Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid


At timescales once deemed immeasurably small by Einstein, the random movement of Brownian particles in a liquid is expected to be replaced by ballistic motion. So far, an experimental verification of this prediction has been out of reach due to a lack of instrumentation fast and precise enough to capture this motion. Here we report the observation of the Brownian motion of a single particle in an optical trap with 75 MHz bandwidth and sub-ångström spatial precision and the determination of the particle’s velocity autocorrelation function. Our observation is the first measurement of ballistic Brownian motion of a particle in a liquid. The data are in excellent agreement with theoretical predictions taking into account the inertia of the particle and hydrodynamic memory effects.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Schematic diagram of the experiment.
Figure 2: Example MSD for silica particles 1 μm and 2.5 μm in diameter.
Figure 3: Ballistic regime.
Figure 4: Experimental VACF and theoretical description.


  1. 1

    Brown, R. A brief account of microscopical observations made in the months of June, July and August, 1827 on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Phil. Mag. 4, 161–173 (1828).

    Article  Google Scholar 

  2. 2

    Einstein, A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 322, 549–560 (1905).

    Article  Google Scholar 

  3. 3

    Frey, E. & Kroy, K. Brownian motion: A paradigm of soft matter and biological physics. Ann. Phys. 14, 20–50 (2005).

    Article  Google Scholar 

  4. 4

    Black, F. & Scholes, M. S. The pricing of options and corporate liabilities. J. Political Econ. 81, 637–654 (1973).

    MathSciNet  Article  Google Scholar 

  5. 5

    Smoluchowski, M. Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Ann. Phys. 21, 756–780 (1906).

    Article  Google Scholar 

  6. 6

    Einstein, A. Zur Theorie der Brownschen Bewegung. Ann. Phys. 324, 371–381 (1906).

    Article  Google Scholar 

  7. 7

    Langevin, P. Sur la théorie du mouvement brownien. C.R. Acad. Sci. Paris 146, 530–533 (1908).

    MATH  Google Scholar 

  8. 8

    Rahman, A. Correlations in the motion of atoms in liquid argon. Phys. Rev. 136, A404–A411 (1964).

    Article  Google Scholar 

  9. 9

    Rahman, A. Liquid structure and self-diffusion. J. Chem. Phys. 45, 2585–2592 (1964).

    ADS  Article  Google Scholar 

  10. 10

    Alder, B. J. & Wainwright, T. E. Velocity autocorrelations for hard spheres. Phys. Rev. Lett. 18, 988–990 (1967).

    ADS  Article  Google Scholar 

  11. 11

    Alder, B. J. & Wainwright, T. E. Decay of the velocity autocorrelation function. Phys. Rev. A 1, 18–21 (1970).

    ADS  Article  Google Scholar 

  12. 12

    Paul, G. L. & Pusey, P. N. Observation of a long-time tail in Brownian motion. J. Phys. A. 14, 3301–3327 (1981).

    ADS  Article  Google Scholar 

  13. 13

    Weitz, D. A., Pine, D. J., Pusey, P. N. & Tough, R. J. A. Nondiffusive Brownian motion studied by diffusing-wave spectroscopy. Phys. Rev. Lett. 63, 1747–1750 (1989).

    ADS  Article  Google Scholar 

  14. 14

    Vladimirsky, V. & Terletzky, Y. A. Hydrodynamical theory of translational Brownian motion. Zh. Eksp. Teor. Fiz. 15, 258–263 (1945).

    MathSciNet  Google Scholar 

  15. 15

    Hinch, E. J. Application of the Langevin equation to fluid suspensions. J. Fluid Mech. 72, 499–511 (1975).

    ADS  MathSciNet  Article  Google Scholar 

  16. 16

    Einstein, A. Theoretische Bemerkungen über die Brownsche Bewegung. Z. Elektrochem. 13, 41–42 (1907).

    Article  Google Scholar 

  17. 17

    Blum, J. et al. Measurement of the translational and rotational Brownian motion of individual particles in a rarefied gas. Phys. Rev. Lett. 97, 230601 (2006).

    ADS  Article  Google Scholar 

  18. 18

    Li, T., Kheifets, S., Medellin, D. & Raizen, M. G. Measurement of the instantaneous velocity of a Brownian particle. Science 328, 1673–1675 (2010).

    ADS  Article  Google Scholar 

  19. 19

    Uhlenbeck, G. E. & Ornstein, L. S. On the theory of Brownian motion. Phys. Rev. 36, 823–841 (1930).

    ADS  Article  Google Scholar 

  20. 20

    Lukić, B. et al. Direct observation of nondiffusive motion of a Brownian particle. Phys. Rev. Lett. 95, 160601 (2005).

    ADS  Article  Google Scholar 

  21. 21

    Jeney, S., Lukić, B., Kraus, J. A., Franosch, T. & Forró, L. Anisotropic memory effects in colloidal diffusion. Phys. Rev. Lett. 100, 240604 (2008).

    ADS  Article  Google Scholar 

  22. 22

    Zhu, J. X., Durian, D. J., Muller, J., Weitz, D. A. & Pine, D. J. Scaling of transient hydrodynamic interactions in concentrated suspensions. Phys. Rev. Lett. 68, 2559–2562 (1992).

    ADS  Article  Google Scholar 

  23. 23

    Kao, M. H., Yodh, A. G. & Pine, D. J. Observation of Brownian-motion on the timescale of hydrodynamic interactions. Phys. Rev. Lett. 70, 242–245 (1993).

    ADS  Article  Google Scholar 

  24. 24

    Henderson, S., Mitchell, S. & Bartlett, P. Propagation of hydrodynamic interactions in colloidal suspensions. Phys. Rev. Lett. 88, 088302 (2002).

    ADS  Article  Google Scholar 

  25. 25

    Liverpool, T. B. & MacKintosh, F. C. Inertial effects in the response of viscous and viscoelastic fluids. Phys. Rev. Lett. 95, 208303 (2005).

    ADS  Article  Google Scholar 

  26. 26

    Atakhorrami, M., Koenderink, G. H., Schmidt, C. F. & MacKintosh, F. C. Short-time inertial response of viscoelastic fluids: Observation of vortex propagation. Phys. Rev. Lett. 95, 208302 (2005).

    ADS  Article  Google Scholar 

  27. 27

    Neuman, K. C. & Block, S. M. Optical trapping. Rev. Sci. Instrum. 75, 2787–2809 (2004).

    ADS  Article  Google Scholar 

  28. 28

    Chavez, I., Huang, R. X., Henderson, K., Florin, E-L. & Raizen, M. G. Development of a fast position-sensitive laser beam detector. Rev. Sci. Instrum 79, 105104 (2008).

    ADS  Article  Google Scholar 

  29. 29

    Gittes, F. & Schmidt, C. F. Interference model for back-focal-plane displacement detection in optical tweezers. Opt. Lett. 23, 7–9 (1998).

    ADS  Article  Google Scholar 

  30. 30

    Pralle, A., Prummer, M., Florin, E.L., Stelzer, E. H. K. & Horber, J. K. H. Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light. Microsc. Res. Tech. 44, 378–386 (1999).

    Article  Google Scholar 

  31. 31

    Clercx, H. J. H. & Schram, P. P. J. M. Brownian particles in shear flow and harmonic potentials: A study of long-time tails. Phys. Rev. A 46, 1942–1950 (1992).

    ADS  MathSciNet  Article  Google Scholar 

  32. 32

    Giterman, M. Sh. & Gertsenshtein, M. E. Theory of the Brownian motion and the possibilities of using it for the study of the critical state of a pure substance. Sov. Phys. JETP 23 (4), 722–728 (1966).

    ADS  Google Scholar 

  33. 33

    Zwanzig, R. & Bixon, M. Compressibility effects in the hydrodynamic theory of Brownian motion. J. Fluid. Mech. 69, 21–25 (1975).

    ADS  Article  Google Scholar 

  34. 34

    Flyvbjerg, H. & Petersen, H. G. Error estimates on average of correlated data. J. Chem. Phys. 91, 461–466 (1989).

    ADS  MathSciNet  Article  Google Scholar 

Download references


This research was supported by NSF grants PHY-0647144 and DBI-0552094. S.J. and B.L. acknowledge support from the NCCR Nanoscale Science. M.G.R. acknowledges support from the Sid W. Richardson Foundation and the R. A. Welch Foundation, grant number F-1258. We thank V. Zyuzin for translating ref. 14.

Author information




R.H. and E-L.F. conceived the experiment. B.L. and S.J. contributed to the planning of the early experiments and provided an early version of the VACF analysis software. I.C., R.H., E-L.F. and M.G.R. developed, built and characterized the fast position detector, and incorporated it into the set-up. R.H. carried out the experiments in part assisted by I.C. R.H. analysed the data. R.H., K.M.T. and E-L.F. interpreted the data and wrote the manuscript. All authors discussed and commented on the final version of the manuscript.

Corresponding author

Correspondence to Ernst-Ludwig Florin.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Huang, R., Chavez, I., Taute, K. et al. Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid. Nature Phys 7, 576–580 (2011).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing