Statistical mechanics of a gas-fluidized particle


Characterization of the microscopic fluctuations in systems that are far from equilibrium is crucial for understanding the macroscopic response. One approach is to use an ‘effective temperature’—such a quantity has been invoked for chaotic fluids1,2, spin glasses3,4, glasses5,6 and colloids7,8, as well as non-thermal systems such as flowing granular materials9,10,11,12,13,14 and foams15. We therefore ask to what extent the concept of effective temperature is valid. Here we investigate this question experimentally in a simple system consisting of a sphere placed on a fine screen in an upward flow of gas; the sphere rolls because of the turbulence it generates in the gas stream. In contrast to many-particle systems, in which it is difficult to measure and predict fluctuations, our system has no particle–particle interactions and its dynamics can be captured fully by video imaging. Surprisingly, we find that the sphere behaves exactly like a harmonically bound brownian particle. The random driving force and frequency-dependent drag satisfy the fluctuation–dissipation relation, a cornerstone of statistical mechanics. The statistical mechanics of near-equilibrium systems is therefore unexpectedly useful for studying at least some classes of systems that are driven far from equilibrium.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Speed (a), radial position (b), and energy (c) probability functions for a sphere that rolls stochastically owing to the turbulence it generates in an upflow of gas.
Figure 2: Velocity autocorrelation for the stochastically rolling sphere.
Figure 3: Dimensionless amplitude (a), and phase (b), of the average response of the sphere to sinusoidal tilting at frequency ω.
Figure 4: Autocorrelation and distribution (inset) of the random force acting on the sphere.


  1. 1

    Hohenberg, P. C. & Shraiman, B. I. Chaotic behavior of an extended system. Physica D 37, 109–115 (1989)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2

    Egolf, D. A. Equilibrium regained: From nonequilibrium chaos to statistical mechanics. Science 287, 101–104 (2000)

    ADS  CAS  Article  Google Scholar 

  3. 3

    Cugliandolo, L. F. & Kurchan, J. Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model. Phys. Rev. Lett. 71, 173–176 (1993)

    ADS  CAS  Article  Google Scholar 

  4. 4

    Herisson, D. & Ocio, M. Fluctuation-dissipation ratio of a spin glass in the aging regime. Phys. Rev. Lett. 88, 257202 (2002)

    ADS  CAS  Article  Google Scholar 

  5. 5

    Grigera, T. S. & Israeloff, N. E. Observation of fluctuation-dissipation-theorem violations in a structural glass. Phys. Rev. Lett. 83, 5038–5041 (1999)

    ADS  CAS  Article  Google Scholar 

  6. 6

    Berthier, L. & Barrat, J. L. Shearing a glassy material: numerical tests of nonequilibrium mode-coupling approaches and experimental proposals. Phys. Rev. Lett. 89, 095702 (2002)

    ADS  Article  Google Scholar 

  7. 7

    Segre, P. N., Liu, F., Umbanhowar, P. B. & Weitz, D. A. An effective gravitational temperature for sedimentation. Nature 409, 594–597 (2001)

    ADS  CAS  Article  Google Scholar 

  8. 8

    Bellon, L., Ciliberto, S. & Laroche, C. Violation of the fluctuation-dissipation relation during the formation of a colloidal glass. Europhys. Lett. 53, 511–517 (2001)

    ADS  CAS  Article  Google Scholar 

  9. 9

    Mueth, D. M. et al. Signatures of granular microstructure in dense shear flows. Nature 406, 385–389 (2000)

    ADS  CAS  Article  Google Scholar 

  10. 10

    Losert, W., Bocquet, L., Lubensky, T. C. & Gollub, J. P. Particle dynamics in sheared granular matter. Phys. Rev. Lett. 85, 1428–1431 (2000)

    ADS  CAS  Article  Google Scholar 

  11. 11

    Reydellet, G., Rioual, F. & Clement, E. Granular hydrodynamics and density wave regimes in a vertical chute experiment. Europhys. Lett. 51, 27–33 (2000)

    ADS  CAS  Article  Google Scholar 

  12. 12

    Lemieux, P. A. & Durian, D. J. From avalanches to fluid flow: a continuous picture of grain dynamics down a heap. Phys. Rev. Lett. 85, 4273–4276 (2000)

    ADS  CAS  Article  Google Scholar 

  13. 13

    Makse, H. A. & Kurchan, J. Testing the thermodynamic approach to granular matter with a numerical model of a decisive experiment. Nature 415, 614–617 (2002)

    ADS  CAS  Article  Google Scholar 

  14. 14

    D'Anna, G., Mayor, P., Barrat, A., Loreto, V. & Nori, F. Observing brownian motion in vibration-fluidized granular matter. Nature 424, 909–912 (2003)

    ADS  CAS  Article  Google Scholar 

  15. 15

    Ono, I. K. et al. Effective temperatures of a driven system near jamming. Phys. Rev. Lett. 89, 095703 (2002)

    ADS  Article  Google Scholar 

  16. 16

    Jaeger, H. M., Nagel, S. R. & Behringer, R. P. Granular solids, liquids, and gases. Rev. Mod. Phys. 68, 1259–1273 (1996)

    ADS  Article  Google Scholar 

  17. 17

    Duran, J. Sands and Powders, and Grains: An Introduction to the Physics of Granular Materials (Springer, New York, 2000)

    Google Scholar 

  18. 18

    Geldart, D. Gas Fluidization Technology (Wiley, New York, 1986)

    Google Scholar 

  19. 19

    Pouligny, B., Malzbender, R., Ryan, P. & Clark, N. A. Analog simulation of melting in two dimensions. Phys. Rev. B 42, 988–991 (1990)

    ADS  CAS  Article  Google Scholar 

  20. 20

    Ippolito, I., Annic, C., Lemaitre, J., Oger, L. & Bideau, D. Granular temperature: experimental analysis. Phys. Rev. E 52, 2072–2075 (1995)

    ADS  CAS  Article  Google Scholar 

  21. 21

    Olafsen, J. S. & Urbach, J. S. Clustering, order, and collapse in a driven granular monolayer. Phys. Rev. Lett. 81, 4369–4372 (1998)

    ADS  CAS  Article  Google Scholar 

  22. 22

    Feitosa, K. & Menon, N. Breakdown of energy equipartition in a 2D binary vibrated granular gas. Phys. Rev. Lett. 88, 198301 (2002)

    ADS  Article  Google Scholar 

  23. 23

    Baxter, G. W. & Olafsen, J. S. Gaussian statistics in granular gases. Nature 425, 680 (2003)

    ADS  CAS  Article  Google Scholar 

  24. 24

    Prentis, J. J. Experiments in statistical mechanics. Am. J. Phys. 68, 1073–1083 (2000)

    ADS  Article  Google Scholar 

  25. 25

    Kubo, R., Toda, M. & Hashitsume, N. Statistical Physics II: Nonequilibrium Statistical Mechanics (Springer, New York, 1991)

    Google Scholar 

  26. 26

    Achenbach, E. Vortex shedding from spheres. J. Fluid Mech. 62, 209–221 (1974)

    ADS  Article  Google Scholar 

  27. 27

    Suryanarayana, G. K. & Prabhu, A. Effect of natural ventilation on the boundary layer separation and near-wake vortex shedding characteristics of a sphere. Exp. Fluids 29, 582–591 (2000)

    Article  Google Scholar 

  28. 28

    Leweke, T., Bearman, P. W. & Williamson, C. H. K. Special issue on bluff body wakes and vortex-induced vibrations—Preface. J. Fluids Struct. 15, 377–378 (2001)

    Google Scholar 

Download references


We thank L. Bocquet, R.F. Bruinsma, P.G. de Gennes, J. B. Freund, D. Levine, and M. A. Rutgers for suggestions. This work was supported by the NSF through grants to D.J.D. and A.J.L.

Author information



Corresponding author

Correspondence to D. J. Durian.

Ethics declarations

Competing interests

The authors declare that they have no competing financial interests.

Supplementary information


Video clip (real-time, 30 frames per second) of the stochastic rolling motion of a sphere due to the turbulence it generates in an otherwise-uniform upflow of air through a fine mesh screen. (MP4 994 kb)

Video clip (real-time, 30 frames per second) of the stochastic rolling motion of a sphere due to the turbulence it generates in an otherwise-uniform upflow of air through a fine mesh screen. (MP4 994 kb)

Supplementary Information and Figure (DOC 58 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ojha, R., Lemieux, P., Dixon, P. et al. Statistical mechanics of a gas-fluidized particle. Nature 427, 521–523 (2004).

Download citation

Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.