- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

When velocity autocorrelations mirror force autocorrelations: Exact noise-cancellation in interacting Brownian systems

Anton Lüders1,*, Suvendu Mandal2, and Thomas Franosch1,†

  • *Contact author: anton.lueders@uibk.ac.at
  • †Contact author: thomas.franosch@uibk.ac.at
PDF

Phys. Rev. E 113, 035305 – Published 9 March, 2026

DOI: https://doi.org/10.1103/8hrb-bkv9

Abstract

Resolving the mean-squared displacement (MSD) and velocity autocorrelation function (VACF) of interacting Brownian particles remains a central challenge in simulations of soft-matter systems, especially at low densities where particle-particle interactions are sparse and signals are dominated by thermal noise. A recently proposed noise-cancellation (NC) algorithm [Mandal et al., Phys. Rev. Lett. 123, 168001 (2019)] addressed this by decomposing particle trajectories into two components: free Brownian motion and interaction-induced displacements. The NC approximation enhances signal clarity by neglecting cross-correlations between the total particle displacements and the extracted interaction-induced contributions of the trajectories—an assumption that has so far lacked rigorous theoretical justification. In this work, we establish an exact theoretical relation between the VACF, the force autocorrelation function (FACF) characterizing the interaction-induced contributions, and these cross-correlations, which is valid for Brownian systems. We show that in thermal equilibrium, the cross-correlations vanish for Brownian systems because the VACF is strictly proportional to the negative FACF, which establishes the NC algorithm as an exact method. In contrast, for Brownian nonequilibrium systems, the cross-correlations remain finite, providing a direct fingerprint of nonequilibrium physics in such systems and a criterion to distinguish equilibrium from nonequilibrium states. Here, suitable corrections must be applied for the NC method to remain accurate. Our results expand the scope of the NC algorithm to a broad range of soft-matter systems in and out of equilibrium, where it has the potential to strongly enhance the resolution of VACF data obtained through simulations in future studies.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (72)

  1. M. Mouas, J.-G. Gasser, S. Hellal, B. Grosdidier, A. Makradi, and S. Belouettar, Diffusion and viscosity of liquid tin: Green-Kubo relationship-based calculations from molecular dynamics simulations, J. Chem. Phys. 136, 094501 (2012).
  2. N. Heptner and J. Dzubiella, Equilibrium structure and fluctuations of suspensions of colloidal dumbbells, Mol. Phys. 113, 2523 (2015).
  3. J. Hansen and I. McDonald, Theory of Simple Liquids: With Applications to Soft Matter (Academic Press, Oxford, UK, 2013).
  4. R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics (Springer Science & Business Media, Berlin, Germany, 2012), Vol. 31.
  5. G. Williams, Use of the dipole correlation function in dielectric relaxation, Chem. Rev. 72, 55 (1972).
  6. B. Cichocki and B. Felderhof, Slow dynamics of linear relaxation systems, Physica A 211, 165 (1994).
  7. B. J. Alder and T. E. Wainwright, Decay of the velocity autocorrelation function, Phys. Rev. A 1, 18 (1970).
  8. M. H. Ernst, E. H. Hauge, and J. M. J. van Leeuwen, Asymptotic time behavior of correlation functions, Phys. Rev. Lett. 25, 1254 (1970).
  9. J. R. Dorfman and E. G. D. Cohen, Velocity correlation functions in two and three dimensions, Phys. Rev. Lett. 25, 1257 (1970).
  10. B. Lukić, S. Jeney, C. Tischer, A. J. Kulik, L. Forró, and E.-L. Florin, Direct observation of nondiffusive motion of a Brownian particle, Phys. Rev. Lett. 95, 160601 (2005).
  11. S. Jeney, B. Lukić, J. A. Kraus, T. Franosch, and L. Forró, Anisotropic memory effects in confined colloidal diffusion, Phys. Rev. Lett. 100, 240604 (2008).
  12. T. Franosch, M. Grimm, M. Belushkin, F. Mor, G. Foffi, L. Forró, and S. Jeney, Resonances arising from hydrodynamic memory in Brownian motion, Nature (London) 478, 85 (2011).
  13. R. Huang, I. Chavez, K. M. Taute, B. Lukić, S. Jeney, M. G. Raizen, and E.-L. Florin, Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid, Nat. Phys. 7, 576 (2011).
  14. S. Mandal, L. Schrack, H. Löwen, M. Sperl, and T. Franosch, Persistent anti-correlations in Brownian dynamics simulations of dense colloidal suspensions revealed by noise suppression, Phys. Rev. Lett. 123, 168001 (2019).
  15. S. Hanna, W. Hess, and R. Klein, The velocity autocorrelation function of an overdamped Brownian system with hard-core intraction, J. Phys. A: Math. Gen. 14, L493 (1981).
  16. B. J. Ackerson and L. Fleishman, Correlations for dilute hard core suspensions, J. Chem. Phys. 76, 2675 (1982).
  17. B. U. Felderhof and R. B. Jones, Cluster expansion of the diffusion kernel of a suspension of interacting Brownian particles, Physica A 121, 329 (1983).
  18. D. Frenkel, Velocity auto-correlation functions in a 2D lattice Lorentz gas: Comparison of theory and computer simulation, Phys. Lett. A 121, 385 (1987).
  19. J. Ramírez, S. K. Sukumaran, B. Vorselaars, and A. E. Likhtman, Efficient on the fly calculation of time correlation functions in computer simulations, J. Chem. Phys. 133, 154103 (2010).
  20. H. C. Oettinger, Variance reduced Brownian dynamics simulations, Macromolecules 27, 3415 (1994).
  21. M. Melchior and H. C. Öttinger, Variance reduced simulations of polymer dynamics, J. Chem. Phys. 105, 3316 (1996).
  22. N. J. Wagner and H. C. Öttinger, Accurate simulation of linear viscoelastic properties by variance reduction through the use of control variates, J. Rheol. 41, 757 (1997).
  23. U. Siems and P. Nielaba, Brownian dynamics simulations of model colloids in channel geometries and external fields, J. Phys.: Conf. Ser. 1012, 012011 (2018).
  24. D. Dubbeldam, D. C. Ford, D. E. Ellis, and R. Q. Snurr, A new perspective on the order-n algorithm for computing correlation functions, Mol. Simul. 35, 1084 (2009).
  25. R. Rusch, T. Franosch, and G. Jung, Noise-cancellation algorithm for simulations of Brownian particles, Phys. Rev. E 109, 015303 (2024).
  26. R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, New York, 2001).
  27. C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Rev. Mod. Phys. 88, 045006 (2016).
  28. M. Isele, K. Hofmann, A. Erbe, P. Leiderer, and P. Nielaba, Lane formation of colloidal particles driven in parallel by gravity, Phys. Rev. E 108, 034607 (2023).
  29. D. R. Foss and J. F. Brady, Brownian dynamics simulation of hard-sphere colloidal dispersions, J. Rheol. 44, 629 (2000).
  30. R. B. Jones, in Rotational Diffusion in Dispersive Media, edited by J. J. Telega, Lecture Notes, Vol. 8, (Polish Academy of Sciences Publishing, Warsaw, 2003).
  31. E. Kalz, H. D. Vuijk, J.-U. Sommer, R. Metzler, and A. Sharma, Oscillatory force autocorrelations in equilibrium odd-diffusive systems, Phys. Rev. Lett. 132, 057102 (2024).
  32. S. Dal Cengio, D. Levis, and I. Pagonabarraga, Linear response theory and Green-Kubo relations for active matter, Phys. Rev. Lett. 123, 238003 (2019).
  33. A. Sharma and J. M. Brader, Communication: Green-Kubo approach to the average swim speed in active Brownian systems, J. Chem. Phys. 145, 161101 (2016).
  34. T. Åkesson and B. Jönsson, Brownian dynamics simulation of interacting particles, Mol. Phys. 54, 369 (1985).
  35. W. Härtl, H. Versmold, U. Wittig, and P. Linse, Structure and dynamics of polymer colloid suspensions from dynamic light scattering and Brownian dynamics simulation, J. Chem. Phys. 97, 7797 (1992).
  36. D. M. Heyes and A. C. Brańka, Molecular and Brownian dynamics simulations of self-diffusion in inverse power fluids, Phys. Chem. Liq. 28, 95 (1994).
  37. D. M. Heyes, Mean-field-hydrodynamics Brownian dynamics simulations of viscosity and self-diffusion of near-hard-sphere colloidal liquids, J. Phys.: Condens. Matter 7, 8857 (1995).
  38. D. M. Heyes, Brownian dynamics simulations of self and collective diffusion of near hard sphere colloidal liquids: Inclusion of many-body hydrodynamics, Mol. Phys. 87, 287 (1996).
  39. D. M. Heyes and A. C. Brańka, More efficient Brownian dynamics algorithms, Mol. Phys. 98, 1949 (2000).
  40. A. C. Brańka and D. M. Heyes, The effects of particle softness on the dynamics of molecular and colloidal systems, Mol. Phys. 103, 2359 (2005).
  41. M. Fixman, Inclusion of hydrodynamic interaction in polymer dynamical simulations, Macromolecules 14, 1710 (1981).
  42. D. M. Heyes, J. G. Powles, and G. Rickayzen, The velocity autocorrelation function and self-diffusion coefficient of fluids with steeply repulsive potentials, Mol. Phys. 100, 595 (2002).
  43. S. R. Rastogi, N. J. Wagner, and S. R. Lustig, Rheology, self‐diffusion, and microstructure of charged colloids under simple shear by massively parallel nonequilibrium Brownian dynamics, J. Chem. Phys. 104, 9234 (1996).
  44. B. Liu and B. Dünweg, Translational diffusion of polymer chains with excluded volume and hydrodynamic interactions by Brownian dynamics simulation, J. Chem. Phys. 118, 8061 (2003).
  45. D. M. Heyes and J. R. Melrose, Brownian dynamics simulations of model hard-sphere suspensions, J. Non-Newtonian Fluid Mech. 46, 1 (1993).
  46. R. N. Zia and J. F. Brady, Single-particle motion in colloids: Force-induced diffusion, J. Fluid Mech. 658, 188 (2010).
  47. R. N. Zia and J. F. Brady, Microviscosity, microdiffusivity, and normal stresses in colloidal dispersions, J. Rheol. 56, 1175 (2012).
  48. M. D. Rintoul and S. Torquato, Computer simulations of dense hard‐sphere systems, J. Chem. Phys. 105, 9258 (1996).
  49. A. M. Sunol and R. N. Zia, Confined Brownian suspensions: Equilibrium diffusion, thermodynamics, and rheology, J. Rheol. 67, 433 (2023).
  50. S. Marenne, J. F. Morris, D. R. Foss, and J. F. Brady, Unsteady shear flows of colloidal hard-sphere suspensions by dynamic simulation, J. Rheol. 61, 477 (2017).
  51. M. Gao, Free Ornstein–Uhlenbeck processes, J. Math. Anal. Appl. 322, 177 (2006).
  52. R. Goerlich, M. Li, S. Albert, G. Manfredi, P.-A. Hervieux, and C. Genet, Noise and ergodic properties of Brownian motion in an optical tweezer: Looking at regime crossovers in an Ornstein-Uhlenbeck process, Phys. Rev. E 103, 032132 (2021).
  53. F. Marchesoni and A. Taloni, Subdiffusion and long-time anticorrelations in a stochastic single file, Phys. Rev. Lett. 97, 106101 (2006).
  54. A. Taloni and F. Marchesoni, Single-file diffusion on a periodic substrate, Phys. Rev. Lett. 96, 020601 (2006).
  55. O. Flomenbom and A. Taloni, On single-file and less dense processes, Europhys. Lett. 83, 20004 (2008).
  56. A. Taloni and M. A. Lomholt, Langevin formulation for single-file diffusion, Phys. Rev. E 78, 051116 (2008).
  57. A. Taloni, O. Flomenbom, R. Castañeda-Priego, and F. Marchesoni, Single file dynamics in soft materials, Soft Matter 13, 1096 (2017).
  58. Q.-H. Wei, C. Bechinger, and P. Leiderer, Single-file diffusion of colloids in one-dimensional channels, Science 287, 625 (2000).
  59. M. Kollmann, Single-file diffusion of atomic and colloidal systems: Asymptotic laws, Phys. Rev. Lett. 90, 180602 (2003).
  60. C. Lutz, M. Kollmann, and C. Bechinger, Single-file diffusion of colloids in one-dimensional channels, Phys. Rev. Lett. 93, 026001 (2004).
  61. Y.-G. Tao, W. K. den Otter, J. T. Padding, J. K. G. Dhont, and W. J. Briels, Brownian dynamics simulations of the self- and collective rotational diffusion coefficients of rigid long thin rods, J. Chem. Phys. 122, 244903 (2005).
  62. H. Löwen, Inertial effects of self-propelled particles: From active Brownian to active Langevin motion, J. Chem. Phys. 152, 040901 (2020).
  63. M. Caraglio and T. Franosch, Analytic solution of an active Brownian particle in a harmonic well, Phys. Rev. Lett. 129, 158001 (2022).
  64. O. Granek, Y. Kafri, and J. Tailleur, Anomalous transport of tracers in active baths, Phys. Rev. Lett. 129, 038001 (2022).
  65. R. Ni, M. A. C. Stuart, and M. Dijkstra, Pushing the glass transition towards random close packing using self-propelled hard spheres, Nat. Commun. 4, 2704 (2013).
  66. S. Mandal, C. Kurzthaler, T. Franosch, and H. Löwen, Crowding-enhanced diffusion: An exact theory for highly entangled self-propelled stiff filaments, Phys. Rev. Lett. 125, 138002 (2020).
  67. S. Asghar, Q.-X. Pei, G. Volpe, and R. Ni, Efficient rare event sampling with unsupervised normalizing flows, Nat. Mach. Intell. 6, 1370 (2024).
  68. A. N. Lüders, S. Mandal, and T. Franosch, When velocity autocorrelations mirror force autocorrelations: Exact noise-cancellation in interacting Brownian systems (data) (1.0.0) [Data set] Universität Innsbruck (2026), https://doi.org/10.48323/pnv6f-fb682.
  69. The code of the performed simulations is available via GitHub: https://github.com/AntonLueders/FACF-based_NC_Algorithm. The corresponding permanent DOI is 10.5281/zenodo.18681193.
  70. D. L. Ermak, A computer simulation of charged particles in solution. I. Technique and equilibrium properties, J. Chem. Phys. 62, 4189 (1975).
  71. D. L. Ermak and J. A. McCammon, Brownian dynamics with hydrodynamic interactions, J. Chem. Phys. 69, 1352 (1978).
  72. J. K. G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, Amsterdam, The Netherlands, 1996).

Outline

Information

Phys. Rev. E 113, 035305– Published 9 March, 2026
  • Received 10 September 2025
  • Accepted 17 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation