Figure 9: χ$_{4}$ ( γ ) for different strain-rates ˙ γ and a = 0 . 01. comparison with quasistatic (‘qs’) sim- ulations. Note the different boundary conditions used. MD simulations are with walls, while qua- sistatic simulations have periodic boundary condi- tions. This may explain the difference in the am- plitude. Figure 10: (Left) Peak-height χ$_{m}$ = χ$_{4}$ ( γ$_{m}$ ) as determined from Fig. 9. The saturation at small strain-rates is an indication of the quasistatic limit, in which χ$_{m}$ ∼ N . In contrast, at high strain- rates no significant dependence on system-size is observed. (Right) Peak-height χ$_{m}$ as function of volume-fraction. ently dominated by system-size effects, as already shown in previous works . 14,15,28,47 Note, that this system-size dependence is of different origin than the finite-size effects present within the above men- tioned intermittent regime, which occurs close to φ$_{c}$ . The intermittency can be avoided by staying away from φ$_{c}$ . In contrast, the system-size de- pendence encountered here is quite independent of volume-fraction, but rather a generic feature of the quasistatic regime, as we show now. To this end let us turn to the molecular-dynamics simulations. We will show that the dependence on system-size indeed reflects the saturation of a length-scale that is finite for larger strain-rates and increases towards the quasistatic regime 6 . As Fig. 9 shows, the amplitude of χ$_{4}$ increases when reducing the strain-rate ( a = 0 . 01, φ = 0 . 9) and ap- proaches the quasistatic limit for small strain-rates. Also the strain γ$_{m}$ , at which χ$_{4}$ is maximal is very well reproduced in the dynamic simulation. Fig. 10 demonstrates a saturation of the ampli- tude χ$_{m}$ ≡ χ$_{4}$ ( γ$_{m}$ ) at small strain-rates indicating that the quasistatic regime is entered. Comparing with the quasistatic simulation, we find somewhat smaller values for the amplitude. It should be re- membered, however, that different boundary con- ditions have been used. The rough walls used in the molecular dynamics simulations are likely re- sponsible for the reduction of the peak-height as compared to the quasistatic simulations (which are performed with periodic boundary conditions). The presence of the quasistatic regime is also ev- idenced by the fact that the amplitude χ$_{m}$ within the plateau depends on system-size, just as in the quasistatic simulations. Outside this regime, on the other hand, no significant N -dependence is ob- served. In effect this means that the quasistatic regime shrinks with increasing system-size. The strainrate ˙ γ$_{qs}$ ( N ) that describes the crossover to the quasistatic regime decreases with N . This is in line with Refs., 14,15 where a power-law depen- dence γ$_{qs}$ ∼ 1 /N is reported. From our data we cannot make any definitive statement about this dependence. The results are furthermore consistent with those of Ono et al. 48 The lowest strain-rate accessible in the latter study was ˙ γ = 0 . 0001. At this strain- rate the correlation length was observed to be on the order of 3 in agreement with our data. We also probed the volume-fraction dependence, by performing runs at φ = 0 . 85 , 0 . 87 , 0 . 9 , 0 . 95 and 1. The resulting amplitude of χ$_{4}$ is given in Fig. 10. Interestingly we observe a mild increase in the am- $^{6}$A more detailed account of these simulations will be presented in: P. Chaudhuri and L. Bocquet, in preparation (2010). 9  0  50  100  150  200  250 1e-05 1e-04 1e-03 1e-02 1e-01 1e+00 qs 1e-06 1e-05 3e-05 5e-05 1e-04 1e-03  1  2  4  10 1e-06 1e-05 1e-04 1e-03 m 1/2 N=2500 N=1600 N=900  1  2  4 0.84  0.89  0.94  0.99  m 1/2 =1e-04