Superdiffusion of two-dimensional Yukawa liquids due to a perpendicular magnetic field

Stochastic transport of a two-dimensional (2D) dusty plasma liquid with a perpendicular magnetic field is studied. Superdiffusion is found to occur especially at higher magnetic fields with $\beta$ of order unity. Here, $\beta ={\omega }_c/{\omega }_{pd}$ is the ratio of the cyclotron and plasma frequencies for dust particles. The mean-square displacement $\mathrm{MSD}=4D_{\alpha }{t}^{\alpha }$ is found to have an exponent $\alpha >1$, indicating superdiffusion, with $\alpha$ increasing monotonically to $1.1$ as $\beta$ increases to unity. The 2D Langevin molecular dynamics simulation used here also reveals that another indicator of random particle motion, the velocity autocorrelation function, has a dominant peak frequency ${\omega }_{\text{peak}}$ that empirically obeys ${\omega }_{\text{peak}}^2={\omega }_c^2+{\omega }_{pd}^2/4$.

Figure 1

Particle trajectories in the x-y plane for different constant perpendicular magnetic-field strengths, $\beta ={\omega }_c/{\omega }_{pd}=0$ (a), 0.5 (b), and 1.0 (c). The random walk without a magnetic field changes its character, becoming more circular and wandering less as the magnetic field increases in (b) and (c). Color represents time, and only $\approx 10\%$ of the simulated region and $\approx 0.03\%$ of the simulation duration are shown here. Our simulation conditions are $\Gamma =200$ and $\kappa =2$.

Figure 2

Mean-squared displacement MSD for different magnetic fields, in the unit of ${\omega }_{pd}t$ (a), or $f_{c}t$ (b). At long times $100<{\omega }_{pd}t<1000$, well after the ballistic portion, we can fit the MSD time series to Eq. (2). As $\beta$ increases, the MSD curves are lower and lower, indicating that the wandering motion of particles is suppressed by magnetic field, and $D_{\alpha }$ decreases. Oscillations at shorter times are due to cyclotron motion of individual particles, as is best seen in (b), where time is normalized by $1/f_c$. Dips in the MSD time series occur around one cyclotron period, two periods, and so on. Here, the MSD is normalized using the Wigner-Seitz radius $a$. Our simulation conditions are $\Gamma =200$ and $\kappa =2$.

Figure 3

(a) Indication of superdiffusion. The exponent $\alpha$ is $>1$, especially for strong magnetic fields $\beta \approx 1$. These data are the results of fitting the MSD time series in Fig. 2 to the exponential scaling of Eq. (2) in the indicated time ranges. This result $\alpha >1$ for a 2D Yukawa liquid is different from that of Ranganathan et al. who reported normal diffusion, $\alpha =1$, for a 2D Coulomb liquid [49]. As the time range for the fitting is longer, we can see a clear trend that the exponent $\alpha$ is smaller. In (b), as $\beta$ increases from 0 to 1, the coefficient $D_{\alpha }$ decreases monotonically more than $70\%$ as $\beta$ increases, which means that the magnetic field greatly suppresses the wandering of particles. Note that the scatter of our data for each $\beta$ value corresponds to the error bar. Fits of our $D_{\alpha }$ data for the time range of $100<{\omega }_{pd}t<1000$ only to our empirical expressions Eq. (4) and Eq. (5) derived by Ranganathan et al. [49] are shown as solid and dashed lines, respectively.

Figure 4

Velocity autocorrelation function, VACF, for different magnetic fields. Time is normalized by $1/{\omega }_{pd}$ in (a) and $1/f_c$ in (b). The oscillations decay more slowly with higher magnetic field, as seen in (a) for increasing $\beta$. These oscillations are mainly due to the cyclotron motion, since its frequency is nearly the same as $f_c$. The period of oscillation is related to the magnetic-field strength, as seen in (b) where VACF curves for two values of $\beta$ are nearly aligned.

Figure 5

(a) Vibrational density of states, i.e., spectral power of the normalized VACF for different magnetic fields. The curves exhibit a dominant peak; the frequency of this peak is plotted in (b). The peak frequency increases monotonically as $\beta$ increases. The peak frequency fits an empirical curve, ${\omega }_{\text{peak}}^2/{\omega }_{pd}^2=0.25+{\beta }^2$, i.e., ${\omega }_{\text{peak}}^2=0.25{\omega }_{pd}^2+{\omega }_c^2$, shown as a smooth curve. This fit shows how the peak frequency is always greater than the cyclotron frequency. For comparison, we also plot the data from the frictionless simulation of [46] for the peak frequency of the longitudinal waves at the wave number of $ka=5.55$.

Figure 6

Longitudinal and transverse phonon spectra of our 2D Yukawa liquid when $\beta =0$ (a),(b), $\beta =0.5$ (c),(d), and $\beta =1$ (e),(f). These spectra differ from the vibrational density in Fig. 5 because they reflect both spatial and temporal fluctuations as characterized by a current [45], not just the temporal fluctuations characterized by the VACF.