Fast dynamics and emergent topological defects in long-range interacting particle systems

Abstract

Long-range interacting systems exhibit unusual physical properties not shared by systems with short-range interactions. Understanding the dynamical and statistical effects of long-range interactions yields insights into a host of physical systems in nature and industry. In this work, we investigate the classical microscopic dynamics of screened Coulomb interacting particles confined in the disk, and reveal the featured dynamics and emergent statistical regularities created by the long-range interaction. We highlight the long-range interaction driven fast single-particle and collective dynamics, and the emergent topological defect structure. This work suggests the rich physics arising from the interplay of long-range interaction, topology and dynamics.

Appendices

Appendix A: Simulation details

We employ the adaptive Verlet method to construct long-time particle trajectories [19]. The time step dt is dynamically varying for striking a balance of the energy conservation and computational efficiency.

Fig. 5

Plot of the potential energy $E_p$ and the kinetic energy $E_k$ versus $\lambda$ in equilibrium state. $N=5000$. $k_0=10^5$. $V_0=1$

Fig. 6

Short- and long-range interacting particle systems exhibit distinct scenarios of dynamical evolution as represented in the space of $\{y, v_y\}$. a–d $\lambda /a=0.05$. $t/\tau _0=\{0.15, 0.90, 1.35, 5.85\}$. e–h $\lambda /a=10$. $t/\tau _0=\{0.0015, 0.009, 0.036, 6.0\}$. The space is divided into $50\times 50$ cells. The colored legends indicate the number of particles in $\delta y \delta v_y$. $y \in [-y_m, y_m]$ and $v_y \in [-v_m, v_m]$. a–d $y_m= \{1.0, 0.9, 1.0, 1.0\}$. $v_m=\{1.92, 2.6, 2.83, 3.24 \}$. e–h $y_m= \{1.01, 1.12, 1.12, 1.15\}$. $v_m= \{16.40, 28.50, 42.21, 59.23\}$. $k_0=10^5$. $N=5000$. $V_0=1$

Fig. 7

Temporal variation of the kinetic (red curves) and potential (blue curves) energies for typical short- and long-range interacting systems under varying strength $V_0$ of the screened Coulomb potential. $N=1000$. $k_0=10^5$

We denote the trajectory of any particle labelled i as $\{ \mathbf {x}_i(t_j) \}$. From the initial state specified by $\mathbf {x}_i(t_0)$ and $\dot{\mathbf {x}}_i(t_0)$, we obtain $\mathbf {x}_i(t_i=t_0+h)$ by

$$\begin{aligned} \mathbf {x}_i(t+h) = \mathbf {x}_i(t) + \dot{\mathbf {x}}_i(t)h + \frac{1}{2} \ddot{\mathbf {x}}_i(t) h^2 +{{\mathcal {O}}} (h^3). \end{aligned}$$

$\ddot{\mathbf {x}}_i(t)= \mathbf {F}_i(t)/m_0$, $\mathbf {F}_i(t)$ is the force on the particle i at time t, and $m_0$ is the mass of the particle. $\mathbf {F}_i=\sum _{i \ne j} \mathbf {F}_{ij} + \mathbf {F}_{iW}$, where the first term is the interaction force from all the other particles, and the second term arises if $|\mathbf {x}_i| > r_0$. According to the adaptive Verlet method, from $\mathbf {x}_i(t_j)$ and $\mathbf {x}_i(t_{j-1})$, we have

$$\begin{aligned} \mathbf {x}_i(t_{j+1}) = \mathbf {x}_i(t_j) + \left( \mathbf {x}_i(t_j) - \mathbf {x}_i(t_{j-1})\right) \frac{dt_{j}}{dt_{j-1}} + \ddot{\mathbf {x}}_i(t_j) \frac{dt_j+dt_{j-1}}{2} dt_j. \end{aligned}$$

For uniform time step $dt_j=h$, the above equation reduces to the ordinary Verlet integration scheme:

$$\begin{aligned} \mathbf {x}_i(t+2h) = 2 \mathbf {x}_i(t+h) - \mathbf {x}_i(t) + \ddot{\mathbf {x}}_i(t+h)h^2 + {{\mathcal {O}}} (h^4). \end{aligned}$$

We employ the procedure of random disk packing to generate the initial random configuration [30]. Specifically, the disks of radius $r_d$ are placed within the circle of radius $r_0$ in sequence. In this process, each newly added disk shall not overlap any existent disk. The centers of the disks constitute the initial positions of the point particles. Typically, the value of $r_d$ is about 0.3a, where a is the mean distance of nearest particles. The reason of using random disk packing instead of random point packing is as follows. Simulations show that random point packing could lead to aggregation of particles. The resulting large force requires a very fine time step to fulfill the conservation law of energy, which significantly slows down the dynamical evolution of the system in simulations.

The total mechanical energy in our simulations is well conserved at a high precision up to several decimal digits in the energy value. In the main text, we have shown the temporal variation of the kinetic and potential energies. Here, in Fig. 5, we plot the kinetic and potential energies versus $\lambda$ in equilibrium state. From Fig. 5a, we see that the potential energy increases much faster than the kinetic energy. Furthermore, Fig. 5b shows the rapid decline of the ratio $E_k/E_p$ with the increase of $\lambda$.

Appendix B: More information about relaxation kinetics

Typical instantaneous states in the dynamical evolution of the system in the space spanned by x and $v_x$ are presented in Fig. 1 in the main text. In Fig. 6, we further present the identical dynamical evolutions as in Fig. 1 in the complementary space of $\{y, v_y\}$. It is observed that for the case of $\lambda /a=10$, the patterns in both spaces of $\{x, v_x\}$ and $\{y, v_y\}$ become similar after $t/\tau _0 = 0.009$ [see Figs. 1f and 6f]. In other words, the distribution function for the system of $\lambda /a=10$ becomes symmetric with respect to $(x, v_x)$ and $(y, v_y)$ in a much faster fashion in comparison with the system of $\lambda /a=0.05$.

In Fig. 7, we show the temporal variation of the kinetic and potential energies for typical short- and long-range interacting systems under varying $V_0$. The value of $V_0$ reflects the strength of the screened Coulomb potential, as shown in Eq.(3). Since $V_0$ is measured in the unit of $m_0 r_0 v_0^2$, varying $V_0$ is equivalent to changing the value of the initial speed $v_0$. Figure 7 shows that the conversion of kinetic and potential energies in either short- or long-range interacting systems conforms to a common scenario as the value of $V_0$ is varied from $V_0=0.1$ to $V_0 = 5$. We also notice that unlike the case of $\lambda /a=10$ [the lower panels in Fig. 7d–f], the total energy of the systems with $\lambda /a=0.05$ is almost invariant as the value of $V_0$ is varied. This could be attributed to the short-range nature of the interaction, which resembles a hard repulsion at short distance.

In Fig. 8a and b, we present typical instantaneous distributions of the orientation of the particle velocity in the relaxation process for $\lambda /a=0.05$ and $\lambda /a=10$, respectively. $\theta$ is the angle between the direction of the particle velocity and x-axis. Similar to the relaxation of particle speed, the relaxation of $\theta$ in the long-range interacting system is also much faster than that in the short-range interacting system.

In Fig. 9a and b, we show the temporal variation of the H function under a softer confining potential for $\lambda /a=0.05$ and $\lambda /a=10$, respectively. Here, $k_0=10^4$, which is ten times less than the case we have discussed in the main text. In comparison with the H-curves in Fig. 3a and b in the main text, we find that a softer confining potential tends to significantly slow the relaxation kinetics for the long-range interacting system. The relaxation time increases from about $0.07\tau _0$ at $k_0=10^5$ to about $0.2\tau _0$ at $k_0=10^4$. In contrast, the relaxation rate of the short-range interacting system is unaffected by the stiffness of the boundary wall.

Fig. 8

Relaxation of the orientation of the particle velocity for $\lambda /a=0.05$ (a) and $\lambda /a=10$ (b). $\theta$ is the angle between the direction of the particle velocity and x-axis. In a $t/\tau _0= 0.57$ (green), 2.07 (blue), 3.57 (purple), and 5.85 (black). In b $t/\tau _0= 0.0001$ (green), 0.0003 (blue), 0.0006 (purple), and 0.01 (black). $N=5000$. $k_0=10^5$. $V_0=1$

Fig. 9

Temporal variation of the H curve under a softer confining potential in comparison with that in the main text. $k_0=10^4$. $\lambda /a=0.05$ (a) and $\lambda /a=10$ (b). The H curve for $\lambda /a=10$ (b) takes longer time to become stable than that at $k_0=10^5$. In contrast, the H curve for $\lambda /a=0.05$ (a) is almost identical to Fig. 3a in the main text for $\lambda /a=0.05$ and $k_0=10^5$. $N=5000$. $V_0=1$

Appendix C: Distribution of particle density in equilibrium

Figure 10a and b show the square root of the cumulative particle distribution in equilibrium particle configurations. n(r) is the total number of particles inside the circle of radius r. We find that the $\sqrt{n(r)}$ curve for $\lambda /a=0.05$ is linear in the interval $r/r_0 \in [0, 1]$ in both Fig. 10a and b, where $r_0$ is the radius of the disk. The linearity of $\sqrt{n(r)}$ with r indicated a uniform distribution of particles, since $\sqrt{n(r)} = \sqrt{\pi \rho _0}r \propto r$ for a uniform particle distribution of density $\rho _0$. Figure 10 shows that changing the stiffness of the confining potential leads to the variation of the particle density for the long-range interacting system. More particles are accumulated near a softer boundary, which reduces the particle density within the disk. In contrast, for the short-range interacting system, the particle density distribution is almost unaffected by the stiffness of the confining potential.

Fig. 10

Plot of the square root of the cumulative particle distribution $\sqrt{n(r)}$ in equilibrium configurations. $k_0=10^4$ (a). $k_0=10^5$ (b). The solid red lines represent the uniform distribution of particles. $N=5000$. $V_0=1$