 0  0.2  0.4  0.6  0.8  1  0  2  4  6  8 10 Order Parameter S (deg) L= 32 L= 64 L=128 L=256 MFT  0  0.2  0.4  0.6  0.8  1  0  2  4  6  8  10 Frequency Deviation Q (deg) L= 32 L= 64 L=128 L=256 MFT (a) (b)  0  0.001  0.002  0.003  0  2  4  6  8  10 Var(S) (deg) (a) (b) -0.001  0  0.001  0.002  0.003 0 5000 10000 15000 C (t) Time t$^{δ}$ =5.5 6.0 6.5 7.0 S (c)  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7 0 10000 20000 30000 40000 50000 Order Parameter S Time  t =5.0 6.0 7.0 FIG. 2: Transition behavior for system size L = 32 , 64 , 128, and 256 with comparison to the mean-field theory (MFT). (a) Orientational order parameter S = |〈 n 〉| = |〈 e $^{iφ}$〉 | versus δ 0 . (b) Normalized frequency deviation Q = √ 〈 Ω 2 〉 / 〈 ω 2 〉 versus δ 0 versus δ 0 . FIG. 4: (a) Variance of the order parameter Var( S ) as a function of δ 0 . Fluctuation is most enhanced at δ 0 = 6 $^{◦}$. (b) Auto-correlation function of the order parameter C S ( t ). An oscillatory behavior is prominent at δ 0 = 6 $^{◦}$. (c) Time series S ( t ). FIG. 3: Distribution of the normalized actual frequency Ω / √ 〈 ω 2 〉 as functions of δ 0 . For comparison, the distribu- tion of the intrinsic frequency ω i is also shown.  0.001  0.01  0.1  1  10  100  1000  10000 -3 -2 -1  0  1  2  3 Distribution P( ) Scaled Frequency 3 4.5 5 6 10 intrinsic over a wide distance. For δ$_{0}$ > 6 $^{◦}$, on the other hand, the correlation is short-ranged and decays more slowly than exponential. The qualitative change in the correlation function gives another support of the above estimate of the threshold angle, δ$_{th}$ = 6 $^{◦}$. Mean-field theory. - Now we apply a mean-field ap- proximation to our model. When the coupling is weak ( γ ≪ 1) and the force angle is small ( δ$_{0}$ ≪ 1), we can neglect the δ$_{j}$ ’s in the RHS of eq. (2) because they give 3