^{1,a)}, Raffaele Cafiero

^{2}, Annette Zippelius

^{3}, Hans Jürgen Herrmann

^{4}and Stefan Luding

^{5,b)}

### Abstract

We study a homogeneously driven granular gas of inelastic hard particles with rough surfaces subject to Coulomb friction. The stationary state as well as the full dynamic evolution of the translational and rotational granular temperatures are investigated as a function of the three parameters of the frictionmodel. Four levels of approximation to the (velocity-dependent) tangential restitution are introduced and used to calculate translational and rotational temperatures in a mean field theory. When comparing these theoretical results to numerical simulations of a randomly driven monolayer of particles subject to Coulomb friction, we find that already the simplest model leads to qualitative agreement, but only the full Coulomb frictionmodel is able to reproduce/predict the simulation results quantitatively for all magnitudes of friction. In addition, the theory predicts two relaxation times for the decay to the stationary state. One of them corresponds to the equilibration between the translational and rotational degrees of freedom. The other one, which is slower in most cases, is the inverse of the common relaxation rate of translational and rotational temperatures.

This work has been supported by the European network project FMRXCT980183 (RC) and the DFG (Deutsche Forschungsgemeinschaft) through SFB 602 (O.H. and A. Z), Grant No. Zi209/6-1. (A.Z.) S.L. was also supported by the DFG, Grant No. Lu450/9-1, and by FOM (Stichting Fundamenteel Onderzoek der Materie, The Netherlands) as financially supported by NWO (Nederlandse Organisatie voor Wetenschappelijk Onderzoek).

I. INTRODUCTION

II. MODEL

A. Collision rules

B. Driving model

C. Simulations

III. IMPACT-ANGLE PROBABILITY DISTRIBUTION

IV. DIFFERENTIAL EQUATIONS IN MEAN FIELD THEORY APPROXIMATIONS

A. Model A: Constant tangential restitution

B. Model B: Simplified mean tangential restitution

C. Model C: Mean tangential restitution

1. Constant tangential restitution limit

2. Weak friction limit

3. Comparison of model B and model C

D. Model D: Variable (simplified) tangential restitution

1. Constant tangential restitution limit

2. Weak friction limit

E. Model E: Variable (exact) tangential restitution

1. Constant tangential restitution limit

2. Weak friction limit

V. STEADY STATE

A. Analytical results

1. Model A

2. Model B

3. Model D

4. Model E

5. Models C and E for small

6. Discussion

B. Comparison with simulations

1. Variation of

2. Variation of —translational temperature

3. Variation of —rotational temperature

VI. APPROACH TO STEADY STATE

A. Close to steady state

B. Full dynamic evolution

VII. SUMMARY AND DISCUSSION

### Key Topics

- Friction
- 40.0
- Mean field theory
- 36.0
- Field theory models
- 20.0
- Differential equations
- 8.0
- Numerical modeling
- 8.0

## Figures

Schematic drawing of two-particle contact in the center of mass reference frame. Shown are the relative velocity of the contact points, the impact angle of the contact points, and the angle between the relative translational velocity of the particles and their contact normal.

Schematic drawing of two-particle contact in the center of mass reference frame. Shown are the relative velocity of the contact points, the impact angle of the contact points, and the angle between the relative translational velocity of the particles and their contact normal.

Tangential restitution as function of the impact angle for different values of the coefficient of friction .

Tangential restitution as function of the impact angle for different values of the coefficient of friction .

Plots of the probability distribution of from simulations (symbols) and from Eq. (12) with values from the simulations. The arrows indicate the corresponding , while the parameters are (a) , and variable , and (b) , and variable .

Plots of the probability distribution of from simulations (symbols) and from Eq. (12) with values from the simulations. The arrows indicate the corresponding , while the parameters are (a) , and variable , and (b) , and variable .

Expected mean tangential restitution, , as function of the friction coefficient for models A, B, and C. The parameters used are , (for A, B) and different , 0.40, and 0.15 (model C: solid lines from right to left).

Expected mean tangential restitution, , as function of the friction coefficient for models A, B, and C. The parameters used are , (for A, B) and different , 0.40, and 0.15 (model C: solid lines from right to left).

Simulation results (symbols) and theory (lines) for the parameters , , , and , plotted against the maximum tangential restitution . (a) Translational temperature , and (b) rotational temperature , plotted against , and scaled by , the mean field value for smooth particles. (c) Ratio of rotational and translational temperature , plotted against .

Simulation results (symbols) and theory (lines) for the parameters , , , and , plotted against the maximum tangential restitution . (a) Translational temperature , and (b) rotational temperature , plotted against , and scaled by , the mean field value for smooth particles. (c) Ratio of rotational and translational temperature , plotted against .

Translational temperature scaled by the mean field value for smooth particles , plotted against , for the parameters as in Fig. 5. The tangential restitution coefficients are fixed to (a) , and (b) . Data with normal restitution (solid symbols and thick lines) and (open symbols and thin lines) are compared. Models A and B are shown for only.

Translational temperature scaled by the mean field value for smooth particles , plotted against , for the parameters as in Fig. 5. The tangential restitution coefficients are fixed to (a) , and (b) . Data with normal restitution (solid symbols and thick lines) and (open symbols and thin lines) are compared. Models A and B are shown for only.

plotted against , for the same parameters as in Fig. 6. The tangential restitution coefficients are again fixed to (a) , and (b) .

plotted against , for the same parameters as in Fig. 6. The tangential restitution coefficients are again fixed to (a) , and (b) .

Deviation from equipartition, , plotted against the inverse friction coefficient, , for simulations from Fig. 6(b). Note the double-logarithmic scale of this plot.

Deviation from equipartition, , plotted against the inverse friction coefficient, , for simulations from Fig. 6(b). Note the double-logarithmic scale of this plot.

Ratio of rotational and translational temperature, , plotted against , for some simulations from Fig. 6(b). Note the double-logarithmic scale of this plot.

Ratio of rotational and translational temperature, , plotted against , for some simulations from Fig. 6(b). Note the double-logarithmic scale of this plot.

Relaxation rates , close to steady state for as a function of .

Relaxation rates , close to steady state for as a function of .

Evolution of temperatures with rescaled time, with , for simulations with , , , , and (a) , (b) . Note that there are no fit parameters.

Evolution of temperatures with rescaled time, with , for simulations with , , , , and (a) , (b) . Note that there are no fit parameters.

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