Statistical mechanics of a gas-fluidized particle

Abstract

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.

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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.

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Acknowledgements

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.

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Correspondence to D. J. Durian.

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Supplementary information

41586_2004_BFnature02294_MOESM1_ESM.mp4

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)

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Ojha, R., Lemieux, P., Dixon, P. et al. Statistical mechanics of a gas-fluidized particle. Nature 427, 521–523 (2004). https://doi.org/10.1038/nature02294

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