Applied Mathematics and Mechanics

, Volume 39, Issue 1, pp 63–82 | Cite as

A note on hydrodynamics from dissipative particle dynamics

  • X. BianEmail author
  • Z. Li
  • N. A. Adams


We calculate current correlation functions (CCFs) of dissipative particle dynamics (DPD) and compare them with results of molecular dynamics (MD) and solutions of linearized hydrodynamic equations. In particular, we consider three versions of DPD, the empirical/classical DPD, coarse-grained (CG) DPD with radial-direction interactions only and full (radial, transversal, and rotational) interactions between particles. To facilitate quantitative discussions, we consider specifically a star-polymer melt system at a moderate density. For bonded molecules, it is straightforward to define the CG variables and to further derive CG force fields for DPD within the framework of the Mori-Zwanzig formalism. For both transversal and longitudinal current correlation functions (TCCFs and LCCFs), we observe that results of MD, DPD, and hydrodynamic solutions agree with each other at the continuum limit. Below the continuum limit to certain length scales, results of MD deviate significantly from hydrodynamic solutions, whereas results of both empirical and CG DPD resemble those of MD. This indicates that the DPD method with Markovian force laws possibly has a larger applicability than the continuum description of a Newtonian fluid. This is worth being explored further to represent generalized hydrodynamics.

Key words

dissipative particle dynamics (DPD) fluctuating hydrodynamics molecular dynamics (MD) coarse-graining Mori-Zwanzig projection 

Chinese Library Classification


2010 Mathematics Subject Classification

82-08 82C31 76M28 


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Z. LI acknowledges funding support of the U. S. Army Research Laboratory with Cooperative Agreement No. W911NF-12-2-0023.


  1. [1]
    Landau, L. D. and Lifshitz, E. M. Fluid Mechanics. Vol. 6 Course of Theoretical Physics, Pergamon Press, Oxford (1959)Google Scholar
  2. [2]
    Batchelor, G. K. An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge (1967)zbMATHGoogle Scholar
  3. [3]
    Noid, W. G. Perspective: coarse-grained models for biomolecular systems. Journal of Chemical Physics, 139(9), 090901 (2013)CrossRefGoogle Scholar
  4. [4]
    Succi, S. The lattice Boltzmann equation: for fluid dynamics and beyond. Numerical Mathematics and Scientific Computations, Oxford University Press, Oxford (2001)zbMATHGoogle Scholar
  5. [5]
    Dünweg, B. and Ladd, A. J. C. Lattice Boltzmann Simulations of Soft Matter Systems. A Advanced Computer Simulation Approaches for Soft Matter Sciences III (eds. Holm, C. and Kremer, K.), Volume 221 of Advances in Polymer Science, Springer Berlin Heidelberg, Berlin, 89–166 (2009)CrossRefGoogle Scholar
  6. [6]
    Hoogerbrugge, P. J. and Koelman, J. M. V. A. Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhysics Letters, 19(3), 155–160 (1992)CrossRefGoogle Scholar
  7. [7]
    Espa˜nol, P. and Warren, P. Statistical mechanics of dissipative particle dynamics. Europhysics Letters, 30(4), 191–196 (1995)CrossRefGoogle Scholar
  8. [8]
    Malevanets, A. and Kapral, R. Mesoscopic model for solvent dynamics. Journal of Chemical Physics, 110(17), 8605–8613 (1999)CrossRefGoogle Scholar
  9. [9]
    Gompper, G., Ihle, T., Kroll, D. M., and Winkler, R. G. Multi-particle collision dynamics: a particle-based mesoscale simulation approach to the hydrodynamics of complex fluids. Advanced Computer Simulation Approaches for Soft Matter Sciences III (eds. Holm, C, and Kremer, K.) volume 221 of Advances in Polymer Science, Springer Berlin Heidelberg, Berlin, 1–87 (2009)Google Scholar
  10. [10]
    Lyubartsev, A. P. and Laaksonen, A. Calculation of effective interaction potentials from radial distribution functions: a reverse monte carlo approach. Physical Review E, 52, 3730–3737 (1995)CrossRefGoogle Scholar
  11. [11]
    Reith, D., Pütz, M., and Müller-Plathe, F. Deriving effective mesoscale potentials from atomistic simulations. Journal of Computational Chemistry, 24(13), 1624–1636 (2003)CrossRefGoogle Scholar
  12. [12]
    Ercolessi, F. and Adams, J. B. Interatomic potentials from first-principles calculations: the forcematching method. Europhysics Letters, 26(8), 9306054 (1994)CrossRefGoogle Scholar
  13. [13]
    Izvekov, S. and Voth, G. A. Multiscale coarse graining of liquid-state systems. Journal of Chemical Physics, 123(13), 134105 (2005)CrossRefGoogle Scholar
  14. [14]
    Shell, M. S. The relative entropy is fundamental to multiscale and inverse thermodynamic problems. Journal of Chemical Physics, 129(14), 144108 (2008)CrossRefGoogle Scholar
  15. [15]
    Zwanzig, R. Ensemble method in the theory of irreversibility. Journal of Chemical Physics, 33(5), 1338–1341 (1960)MathSciNetCrossRefGoogle Scholar
  16. [16]
    Mori, H. Transport, collective motion, and Brownian motion. Progress of Theoretical Physics, 33, 423–455 (1965)CrossRefzbMATHGoogle Scholar
  17. [17]
    Koelman, J. M. V. A. and Hoogerbrugge, P. J. Dynamic simulations of hard-sphere suspensions under steady shear. Europhysics Letters, 21, 363–368 (1993)CrossRefGoogle Scholar
  18. [18]
    Groot, R. D. and Warren, P. B. Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. Journal of Chemical Physics, 107(11), 4423–4435 (1997)CrossRefGoogle Scholar
  19. [19]
    Yamamoto, S., Maruyama, Y., and Hyodo, S. Dissipative particle dynamics study of spontaneous vesicle formation of amphiphilic molecules. Journal of Chemical Physics, 116(13), 5842–5849 (2002)CrossRefGoogle Scholar
  20. [20]
    Fan, X., Phan-Thien, N., Yong, N. T., Wu, X., and Xu, D. Microchannel flow of a macromolecular suspension. Physics of Fluids, 15, 11–21 (2003)CrossRefzbMATHGoogle Scholar
  21. [21]
    Pivkin, I. V. and Karniadakis, G. E. Accurate coarse-grained modeling of red blood cells. Physical Review Letters, 101, 118105 (2008)CrossRefGoogle Scholar
  22. [22]
    Fedosov, D. A., Caswell, B., and Karniadakis, G. E. A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophysical Journal, 98, 2215–2225 (2010)CrossRefGoogle Scholar
  23. [23]
    Espa˜nol, P. and Warren, P. B. Perspective: dissipative particle dynamics. Journal of Chemical Physics, 146(15), 150901 (2017)CrossRefGoogle Scholar
  24. [24]
    Li, Z., Li, X., Bian, X., Deng, M., Tang, Y. H., Caswell, B., and Karniadakis, G. E. Dissipative particle dynamics: foundation, evolution, implementation, and applications. Particles in Flows, Springer, Berlin (2017)Google Scholar
  25. [25]
    Kinjo, T. and Hyodo, S. Equation of motion for coarse-grained simulation based on microscopic description. Physical Review E, 75, 051109 (2007)CrossRefGoogle Scholar
  26. [26]
    Lei, H., Caswell, B., and Karniadakis, G. E. Direct construction of mesoscopic models from microscopic simulations. Physical Review E, 81, 026704 (2010)CrossRefGoogle Scholar
  27. [27]
    Hijón, C., Espa˜nol, P., Vanden-Eijnden, E., and Delgado-Buscalioni, R. Mori-Zwanzig formalism as a practical computational tool. Faraday Discuss, 144, 301–322 (2010)CrossRefGoogle Scholar
  28. [28]
    Li, Z., Bian, X. Caswell, B., and Karniadakis, G. E. Construction of dissipative particle dynamics models for complex fluids via the Mori-Zwanzig formulation. Soft Matter, 10, 8659–8672 (2014)CrossRefGoogle Scholar
  29. [29]
    Espa˜nol, P. and Revenga, M. Smoothed dissipative particle dynamics. Physical Review E, 67(2), 026705 (2003)CrossRefGoogle Scholar
  30. [30]
    Vázquez-Quesada, A., Ellero, M., and Espa˜nol, P. Smoothed particle hydrodynamic model for viscoelastic fluids with thermal fluctuations. Physical Review E, 79(5), 056707 (2009)MathSciNetCrossRefGoogle Scholar
  31. [31]
    Marsh, C. A., Backx, G., and Ernst, M. H. Fokker-Planck-Boltzmann equation for dissipative particle dynamics. Europhysical Letters, 38(6), 411–415 (1997)CrossRefGoogle Scholar
  32. [32]
    Espa˜nol, P. Hydrodynamics from dissipative particle dynamics. Physical Review E, 52(2), 1734–1742 (1995)MathSciNetCrossRefGoogle Scholar
  33. [33]
    Ripoll, M., Ernst, M. H., and Espa˜nol, P. Large scale and mesoscopic hydrodynamics for dissipative particle dynamics. Journal of Chemical Physics, 115, 7271–7284 (2001)CrossRefGoogle Scholar
  34. [34]
    Bian, X., Li, Z., Deng, M., and Karniadakis, G. E. Fluctuating hydrodynamics in periodic domains and heterogeneous adjacent multidomains: thermal equilibrium. Physical Review E, 92, 053302 (2015)CrossRefGoogle Scholar
  35. [35]
    Azarnykh, D., Litvinov, S., Bian, X., and Adams, N. A. Determination of macroscopic transport coefficients of a dissipative particle dynamics solvent. Physical Review E, 93, 013302 (2016)CrossRefGoogle Scholar
  36. [36]
    Vázquez-Quesada, A., Ellero, M., and Espa˜nol, P. Consistent scaling of thermal fluctuations in smoothed dissipative particle dynamics. Journal of Chemical Physics, 130(3), 034901 (2009)CrossRefGoogle Scholar
  37. [37]
    Hansen, J. P. and McDonald, I. R. Theory of Simple Liquids, 4th ed., Elsevier, Burlington (2013)zbMATHGoogle Scholar
  38. [38]
    Boon, J. P. and Yip, S. Molecular Hydrodynamics, Dover Publications, New York (1991)Google Scholar
  39. [39]
    Zwanzig, R. Memory effects in irreversible thermodynamics. Physical Review, 124, 983–992 (1961)CrossRefzbMATHGoogle Scholar
  40. [40]
    Kawasaki, K. Simple derivations of generalized linear and nonlinear Langevin equations. Journal of Physics A: Mathematical Nuclear and General, 6, 1289–1295 (1973)MathSciNetCrossRefGoogle Scholar
  41. [41]
    Nordholm, S. and Zwanzig, R. A systematic derivation of exact generalized Brownian motion theory. Journal of Statistical Physics, 13(4), 347–371 (1975)MathSciNetCrossRefGoogle Scholar
  42. [42]
    Weeks, J. D., Chandler, D., and Andersen, H. C. Role of repulsive forces in determining the equilibrium structure of simple liquids. Journal of Chemical Physics, 54(12), 5237–5247 (1971)CrossRefGoogle Scholar
  43. [43]
    Kremer, K. and Grest, G. S. Dynamics of entangled linear polymer melts: a molecular-dynamics simulation. Journal of Chemical Physics, 92(8), 5057–5086 (1990)CrossRefGoogle Scholar
  44. [44]
    Tuckerman, M. E. Statistical Mechanics: Theory and Molecular Simulation, Oxford University Press, Oxford (2010)zbMATHGoogle Scholar
  45. [45]
    Backer, J. A., Lowe, C. P., Hoefsloot, H. C. J., and Iedema, P. D. Poiseuille flow to measure the viscosity of particle model fluids. Journal of Chemical Physics, 122, 154503 (2005)CrossRefGoogle Scholar
  46. [46]
    Li, Z., Bian, X., Yang, X., and Karniadakis, G. E. A comparative study of coarse-graining methods for polymeric fluids: Mori-Zwanzig vs. iterative Boltzmann inversion vs. stochastic parametric optimization. Journal of Chemical Physics, 145(4), 044102 (2016)CrossRefGoogle Scholar
  47. [47]
    Lei, H., Yang, X., Li, Z., and Karniadakis, G. E. Systematic parameter inference in stochastic mesoscopic modeling. Journal of Computational Physics, 330, 571–593 (2017)MathSciNetCrossRefGoogle Scholar
  48. [48]
    Kirkwood, J. G. The statistical mechanical theory of transport processes, i. general theory. Journal of Chemical Physics, 14(3), 180–201 (1946)CrossRefGoogle Scholar
  49. [49]
    Berne, B. J. and Pecora, R. Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics, Dover Publications, New York (2000)Google Scholar
  50. [50]
    Espa˜nol, P. Fluid particle model. Physical Review E, 57(3), 2930–2948 (1998)CrossRefGoogle Scholar
  51. [51]
    Berne, B. J. Statistical Mechanics, Part B: Time-dependet Process, chapter 5, Plenum Press, New York, 233–257 (1977)CrossRefGoogle Scholar
  52. [52]
    Kreyszig, E. Advanced Engineering Mathematics, 10th ed., John Wiley & Sons, Hoboken (2011)zbMATHGoogle Scholar
  53. [53]
    Palmer, B. J. Transverse-current autocorrelation-function calculations of the shear viscosity for molecular liquids. Physical Review E, 49, 359–366 (1994)CrossRefGoogle Scholar
  54. [54]
    Li, Z., Bian, X., Li, X., and Karniadakis, G. E. Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism. Journal of Chemical Physics, 143(24), 243128 (2015)CrossRefGoogle Scholar
  55. [55]
    Li, Z., Lee, H., Darve, E., and Karniadakis, G. E. Computing the non-Markovian coarse-grained interactions derived from the Mori-Zwanzig formalism in molecular systems: application to polymer melts. Journal of Chemical Physics, 146, 014104 (2017)CrossRefGoogle Scholar
  56. [56]
    Lei, H., Baker, N. A., and Li, X. Data-driven parameterization of the generalized Langevin equation. Proceedings of the National Academy of Sciences of the United States of America, 113(50), 14183–14188 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Chair of Aerodynamics and Fluid Mechanics, Department of Mechanical EngineeringTechnical University of MunichMünchenGermany
  2. 2.Division of Applied MathematicsBrown UniversityRhode IslandUSA

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