Weak chaos and the “melting transition” in a confined microplasma system

We present results demonstrating the occurrence of changes in the collective dynamics of a Hamiltonian system which describes a confined microplasma characterized by long-range Coulomb interactions. In its lower energy regime, we first detect macroscopically the transition from a “crystallinelike” to a “liquidlike” behavior, which we call the “melting transition.” We then proceed to study this transition using a microscopic chaos indicator called the smaller alignment index (SALI), which utilizes two deviation vectors in the tangent dynamics of the flow and is nearly constant for ordered (quasiperiodic) orbits, while it decays exponentially to zero for chaotic orbits as $\text{exp}\left[-\left({\lambda }_1-{\lambda }_2\right)t\right]$, where ${\lambda }_1>{\lambda }_2>0$ are the two largest Lyapunov exponents. During the melting phase, SALI exhibits a peculiar stairlike decay to zero, reminiscent of “sticky” orbits of Hamiltonian systems near the boundaries of resonance islands. This alerts us to the importance of the $\Delta \lambda ={\lambda }_1-{\lambda }_2$ variations in that regime and helps us identify the energy range over which “melting” occurs as a multistage diffusion process through weakly chaotic layers in the phase space of the microplasma. Additional evidence supporting further the above findings is given by examining the ${\text{GALI}}_k$ indices, which generalize SALI $\left(={\text{GALI}}_2\right)$ to the case of $k>2$ deviation vectors and depend on the complete spectrum of Lyapunov exponents of the tangent flow about the reference orbit.

Figure 1

Plot of typical orbits for $N=5$ and $\gamma =0.07$ (prolate quasi 1–dimensional geometry): (a) At $E=2$ the motion is dominated by its stable normal modes and “crystalline” behavior is observed. (b) At $E=2.35$ a transition to chaotic motion occurs, where “melting” is expected to take place. (c) At $E=9$ the onset of “thermal cloud” behavior is evident.

Figure 2

Plot of the dependence of the temperature $T$ [as in Eq. (9), dimensionless scale] on the energy $E$ for the microplasma system (5) of $N=5$ ions in a prolate Penning trap $\left(\gamma =0.07\right)$. The error bars represent one standard deviation from the time averaged temperature $T$. Clearly, the crossover from ordered to weakly chaotic motion around $E∊\left(2,2.5\right)$ does not occur in a manner justifying the use of statistical mechanics considerations. In the inset, the ratio of the standard deviation of the time averaged $T$ to the time averaged temperature $T$ shows that they are comparable.

Figure 3

The Kolmogorov-Sinai entropy $H_{KS}$, for $\gamma =0.07$ and $N=5$, computed by the Eq. (16) (dashed highest line) and the Lyapunov exponents (black solid lines) are plotted from the maximum down to the lowest one (15th) as a function of the energy. Note that, as shown in the inset, their maximum occurs at energies close to $E\approx 6.2$, which is much higher than the regime where the “melting transition” takes place (see text).

Figure 4

(a) Plot of the logarithms of the ratios ${\mathcal{P}}_i\left(E\right),i=1,\dots ,9$ [see Eq. (17)], for $\gamma =0.07$ and $N=5$, as a function of the energy $E$. Note the characteristic gaps in the spectrum appearing for large values of $E$. (b): Same as in panel (a) for $i=1,\dots ,4$ and small values of $E$, with the characteristic gaps in the spectrum becoming erratic in the regime of weak chaos, settling down to more clearly defined values around $E\ge 2.3$.

Figure 5

Plot of the evolution of SALI as a function of time $t$ for $N=5$, $\gamma =0.07$ and five typical energies: (a) $E=1.9$ where the motion is ordered. (b) $E=3.5$ (strong chaotic behavior) where we also plot the theoretical prediction $\text{SALI}\left(t\right)\propto e^{-\left({\lambda }_1-{\lambda }_2\right)t}$ for comparison. (c) $E=2.033$, $E=2.05$, and $E=2.283$, plotting also the corresponding theoretical predictions. Note that all three vertical axes are logarithmic.

Figure 6

Plot of the quantity $Q=\frac{\Delta \lambda }{K}$, for $\gamma =0.07$ and $N=5$, as a function of the energy $E∊\left[E_0,E_{\text{max}}\right]$, where $\Delta \lambda$ is the difference of the two largest Lyapunov exponents and $K$ is the slope calculated by linear regression performed on the $\text{log}\mathbf{\left(}\text{SALI}\left(t\right)\mathbf{\right)}$ vs $t$ curves. We also plot the line $Q=1$ to indicate the energy range where $\Delta \lambda ={\lambda }_1-{\lambda }_2\approx K$ and Eq. (14) are satisfied.

Figure 7

Plot of the ${\text{LDI}}_k\left(={\text{GALI}}_k\right)$, $k=2,\dots ,10$ as a function of time $t$ for the microplasma system (5) of $N=5$ ions in a prolate trap $\left(\gamma =0.07\right)$ for three typical energies. (a) ${\text{LDI}}_2-{\text{LDI}}_{10}$ for the energy $E=1.9$. (b) Same as in panel (a) for the slightly bigger energy $E=2$. (c) Same as in panel (a) for the bigger energy $E=2.283$. Note that all axes are logarithmic.