Non-Gaussian statistics and superdiffusion in a driven-dissipative dusty plasma

Particle random motion can exhibit both anomalous diffusion and non-Gaussian statistics in some physical systems. Anomalous diffusion is quantified by a deviation from $\alpha =1$ in a power law for a particle’s mean-square displacement, $\mathrm{MSD}\propto {\left(\Delta t\right)}^{\alpha }$. A deviation from Gaussian statistics for a probability distribution function (PDF) is quantified by fitting to a $\kappa$ function or Tsallis distribution, with a fit parameter $q$. We report an experiment and simulations to test a theory that connects anomalous diffusion and non-Gaussian statistics. In the experiment, a single-layer dusty plasma, which behaved as a two-dimensional (2D) driven-dissipative system, had a non-Gaussian PDF. By adjusting an externally applied laser heating, $q$ was varied over a wide range. A correlation between the deviations from Gaussian statistics and normal diffusion for a 2D liquid was found in the experiment. This correlation indicates a connection between anomalous diffusion and non-Gaussian statistics. However, such a connection is lacking in equilibrium 2D Yukawa liquids, as demonstrated in numerical simulations.

Figure 1

Particle trajectories in our main experiment, for a suspension with liquid structure, at the heating laser power $P_L=4.2\mathrm{W}$ and temperature $T=51000\mathrm{K}$. Trajectories shown here are for one cell, representing $1∕6$ of the camera’s full field-of-view (FOV), for a time interval of $0.22\mathrm{s}$. We use much longer trajectories to calculate the MSD (mean-square displacement) and PDF (probability distribution function).

Figure 2

(Color online) PDF for particle displacement. (a) Our main experiment, with nonequilibrium conditions. Data shown here are for three time intervals, $\tau$. Using a logarithmic scale for PDF plotted vs the squared displacement, a Gaussian function would be a straight line; here, the PDF is non-Gaussian with a fat tail. The smooth curves, from fitting to Eq. (2), yield values for $q$. Averaging all values of $q$ for various time intervals, including the three shown here and 197 more in the range from $1\text{to}5\mathrm{s}$, yields a measurement for $q$ for one movie and one data point in Fig. 3a. (b) Equilibrium simulations, for a $\Gamma$ corresponding to the experimental temperature in (a), are nearly Gaussian. The experimental kinetic temperature $T$ is computed using Eq. (8), and can be varied by adjusting the laser power. Data shown here are for a liquid structure.

Figure 3

Experimental results. Each data point corresponds to one cell in one movie, at one $P_{\mathrm{L}}$. (a) The fit parameter $q$ would be unity for Gaussian statistics, while our fat-tail PDF has $q>1$; i.e., this driven-dissipative system exhibits non-Gaussian statistics. (b) Diffusion exponent $\alpha$, as calculated by fitting the MSD to a power law. Normal diffusion has $\alpha =1$, while anomalous diffusion has $\alpha \ne 1$. (c) Temperature fluctuations for the experiment. For an equilibrium system, the ratio would be unity. Temperature fluctuations can arise from finite-size effects [characterized by the “canonical fluctuation,” ${(2∕N)}^{1∕2}\overline{T}$] and due to nonequilibrium effects. Here, the fluctuations exceed the “canonical fluctuation level,” which is an indicator of a nonequilibrium system.

Figure 4

(Color online) Correlation of superdiffusion with non-Gaussian statistics. (a) Variation of $\alpha$ vs $2∕(3-q)$. (b) For a more meaningful test of correlation, we plot the data with shifted origin so that the deviation from normal diffusion ${\delta }_{\mathrm{diff}}=\alpha -1$ is plotted vs the deviation from Gaussian statistics ${\delta }_{\mathrm{Gauss}}=2∕(3-q)-1$. Note that for simulations, which modeled 2D thermal equilibrium systems, superdiffusion is not connected to non-Gaussian statistics, unlike the experiment, which was driven dissipative.