Granular gas dynamics: how Maxwell's demon rules in a non-equilibrium system

Abstract

The main characteristic of a granular gas, which makes it fundamentally different from ordinary molecular gases, is its tendency to form clusters, i.e. to spontaneously separate into dense and dilute regions. This can be interpreted as a separation in cold and hot regions, meaning that Maxwell's demon is at work: this demon – notoriously powerless in any system in thermodynamic equilibrium – makes clever use of the non-equilibrium state of affairs that reigns in a granular gas, with on the one hand an external energy source and on the other a continuous loss of energy due to the inelastic particle collisions.

 We focus on vibrated compartmentalised systems, because these give a particularly clear-cut view of the clustering process and also because they resemble the typical machinery used in industrial applications to sort and transport granular materials. We discuss how the clustering can be exploited to build a Brownian motor, a fountain, a granular clock, and how it gives insight into a related clustering problem of prime importance in modern society, namely the formation of traffic jams.

Keywords:

• granular gas
• clustering
• Maxwell's demon
• hydrodynamic description
• Brownian motor
• traffic jams

Acknowledgements

I am grateful to Jens Eggers, Isaac Goldhirsch, and Mario Markus for kindly permitting me to reproduce their figures, and for their positive and extremely valuable feedback on an early draft of the paper. Many thanks are also due to the referees for their anonymous help in improving the paper via a series of wonderfully insightful comments. Finally, I want to thank Devaraj van der Meer and Detlef Lohse, with whom I had the pleasure to work together on many fascinating aspects of Maxwell's demon in granular gases.

Notes

1. Due to the redirected forces, the pressure on the side walls can become uncomfortably high. In the United States alone some 1000 grain silos collapse every year due to overpressure on the side walls.

2. The dimensionless number that measures the influence of the ambient medium is the Bagnold number Ba, defined as the ratio between a typical Newtonian force acting on the particle (gravity, friction, collisions) and the most relevant force from the medium (drag, lift). A good choice in many cases is to take the gravitational force and the Stokes drag force: Ba = mg/3πηdv =ρ_sd ² g/18ηv, where  η  is the dynamic viscosity of the medium, d and ρ_s are the diameter and material density of the particles, and v is their characteristic velocity. For Ba ≫ 1 the influence of the medium may be neglected; when Ba becomes of the order of 1 or smaller the surrounding medium must be taken into account. For the systems discussed in the present review, with glass beads (ρ_s  = 2.5 103 kg m⁻³) of diameter d = 2 10⁻³ m and typical velocity v = 1 ms⁻¹, moving through air at room temperature and atmospheric pressure (η = 1.8× 10⁻⁵ kg/sm), the value of the Bagnold number is roughly 300.

3. Barchans are crescent-shaped dunes that form in desert areas with a firm underground and a limited sand supply where the wind blows in one prevailing direction. They propagate in the direction of the wind at a typical velocity of several tens of metres per year, with smaller dunes moving faster than big ones and occasionally overtaking them. For an introduction to the extensive literature on barchan dunes we refer to 16. Also recommended are the beautiful photographs by NASA of the barchan fields on Mars, see e.g. 17.

4. An additional and slightly deeper reason for the prominence of fluctuations is the fact that granular matter has weak scale separation (or no separation at all) between the microscopic and macroscopic scales, see e.g. 20.

5. Since the setup is symmetric, it is a matter of chance which of the two compartments will be preferred.

6. The analogy with thermodynamic temperature should be handled with care, since the ensemble averages implied in Equation (1) may not directly apply to single realisations 20,30 and – if the system contains non-identical particles – the lack of energy equipartition between the various species may be a complicating factor, with different species having different granular temperatures 31-33.

7. Recent experiments suggest that this identification holds quite generally for vibrated granular gases of identical particles, and that (except for the bottom layer) the velocity distribution is very nearly Maxwellian throughout the system, see 34.

8. In this situation, the mean free path length  ℓ  of the particles is of the same order as the system size and the gas is called a Knudsen gas. If the dimensionless Knudsen number Kn (defined as the ratio of the mean free path to the system size) is of the order of 1 or higher, the continuum assumption underlying fluid mechanics is no longer a good approximation and statistical methods should be used instead.

9. A non-cyclic array is described by the same Equation (13), only modified at the end compartments k = 1 and k = K. The results do not differ significantly from those for a cyclic array 61.

10. Note that for K = 2 the Brownian motor setup is simply equivalent to the granular fountain of Figure 10.

11. For the analogy, the compartments of the ratchet system should be compared pairwise (a cluster and an adjacent dilute compartment) to the individual magnetic domains in the piece of ferromagnetic material.

12. The interactions in a granular gas conserve mass and momentum (but not energy), whereas in traffic only mass is conserved. This is a source of difference between granular and traffic flow – and obviously it is not the only one. So the resemblance between the two types of flow may be strong but it is not quite perfect, and one should remain cautious as to how far the analogy can be stretched.

13. This backward velocity can be understood as follows: on the average, the cars in a dense jam occupy 7.5 m each, and they leave the front of the jam at a rate of one per 1.5 s (the combined reaction time of driver and car). So the front of the jam moves backward at a speed of 7.5 m per 1.5 s = 5 ms⁻¹, which is 18 km h⁻¹.