SIAM J. Finan. Math., 4 (2013), pp. 151--184, https://doi.org/10.1137/12087387X.
J. Garnier, G. Papanicolaou, and T.-W. Yang, Consensus convergence with stochastic effects, Vietnam J. Math., 45 (2017), pp. 51--75, https://doi.org/10.1007/s10013-016-0190-2.
B. D. Goddard, A. Nold, N. Savva, G. A. Pavliotis, and S. Kalliadasis, General dynamical density functional theory for classical fluids, Phys. Rev. Lett., 109 (2012), 120603, https://doi.org/10.1103/PhysRevLett.109.120603.
B. D. Goddard, G. A. Pavliotis, and S. Kalliadasis, The overdamped limit of dynamic density functional theory: rigorous results, Multiscale Model. Simul., 10 (2012), pp. 633--663, https://doi.org/10.1137/110844659.
S. N. Gomes, S. Kalliadasis, G. A. Pavliotis, and P. Yatsyshin, Dynamics of the Desai-Zwanzig model in multiwell and random energy landscapes, Phys. Rev. E, 99 (2019), 032109, https://doi.org/10.1103/PhysRevE.99.032109.
S. N. Gomes and G. A. Pavliotis, Mean field limits for interacting diffusions in a two-scale potential, J. Nonlinear Sci., 28 (2018), pp. 905--941, https://doi.org/10.1007/s00332-017-9433-y.
S. N. Gomes, G. A. Pavliotis, and U. Vaes, Mean-field Limits for Interacting Diffusions with Colored Noise: Phase Transitions and Spectral Numerical Methods, preprint, https://arxiv.org/abs/1904.05973, 2020.
P. Hanggi and P. Jung, Colored noise in dynamical systems, Adv. in Chem. Phys., 89 (1995), pp. 239--326.
D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), pp. 525--546, https://doi.org/10.1137/S0036144500378302.
D. J. Higham, X. Mao, and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), pp. 1041--1063, https://doi.org/10.1137/S0036142901389530.
W. Horsthemke and R. Lefever, Noise-induced Transitions: Theory and Application in Physics, Chemistry, and Biology, Springer Ser. Synergetics 15, Springer-Verlag, Berlin, Heidelberg, 1984.
A. Igarashi, P. V. E. McClintock, and N. G. Stocks, Velocity spectrum for non-Markovian Brownian motion in a periodic potential, J. Statist. Phys., 66 (1992), pp. 1059--1070, https://doi.org/10.1007/BF01055716.
A. Igarashi and T. Munakata, Non-Markovian Brownian motion in a periodic potential, J. Phys. Soc. Jpn., 57 (1988), pp. 2439--2447.
B. Krauskopf, H. M. Osinga, and J. Galán-Vioque, eds., Numerical Continuation Methods for Dynamical Systems, Understanding Complex Systems, Springer, Dordrecht, 2007, https://doi.org/10.1007/978-1-4020-6356-5.
L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups, in Pure Appl. Math. (Boca Raton) 283, Chapman & Hall/CRC, Boca Raton, FL, 2007.
J. Lu, Y. Lu, and J. Nolen, Scaling Limit of the Stein Variational Gradient Descent Part I: The Mean Field Regime, preprint, https://arxiv.org/abs/1805.04035, 2018.
E. Luçon and W. Stannat, Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction, Ann. Appl. Probab., 26 (2016), pp. 3840--3909, https://doi.org/10.1214/16-AAP1194.
P. A. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: an interplay between physics and functional analysis, VI Workshop on Partial Differential Equations, Part II (Rio de Janeiro, 1999), Mat. Contemp., 19 (2000), pp. 1--29.
P. Monmarché, Long-time behaviour and propagation of chaos for mean field kinetic particles, Stochastic Process. Appl., 127 (2017), pp. 1721--1737, https://doi.org/10.1016/j.spa.2016.10.003.
S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), pp. 577--621, https://doi.org/10.1137/120901866.
K. Oelschläger, A martingale approach to the law of large numbers for weakly interacting stochastic processes, Ann. Probab., 12 (1984), pp. 458--479, www.jstor.org/stable/2243483.
G. A. Pavliotis, Stochastic Processes and Applications, Diffusion Processes, the Fokker-Planck and Langevin Equations, Texts Appl. Math. 60, Springer, New York, 2014, https://doi.org/10.1007/978-1-4939-1323-7.
G. A. Pavliotis and A. M. Stuart, Multiscale Methods, Averaging and Homogenization, Texts Appl. Math. 53, Springer, New York, 2008.
R. Pinnau, C. Totzeck, O. Tse, and S. Martin, A consensus-based model for global optimization and its mean-field limit, Math. Models Methods Appl. Sci., 27 (2017), pp. 183--204, https://doi.org/10.1142/S0218202517400061.
G. M. Rotskoff and E. Vanden-Eijnden, Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach, preprint, https://arxiv.org/abs/1805.00915, 2018.
J. Shen, T. Tang, and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Ser. Comput. Math. 41, Springer, Heidelberg, 2011, https://doi.org/10.1007/978-3-540-71041-7.
M. Shiino, Dynamical behavior of stochastic systems of infinitely many coupled nonlinear oscillators exhibiting phase transitions of mean-field type: H theorem on asymptotic approach to equilibrium and critical slowing down of order-parameter fluctuations, Phys. Rev. A, 36 (1987), pp. 2393--2412, https://doi.org/10.1103/PhysRevA.36.2393.
J. Sirignano and K. Spiliopoulos, Mean Field Analysis of Neural Networks: A Law of Large Numbers, preprint, https://arxiv.org/abs/1805.01053, 2018.
J. Tugaut, Phase transitions of McKean-Vlasov processes in double-wells landscape, Stochastics, 86 (2014), pp. 257--284, https://doi.org/10.1080/17442508.2013.775287.
U. Vaes, Topics in Multiscale Modelling: Numerical Analysis and Applications, Ph.D. thesis, Department of Mathematics, Imperial College London, London, UK, 2019.

Information & Authors


Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 1343 - 1370
ISSN (online): 1540-3467


Submitted: 24 April 2019
Accepted: 28 April 2020
Published online: 2 September 2020


  1. McKean--Vlasov PDEs
  2. nonlocal Fokker--Planck equations
  3. interacting particles
  4. Desai--Zwanzig model
  5. colored noise
  6. Hermite spectral methods
  7. phase transitions

MSC codes

  1. 35Q70
  2. 35Q83
  3. 35Q84
  4. 65N35
  5. 65M70
  6. 82B26



Funding Information

Leverhulme Trust https://doi.org/10.13039/501100000275 : ECF-2018-536
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/P031587/1, EP/L024926/1, EP/L020564/1, EP/K034154/1

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


View PDF







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.